Risk and Statistics,
3rd Tohoku-ISM-UUlm Joint
Workshop
Program and Abstracts

October 12–14, 2022
Tohoku Forum for Creativityᇵʴ᯼ ƌF

Graduate School of Economics and Management
Tohoku University

Banquet place information
restaurant on 3F at Hotel Metropolitan East, 18:30-20:00, Oct. 13th

Visit the hotel located at East Exit in Sendai Station. It takes 20 mins on foot from the venue.
You can use “Aobadori-Ichibancho station” in TOZAI subway line as the nearest station fromthe workshop venue.

Program
October 12, 2022
9:30 Li, Yingxing (Tohoku Univ.)
Product Embedding with Receipt context for Disaggregated Sales Data
10:00 Nomura, Shunichi (Waseda Univ.)
Hierarchical Topic Model for Tensor Data and Extraction of Weekly and Daily
Activity Patterns
10:30 Wu, Stephen (ISM)
Performance-Based Earthquake Early Warning for Regional Seismic Risk
MitigationBreak
11:15 Yoshiba, Toshinao (Tokyo Metropolitan Univ.)
Portfolio Risk Valuation Using Asymmetric Copulas
11:45 Ogihara, Teppei (Univ. Tokyo)
Local Asymptotic Normality for Jump-Diffusion Processes with Discrete
Observations
Lunch
13:30 Lindner, Alexander (Univ. Ulm)
Quasi-Infinitely Divisible Distributions
14:00 Matsui, Muneya (Nanzan Univ.)
Subexponentialiy of Densities of Infinitely Divisible Distributions
Break
14:45 Berger, David (Univ. Dresden)
Existence and Distributional Properties of Solutions for SPDEs Driven by Lévy
White Noise
15:15 Kurisu, Daisuke (Yokohama National Univ.)
Nonparametric Regression for Locally Stationary Random Fields on R^d
Break16:00 Zhang, Fangyuan (EURECOM)Intergenerational Risk Sharing in Defined Contribution Pension Systems:
Analysis with Bayesian Optimization
16:30 Klicnarová, Jana (Univ. South Bohemia)
On Modified Interdirections

October 13, 2022
AI Yotta Special Session
9:30 Shioiri, Satoshi (Tohoku Univ.)
Extracting Valuable Information based on Human Behavior
10:00 Sakai, Nobuyuki (Tohoku Univ.)
Measurement and Statistics of Taste and Deliciousness
10:30 Ishigaki, Tsukasa (Tohoku Univ.)
Recommendation Systems with Network Structure and Big Data
Break
11:15 Chen, An (Univ. Ulm)
Non-Concave Optimization Under Risk Constraints
11:45 Araki, Yuko (Tohoku Univ.)
Statistical Modeling for the Nonlinear Structure of Quantiles in Hierarchical
Data via Regularization
Lunch
13:30 Spodarev, Evgeny (Univ. Ulm)
Prediction of Heavy-Tailed Random Functions
14:00 Kuriki, Satoshi (ISM)
Expected Euler Characteristic Heuristic for Smooth Gaussian Random Fields
with Inhomogeneous Marginals
Break14:45 Kanagawa, Motonobu (EURECOM)Counterfactual Mean Embeddings
15:15 Koike, Yuta (Univ. Tokyo)
Central Limit Theorems in High-Dimensions: Recent DevelopmentsBreak
16:00 Kariya, Takeaki (Nagoya Univ. of Commerce & Business)
A Modelling Framework for Regression with Collinearity
16:30 Miura, Ryozo (Hitotsubashi Univ.)
Occupation Time, Quantiles and Rank on Vasicek Process with Applications
to Exotic Options
BanquetOctober 14, 2022
9:30 Sawada, Masayuki (Hitotsubashi Univ.)
Spatial Regression Discontinuity Designs
10:00 Ishihara, Takuya (Tohoku Univ.)
Shrinkage Methods for Treatment Choice
10:30 Ko, Stanley Iat-Meng (Tohoku Univ.)
Forecasting Stock Returns with Conditional Quantile-level Dependence
Break11:15 Robinson, Peter M (LSE)
Issues in Spatial Processes with Long Range Dependence
11:45 Yajima, Yoshihiro (Univ. Tokyo)
On Estimation of Fractional Brownian Fields and Sheets
12:15 Taniguchi, Masanobu (Waseda Univ.)
Joint Circular Distributions in View of Higher Order Spectra of Time Series
and Copula

Product Embedding with Receipt context for Disaggregated Sales Data
Yinxing Li and Nobuhiko Terui

Marketing data are expanding in several modes nowadays, as the number of variables
explaining customer behavior has greatly increased, a Product Embedding for Large-Scale
Disaggregated Sales Data Yinxing Li and Nobuhiko Terui Marketing data are expanding in several
modes nowadays, as the number of variables explaining customer behavior has greatly
increased, and automated data collection in the store has also led to the recording of customer
choice decisions from large sample sizes. Thus, high-dimensional models have recently gained
considerable importance in several areas, including marketing. Although some models, such as
Prod2Vec, involve various marketing variables such as price and customer demographic data,
the role of the variables in forecasting is still not discussed. In light of the limitations mentioned
above, our study not only aims to propose a model with better forecasting precision but also toreveal how customer demographics affect customer behavior. we propose a Bayesian
Word2Vec based framework that incorporates marketing variables and environment byconsidering following situations in our model. When considering the market basket, our study
incorporates the receipt vector into the model as the prior information of each purchased
product in a basket, which means a preferred purchasing pattern for a certain shopping. It
assumes that the customer will consider the whole purchasing context before choosing a
product. We also assume a state space model for the receipt vector through the trips for eachcustomer. We use the weekday, promotion information as the data for the state space modelfor higher interpretability of the prior structure such as the purchasing scenarios of eachcustomer. Besides, we consider the purchasing probability of a certain product conditional on an
existing market basket is influenced by the following three factors - 1) The compatibility with themarketing basket, which is represented by the inner product of product vectors, 2) customer
utility for the product, which incorporate the customer heterogeneity structure, and 3) Thinking
ahead algorithm, which represents one-step ahead forecasting before purchasing the product.
Our proposed model contributes both to higher precision for forecasting by incorporating themarketing environment and customer heterogeneity into the model, and better interpretability.
We use receipt data from a retailer for our empirical analysis, containing the information ofcustomer demographic, promotion and other marketing information. We show not only theeffectiveness of marketing environment for the forecasting by using the Hit Rate@K for the
hold-out sample comparing to the several benchmark models , but also the high interpretabilityof our proposed model in the empirical studynd automated data collection in the store has

also led to the recording of customer choice decisions from large sample sizes. Thus,
high-dimensional models have recently gained considerable importance in several areas,
including marketing. Although some models, such as Prod2Vec, involve various
marketing variables such as price and customer demographic data, the role of thevariables in forecasting is still not discussed. In light of the limitations mentioned above,
our study not only aims to propose a model with better forecasting precision but also toreveal how customer demographics affect customer behavior. we propose a Bayesian
Word2Vec based framework that incorporates marketing variables and environment byconsidering following situations in our model.

When considering the market basket, our study incorporatesWKHreceipt vector into
the model as the prior information of each purchased product in a basket, which means apreferred purchasing pattern for a certain shopping. It assumes that the customer willconsider the whole purchasing context before choosing a product. We also assume a state
space model for the receipt vector through the trips for each customer. We use theweekday, promotion information as the data for the state space model for higher
interpretability of the prior structure such as the purchasing scenarios of each customer.
Besides, we consider the purchasing probability of a certain product conditional on an
existing market basket is influenced by the following three factors - 1) The compatibility
with the marketing basket, which is represented by the inner product of productvectors, 2) customer utility for the product, which incorporate the customer
heterogeneity structure, and 3) Thinking ahead algorithm, which represents one-stepahead forecasting before purchasing the product.
Our proposed model contributes both to higher precision for forecasting by
incorporating the marketing environment and customer heterogeneity into the model, andbetter interpretability. We use receipt data from a retailer for our empirical analysis,containing the information of customer demographic, promotion and other marketing
information. We show not only the effectiveness of marketing environment for theforecasting by using the Hit Rate@K for the hold-out sample comparing to the several
benchmark models , but also the high interpretability of our proposed model in theempirical study.

Hierarchical Topic Model for Tensor Data and Extraction of
Weekly and Daily Activity Patterns
Shunichi Nomura (Waseda University)

Latent Dirichlet allocation (LDA) is a popular topic model for
extracting common patterns from discrete datasets. It is extended tothe pachinko allocation model (PAM) with a hierarchical topic
structure. This study presents a combination meal allocation (CMA)
model, which is a further enhanced topic model from the PAM thathas both hierarchical categories and hierarchical topics. We consider
count datasets in multiway arrays, i.e., tensors, and introduce a setof topics to each mode of the tensors. The topics in each mode areinterpreted as patterns in the topics and categories in the next mode.
Despite there being a vast number of combinations in multilevel
categories, our model provides simple and interpretable patterns by
sharing the topics in each mode. Latent topics and their membershipare estimated using Markov chain Monte Carlo (MCMC) methods.
We apply the proposed model to step-count data recorded by activity
monitors to extract some common activity patterns exhibited by theusers. Our model identifies four daily patterns of ambulatory activities(commuting, daytime, nighttime, and early-bird activities) as subtopics, and six weekly activity patterns as super-topics. We also
investigate how the amount of activity in each pattern dynamicallyaffects body weight changes.
This is joint work with Michiko Watanabe at Rissho University and
Yuko Oguma at Keio University.

Performance-Based Earthquake Early Warningfor Regional Seismic Risk Mitigation
Stephen Wu, The Institute of Statistical Mathematics, Japan
Keisuke Yano, The Institute of Statistical Mathematics, Japan
Masumi Yamada, DPRI, Kyoto University, Japan
Koji Tamaribuchi, Meteorological Research Institute, Japan
Sarah Minson, US Geological Survey, USA
Elizabeth S. Cochran, US Geological Survey, USA
Talk summary:
Earthquake early warning system (EEWS) aims to provide few seconds to a minute ofshaking alert before strong shaking of an on-going seismic event reaches a target site.
Developing a reliable EEWS typically involves rapidly solving a highly ill-posed seismic
source inversion problem using only few seconds of seismic wave data. Such aproblem is inherently highly uncertain, which means that false and missed alarm isunavoidable. Therefore, a performance-based approach for the system design isneeded in order to develop a practically useful EEWS. Here, we propose a fully
probabilistic framework that covers three important aspects of EEWS – real-timeground motion prediction, spatially correlated warning criteria, and temporally
correlated warning criteria. In each aspect, exploiting relevant prior knowledge can help
reducing the uncertainty of warnings, thus, developing a more robust EEWS. This talkwill mainly focus on discussion of the ideology and design strategy of such practicallyuseful EEWS from a statistical perspective.

Portfolio risk valuation using asymmetric copulas∗
Toshinao Yoshiba†
October 12, 2022
The multivariate Student-t copula is frequently used in ﬁnancial portfolio risk management andother statistical areas when there is tail dependence in the data. It often is a good-ﬁtting copula
but can be improved on when there is tail asymmetry. We propose to use Generalized Hyperbolic(GH) and Azzalini–Capitanio (AC) skew-t copulas to incorporate asymmetric tail dependence ofrisk factors using the numerical implementation for maximum likelihood estimation proposed in
Yoshiba (2018). The GH skew-t copula was proposed by Demarta and McNeil (2005).
The GH skew-t random vector X is constructed by using the Gamma random variable V ∼G(ν/2, ν/2) and the d-variate normal random vector Z = (Z1 , . . . , Zd ) ∼ Nd (0, Ψ) with thecorrelation matrix Ψ as:Z
X = γV −1 + √ ,(1)
Vwhere γ ∈ Rd is the skewness paramemter vector. On the other hand, AC skew-t copula implicit in
Azzalini and Capitanio (2003) multivariate skew-t distribution was proposed by Joe (2006). The
AC skew-t random vector X is given as:
Y
X=√ ,
V

(2)

where Y = (Y1 , . . . , Yd ) is the skew Normal random vector proposed by Azzalini and Dalla Valle(1996), which is constructed as:
Yj = δj |Z0 | + 1 − δj2 Zj , j = 1, . . . , d, Z0 ∼ N (0, 1),(3)where δ ≡ (δ1 , . . . , δd ) ∈ (−1, 1)d is the skewness paramemter vector and Z = (Z1 , . . . , Zd ) ∼Nd (0, Ψ) as the GH skew-t random vector.
We compare the parameters of AC skew-t, GH skew-t, Student-t, Normal copulas using Akaike
and Bayesian information criteria for the two groups of daily stock returns in stress and in peacetime. For skew-t copulas, we assume equi-skewness settings, that is, δ1 = · · · = δd = δ for AC and
γ1 = · · · = γd = γ for GH. Each group is given by the equally weighted portfolio which consists ofthree indices from TOPIX33 Sector Indices. The ﬁrst portfolio consists of ﬁnancial sectors portfolio with high correlation including bank, insurance, and securities sectors. The second consists ofbank, air transportation, electricity sectors with low correlation.
In the empirical analyses, we show the skewness δ, γ of skew-t copulas are signiﬁcantly negative
for both the unﬁltered returns and the ﬁltered returns by EGARCH(1,1). We also show that themore sophisticated the marginal model, the more signiﬁcant the negativity. The tendency holds
both in stress and in peacetime.
With validating the skewness, we investigate the behavior of the value-at-risk and expectedshortfall of the two types of stock portfolio by employing several backtesting methods. The backtesting methods include the unconditional coverage test of Kupiec (1995), the independence test,
and the conditional coverage test of Christoﬀersen (1998) for value-at-risk, and the discrepancy
measurement of Embrechts, Kaufmann and Patie (2005) for expected shortfall.
∗†This research is supported by JSPS KAKENHI Grant Number JP21K01581.
Tokyo Metropolitan University / Institute of Statistial Mathematics (E-mail: tyoshiba@tmu.ac.jp)

1

The unconditional coverage tests for the value-at-risk with unﬁltered return are rejected instress for both the ﬁnancial sectors portfolio and the low-correlated portfolio. On the other hand,
the tests with EGARCH ﬁltered return are not rejected.
The independence tests for the value-at-risk with unﬁltered return are rejected in peacetime forboth the ﬁnancial sectors portfolio and the low-correlated portfolio. On the other hand, the tests
with EGARCH ﬁltered return are not rejected.
The discrepancies of expected shortfall by skew-t copulas are small especially in stress. Evenin peacetime, the discrepancies of expected shortfall by skew-t or Student-t copulas are small.
We conclude that the value-at-risk with high conﬁdence level and the expected shortfall of astock portfolio are well captured by skew-t copulas with ﬁltered return.

References
Azzalini, A. and Dalla Valle, A. (1996) “The Multivariate Skew-Normal Distribution,” Biometrika, 83(4),715–726.
Azzalini, A. and Capitanio, A. (2003) “Distributions generated by perturbation of symmetry with emphasis
on a multivariate skew t-distribution,” Journal of the Royal Statistical Society Series B, 65(2), 367–389.
Christoﬀersen, P. F. (1998) “Evaluating Interval Forecasts,” International Economic Review, 39(4), 841–862.
Demarta, S. and McNeil, A. J. (2005) “The t Copula and Related Copulas,” International Statistical Review,73(1), 111–129.
Embrechts, P., Kaufmann, R., and Patie, P. (2005) “Strategic Long-Term Financial Risks: Single Risk
Factors,” Computational Optimization and Applications, 32(1), 61–90.
Joe, H. (2006) “Discussion of “Copulas: Tales and facts”, by Thomas Mikosch,” Extremes, 9(1), 37–41.
Kupiec, P. H. (1995) “Techniques for Verifying the Accuracy of Risk Measurement Models,” The Journal of
Derivatives, 3(2), 73–84.
Yoshiba, T. (2018) “Maximum likelihood estimation of skew-t copulas with its applications to stock returns,”
Journal of Statistical Computation and Simulations, 88(13), 2489–2506.

2

Local asymptotic normality for jump-diﬀusion processes withdiscrete observations
Teppei Ogihara (University of Tokyo) and Yuma Uehara (Kansai University)
We study local asymptotic normality (LAN) of jump-diﬀusion processes with discrete observations.
Local asymptotic normality (LAN) is an important property in asymptotic statistical theory because it
enables us to discuss asymptotic eﬃciency of parameter estimators for parametric models. Let Xtα be aparametrized m-dimensional stochastic process satisfying
Xtα = x0 + t0a(Xsα , θ)ds + t0b(Xsα , σ)dWs +

Nt


Yi ,

(1)

j=1

where a, b : Rm - and Rm ⊗ Rm -valued continuous functions, Nt is a Poisson process with intensity λ(θ),
Wt : an m-dim standard Brownian motion, (Yi )∞i=1 is an i.i.d. sequence with the density function Fθ , andd
α = (σ, θ) ∈ R is a parameter to be estimated. Let α0 = (σ0 , θ0 ) be the true value of the parameter,
and let Xt = Xtα0 . We observe (Xtk )nk=0 , where tk = khn , hn → 0 and nhn → ∞.
Let {Pα,n }α,n be a family of probability measures generated by the observation (Xtαk )nk=0 . {Pα,n }α,n issaid to satisfy local asymptotic mixed normality (LAMN) at α = α0 if there exist d × d random, positivedeﬁnite matrices Γn and Γ, d-dim random vectors Nn and N , and a positive deﬁnite matrix n such that
 

1 
dPα0 +n u,n− uΓn N n − u Γn u → 0
log
dPα0 ,n2
in Pα0 ,n -probability, N ∼ N (0, Id ), N and Γ are independent, and L((Nn , Γn )|Pα0 ,n ) → L((N , Γ)) asn → ∞ for any u ∈ Rd . If Γ is nonrandom, {Pα,n }α,n satisﬁes the LAN property. Under LAMN, any
regular estimator {Vn } satisﬁes the inequality−1/2
lim inf Eα0 ,n [l(| −1N |)]n (Vn − α0 )|)] ≥ E[l(|Γn→∞for any increasing function l : [0, ∞) → R with l(0) = 0 (Proposition 3 in Jeganathan [3]). An estimatorwhich attains the lower bound of the above inequality is called asymptotically eﬃcient.
To show the LAN property, we need to specify the limit of log(dPα0 +n u,n /dPα0 ,n ). It is diﬃcult todeal with transition density ratio for two diﬀerent jump-diﬀusion processes. In the proof of the LA(M)N
properyt for diﬀusion processes in Gobet [1, 2], he used Aronson estimates
C1 G1 (x, y) ≤ pk (x, y) ≤ C2 G2 (x, y)
for transition density pk of the diﬀusion process to control transition density ratios, where G1 , G2 are
Gaussian density functions and C1 , C2 are positive constants. However, it is diﬃcult to obtain Aronsontype estimates for jump-diﬀusion processes.
Instead, we use a scheme with the so-called L2 regularity condition developed in Jeganathan [3]. Let
pk,α (xk−1 , xk ) be the transition density function. We assume that pk,α ∈ C 2 (Θ) and the zero points ofpk,α do not depend on the parameter α. A simple version of the L2 regularity condition is the conditionof the following convergence:
n

k=1

 
Eα 0

√

pk,αu −√u n ∂θ pk,α0
pk,α0 −√
2 pk,α02(xk−1 , xk )dxk → 0,

where αu = α0 + n u. The integrand in the left-hand side of the above equation can be rewritten as
follows.

 10
(1 − s)un∂α pk,αsu ∂α p
∂α2 pk,αsuk,αsu−
2pk,αsu4(pk,αsu )2



2
nu

pk,αsu dxk ds.

In the last expression, only the transition density at αsu appears, and hence we do not need Aronson-type
estimates for transition density functions. Jeganathan [3] showed LAMN for Markov processes under theL2 regularity condition and some conditions for ∂αl log pk,α0 (l ∈ {1, 2}). The original scheme cannot
be applied for jump-diﬀusion processes because of their fat-tailed behaviors. Therefore, we extend thescheme so that it can be applied to jump-diﬀusion processes.
Another problem to show the LAN property for jump-diﬀusion processes is that the transition probability for no jump is quite diﬀerent from that for the presence of jumps. This fact makes it diﬃcult to
identify the asymptotic behavior of the likelihood function. To deal with this problem, we approximate
the original likelihood function by using a thresholding likelihood function that detects the existenceof jumps. By using these techniques, we obtain the LAN property for jump-diﬀusion processes. The
matrices n and Γ in the deﬁnition of the LAMN property is given byn =n−1/2 Id100(nhn )−1/2 Id2and Γ =

Γ1
0

0
Γ2

,

where π(dx) is the limit distribution of Xt , fθ (z) = λ(θ)Fθ (z), S(x, σ) = bb (x, σ),1[Γ1 ]ij =tr(∂σi SS −1 ∂σj SS −1 )(x, σ0 )dπ(x),2∂θi fθ0 ∂θj fθ0 −1(z)dz.
[Γ2 ]ij = (∂θi a) S (∂θj a)(x, α0 )dπ(x) +
f θ0
These

n and Γ are the same as the ones associated with the quasi-maximum-likelihood and Bayes-type

estimators proposed in Shimizu and Yoshida [5] and Ogihara and Yoshida [4], and then we can showthese estimators are asymptotically eﬃcient in this model.

参考文献[1] E. Gobet. Local asymptotic mixed normality property for elliptic diﬀusion: a Malliavin calculus approach.
Bernoulli, 7(6):899–912, 2001.
[2] E. Gobet. LAN property for ergodic diﬀusions with discrete observations. Ann. Inst. H. Poincaré Probab.
Statist., 38(5):711–737, 2002.
[3] P. Jeganathan. On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed
normal. Sankhyā Ser. A, 44(2):173–212, 1982.
[4] T. Ogihara and N. Yoshida. Quasi-likelihood analysis for the stochastic diﬀerential equation with jumps. Stat.
Inference Stoch. Process., 14(3):189–229, 2011.
[5] Y. Shimizu and N. Yoshida. Estimation of parameters for diﬀusion processes with jumps from discreteobservations. Stat. Inference Stoch. Process., 9(3):227–277, 2006.

Quasi-inﬁnitely divisible distributions
Alexander Lindner
Inﬁnitely divisible probability distributions on Rd constitute a well-studied classof probability distributions. By deﬁnition, a probability distribution μ on Rd is inﬁnitely divisible if and only if it has convolution roots of all orders, i.e. if for every n ∈ Nthere exists some probability measure μn on Rd such that μ∗nn , the n-fold convolutionof μn with itself, is equal to μ. By the famous Lévy-Khintchine formula, a probability
, deﬁned
measure μ on Rd is inﬁnitely divisible if and only its characteristic function μ

izx
d
by μ
(z) = Rd e μ(dx) for z ∈ R , has a Lévy–Khintchine representation, i.e. if andonly if there are γ ∈ Rd , a symmetric non-negative deﬁnite matrix A ∈ Rd×d and aLévy measure ν on Rd such that


(1)

1
μ
(z) = exp iz, γ − z, Az +
2




(e
Rd

iz,x

− 1 − iz, x) ν(dx)∀ z ∈ Rd ;recall that a Lévy measure is a measure on Rd such that ν({0}) = 0 and


Rd

(1 ∧

|x|2 ) ν(dx) < ∞. The triplet (A, ν, γ) is called the characteristic triplet of ν and is
known to be unique; further, to any given triplet (A, ν, γ) there exists an inﬁnitely
divisible distribution having this triplet as characteristic triplet. The importance ofinﬁnitely divisible distributions lies in their one-to-one correspondence to Lévy processes (in law), i.e. stochastic processes with independent and increments that startin 0 and have càdlàg paths. Properties and examples of inﬁnitely divisible distributions are well-studied. Normal distributions, compound Poisson distributions, stable
distributions, geometric distributions and many other distributions are known to beinﬁnitely divisible. On the other hand, probability distributions with compact support
or not inﬁnitely divisible unless they are Dirac measures, and by the Lévy–Khintchine
representation, the characteristic function of an inﬁnitely divisible distribution mustbe zero-free.
The goal of the present talk is to extend the class of inﬁnitely divisible distributions to a larger class. Inspired by the Lévy–Khintchine formula, we call a probability
distribution μ on Rd quasi-inﬁnitely divisible, if its characteristic function has a Lévy–
Khintchine representation as in (1), however with a ‘signed Lévy measure’ and somesymmetric, but not necessarily non-negative deﬁnite, matrix A ∈ Rd×d . Here, by asigned Lévy measure we mean a ‘signed measure’ that can be written as a diﬀerenceof two Lévy measures, which will henceforth be called a quasi-Lévy measure. The
1

corresponding triplet (A, ν, γ) can again be shown to be unique and is again called
the characteristic triplet of the quasi-inﬁnitely divisible distribution.
We will study quasi-inﬁnitely divisible distributions to some extent and present
examples. In particular, we will see that not every triplet gives rise to a probability distribution, and even more, that the matrix A must be positive semi-deﬁnite.
The quasi-Lévy measure however can be truly signed, although not all quasi-Lévy
measures can appear. We then study certain subclasses of distributions. E.g., it canbe shown that a probability distribution that has an atom of mass greater than 1/2
must be quasi-inﬁnitely divisible. Further, we study probability distributions on thegrid Zd and obtain that such a distribution is quasi-inﬁnitely divisible if and only if
its characteristic function is zero-free. We deduce that the class of quasi-inﬁnitely
divisible distributions is dense with respect to weak convergence in dimension d = 1,
but fails to be dense in higher dimensions. We also mention some further results
regarding characterisations of quasi-inﬁnite divisibility for subclasses of probability
distributions, such as discrete distributions on R or probability distributions thathave a non-trivial absolutely continuous part and non-trivial discrete part but nocontinuous singular part, as discussed in [1, 2, 3, 5]. The talk is based on [4, 6, 7],
which are (joint) works with/of Berger, Kutlu, Pan and Sato.
References[1] Alexeev, I.A. and Khartov, A.A. (2021) Spectral representations of characteristic functions of
discrete probability laws. Bernoulli, to appear.
[2] Berger, D. (2019) On quasi-inﬁnitely divisible distribuitons with point mass. Math. Nachr. 292,
1674–1684.
[3] Berger, D. and Kutlu, M. (2022) Quasi-inﬁnite divisibility of a class of distributions with discrete
part. Proc. Amer. Math. Soc., to appear.
[4] Berger, D. and Lindner, A. (2022) A Cramér–Wold device for inﬁnite divisibility of Zd -valued
distributions. Bernoulli 28, 1276–1283.
[5] Khartov, A.A. (2022) A criterion of quasi-inﬁnite divisibility for discrete laws. Stat. Probab. Lett.
185, Paper No. 109436, 4 pp.
[6] Kutlu, M. (2021) On a denseness result for quasi-inﬁnitely divisible distributions. Stat. Probab.
Lett. 176, Paper No. 109139, 4 pp.
[7] Lindner, A., Pan, L., and Sato, K. (2018) On quasi-inﬁnitely divisible distributions. Trans. Amer.
Math. Soc. 370, 8483-8520.

2

Subexponentiality of densities of inﬁnitely divisible distributions
Nanzan University

Muneya Matsui

Abstract
We show the equivalence of three properties for an inﬁnitely divisible distribution:
the subexponentiality of the density, the subexponentiality of the density of its
Lévy measure and the tail equivalence between the density and its Lévy measuredensity, under monotonic-type assumptions on the Lévy measure density. The key
assumption is that tail of the Lévy measure density is asymptotic to a non-increasing
function or is almost decreasing. Our conditions are natural and cover a rather wideclass of inﬁnitely divisible distributions. Several signiﬁcant properties for analyzing
the subexponentiality of densities have been derived such as closure properties of[ convolution, convolution roots and asymptotic equivalence ] and the factorization
property. Moreover, we illustrate that the results are applicable for developing
the statistical inference of subexponential inﬁnitely divisible distributions which are
absolutely continuous.

Introduction
Let f, g be probability density functions on R and denote by f ∗ g the convolution of f and g: ∞f ∗ g(x) =f (x − y)g(y)dy,−∞

and denote by f ∗n the nth convolutions with itself. Throughout the paper, for functions α, β : R →R+ , α(x) ∼ β(x) means that limx→∞ α(x)/β(x) → 1. We study the following characteristics fordensities.
Deﬁnition 0.1. (i) f is (right-side) long-tailed, denoted by f ∈ L, if there exists x0 > 0 such thatf (x) > 0, x ≥ x0 and for any ﬁxed y > 0 f (x + y) ∼ f (x).
(ii) f is (right-side) subexponential on R, denoted by S, if f ∈ L and f ∗2 (x) ∼ 2f (x).
(iii) f with dist. F is weakly (right-side) subexponential on R, denoted by S+ , if f ∈ L and thefunction f+ (x) = 1R+ (x)f (x)/F (0), x ∈ R is subexponential, i.e. f+ ∈ S. Here F (x) = 1 − F (x).
Deﬁnition 0.2. (i) We say that a density f : R → R+ is asymptotic to a non-increasing function(a.n.i. for short) if f is locally bounded and positive on [x0 , ∞) for some x0 > 0, and(0.1)

sup f (t) ∼ f (x)
t≥x

and

inf

x0 ≤t≤xf (t) ∼ f (x).

(ii) We say that a density f : R → R+ is almost decreasing (al.d. for short) if there exists x0 > 0and K > 0 such thatf (x + y) ≤ Kf (x),for all x > x0 , y > 0.

Notice that the al.d. property includes the a.n.i. property, and the latter is satisﬁed by theregularly varying functions with negative indices.
We will investigate properties of the above sort, particularly on inﬁnitely divisible distributions
μ on R. The characteristic function (ch.f.) of μ is
 ∞

1
μ
(z) = exp(eizy − 1 − izy1{|y|≤1} )ν(dy) + iaz − b2 z 2 ,
(0.2)
2
−∞
∞
where a ∈ R, b ≥ 0 and ν is the Lévy measure satisfying ν({0}) = 0 and −∞ (1 ∧ x2 )ν(dx) < ∞.
Throughout this paper, we always assume that the Lévy measure ν of μ has a density, and wedenote by ID(R) the class of all inﬁnitely divisible distributions on R.
1

Main contents
We separate the cases depending on weather ν(R) < ∞ or ν(R) = ∞. Note that we use notationg also for the (non-proper) density of a Lévy measure.
Theorem 0.3. Let μ ∈ ID(R) with ν(dx) = g(x)dx such that ν(R) < ∞. Denote the non-Gaussianpart μ , which is a γ-shifted compound Poisson given byμ (dx) = e−λ δγ (dx) + (1 − e−λ )f (x − γ)dx,

γ ∈ R,

where δγ is Dirac measure at γ, f is a proper density and λ > 0 is the Poisson parameter. Thenthe following are equivalent.
(i)f ∈ S+ and f is al.d.

(ii)

g ∈ S+ and g is al.d.

(iii)

g ∈ L, g is al.d. and lim f (x)/g(x) = λ/(1 − e−λ ).
x→∞Theorem 0.4. Let μ ∈ ID(R) with ν(dx) = g(x)dx such that ν(R) = ∞. Suppose that g1 (x) =1{x>1} g(x)/ν((1, ∞)) is bounded. For a density f of μ we consider the following properties.
(i)f ∈ S+(ii)

g1 ∈ S +(iii)

g1 ∈ Land&f is al.d.
lim f (x)/g1 (x) = ν((1, ∞)).

x→∞(a) If g is a.n.i., then we can choose f such that (i), (ii) and (iii) are equivalent.
(b) If g is al.d., then we can choose f such that (ii) ⇔ (iii) implies (i).
Since the a.n.i. property includes regular variation, the following is immediate.
Corollary 0.5. Let μ ∈ ID(R) with ν(dx) = g(x)dx such that ν(R) = ∞. Suppose that g1 (x) =1{x>1} g(x)/ν((1, ∞)) is bounded. Then, g1 (x) is a.n.i. and we can choose a density f of μ suchthat it is regularly varying if and only if g is regularly varying, and in this case f (x) ∼ g(x).
We apply our results to the consistency proof of the maximum likelihood estimation (MLE forshort) for μ ∈ ID(R) which is absolutely continuous. For simplicity we put a = b = 0 in μ(z) of(0.2) and assume that μ(z) is absolutely integrable.
Let f (x; θ) be the density of μ with θ a parameter vector and g(x; θ) be a density of the corresponding Lévy measure ν. Let (X1 , . . . , Xn ) be a random sample from f (x; θ0 ) with θ0 ∈ Θ whereΘ is a compact parameter space. Deﬁne the likelihood functionn−1
Mn (θ) = n
log f (Xi ; θ).
i=1

MLE θ̂n maximizes the function θ → Mn (θ). We say that a function α(x; θ) is identiﬁable ifa.e.
α(· ; θ) = α(· ; θ ) every θ = θ ∈ Θ, i.e. α(x; θ) = α(x; θ ) does not hold. For convenience, weonly consider the symmetric or positive-half case, but we can easily generalize the result in the
non-symmetric two-sided case. We use the function g1 deﬁned in Theorem 0.4.
Proposition 0.6. Let μ ∈ ID(R) given by (0.2) with a = b = 0 such that μ(z) is absolutely integrable. Let g(x; θ) be a symmetric or positive-half density of ν. Suppose (i) : g(x; θ) is identiﬁable,
θ → g(x; θ) is continuous in θ for every x, and (supθ∈Θ | log g1 (x; θ)|)g1 (x; θ0 )dx < ∞ with Θ acompact set such that θ0 ∈ Θ. Suppose (ii) : g1 (x; θ) is bounded and a.n.i., and g1 ∈ S. Then MLEpθ̂n satisﬁes θ̂n → θ0 .
References[1] Matsui, M. (2022) Subexponentialiy of densities of inﬁnitely divisible distributions, arXiv:2205.02074.
2

Existence and distributional properties of solutions for
SPDEs driven by Lévy white noiseDavid Berger
In this talk we are discussing solutions of linear stochastic partial diﬀerential
equations (SPDEs) driven by so-called Lévy white noise, where we are interested
in distributional solutions, i.e. our solutions are generalized stochastic processes on
the space of inﬁnitely diﬀerentiable functions with compact support D(Rd ). Let us
shortly recall the deﬁnition of a generalized process.
Deﬁnition 0.1 (see [1]). A generalized random process is a linear and continuousfunction s : D(Rd ) → L0 (Ω). The linearity means that, for every ϕ1 , ϕ2 ∈ D(Rd ) andγ ∈ R,s(ϕ1 + γϕ2 ) = s(ϕ1 ) + γs(ϕ2 ) almost surely.
The continuity means that if ϕn → ϕ in D(Rd ), then s(ϕn ) → s(ϕ) in L0 (Ω).
A generalized process s can also be seen as a random variable on the spaceof distributions D (Rd ), which then gives rises to deﬁne the so-called characteristicfunctional Ps (ϕ) :=eiu,ϕ dPs (u) for ϕ ∈ D(Rd ). The characteristic functionalD  (Rd )describes uniquely the probability measure generated by the generalized random process. We deﬁne a Lévy white noise as a generalized random process with a special
characteristic functional.
Deﬁnition 0.2. A Lévy white noise L̇ is a generalized random process, where thecharacteristic functional is given by⎛⎞ (ϕ) = exp ⎝ ψ(ϕ(x))λd (dx)⎠PL̇
Rdfor every ϕ ∈ D(Rd ), where ψ : R → C is given by
1 2
ψ(z) = iγz − az + (eixz − 1 − ixz1|x|≤1 )ν(dx)2R+

where a ∈ R , γ ∈ R and ν is a Lévy-measure, i.e. a measure such that ν({0}) = 0
and min(1, x2 )ν(dx) < ∞. We say that L̇ has the characteristic triplet (a, γ, ν).
R

We are interested in solutions of the equation(0.1)

p(D)s = q(D)L̇,
1

where p(D) and q(D) are linear partial diﬀerential operators. We say that a generalized process s solves (0.1) if
s, p(D)∗ ϕ = L̇, q(D)∗ ϕ for every ϕ ∈ D(Rd ).
Constructing an extension of the Lévy white noise to a greater domain we show thefollowing.
Theorem 0.3 (see [1]). Let p and q be polynomials such that the rational functionq(i·)/p(i·) has a holomorphic extension in a strip {z ∈ Cd : z < ε} for some ε > 0
and let L̇ be a Lévy white noise with characteristic triplet (a, γ, ν) such that
1|r|>1 log(|r|)d ν(dr).
R

Then there exists a stationary solution s of (0.1), which we call a stationary CARMA(p, q)
generalized process.
We also discuss distributional properties of the solution s.
Proposition 0.4 (see [1]). Let L̇ have existing β-moment (β > 0) and let p andq be polynomials satisfying the condition from Theorem 0.3. Then the stationary
CARMA(p, q) generalized process s constructed in Theorem 0.3 has existing β-moment,
too.
References[1] D. Berger, Lévy driven CARMA generalized processes and stochastic partial diﬀerential equations, Stoch. Process. Their Appl. (2020), 130 (10), 5865-5887.

2

Nonparametric regression for locally stationary
random ﬁelds on Rd
Daisuke Kurisu (Yokohama National University)

1 Introduction
In this study, we develop an asymptotic theory of nonparametric regression for locally stationary
random ﬁelds (LSRFs) {Xs,An : s ∈ Rn } in Rp observed at irregularly spaced locations in Rn = [0, An ]d ⊂Rd . We ﬁrst derive the uniform convergence rate of general kernel estimators, followed by the asymptotic
normality of an estimator for the mean function of the model. Moreover, we consider additive models
to avoid the curse of dimensionality arising from the dependence of the convergence rate of estimatorson the number of covariates. Subsequently, we derive the uniform convergence rate and joint asymptotic
normality of the estimators for additive functions. We also introduce approximately mn -dependent RFs toprovide examples of LSRFs. We ﬁnd that these RFs include a wide class of Lévy-driven moving averageRFs.

2 ModelIn this study, we consider the following model:sjYsj ,An = m, Xsj ,An + sj ,An , sj ∈ Rn , j = 1, . . . , n,An(1)where E[ s,An |Xs,An ] = 0 and Rn = [0, An ]d ⊂ Rd is a sampling region with An → ∞ as n → ∞.
Here, Ysj ,An and Xsj ,An are random variables of dimensions 1 and p, respectively. We assume that
{Xs,An : s ∈ Rn } is a locally stationary random ﬁeld on Rn ⊂ Rd (d ≥ 2).

3 Main Results
The objectives of this study are to (i) derive the uniform convergence rate of kernel estimators forthe density function of Xs,An and the mean function m in the model (1) over compact sets; (ii) derive
the asymptotic normality of the estimators at a speciﬁed point; and (iii) provide examples of locallystationary random ﬁelds on Rd with a detailed discussion of their properties. To attain the ﬁrst andsecond objectives, we ﬁrst derive the uniform convergence rate of the important general kernel estimators;
the result is crucial for demonstrating our main results. As general estimators include a wide range of
kernel-based estimators such as the Nadaraya-Watson estimators, the general results are of independentinterest. Although these results are general, the estimators are aﬀected by dimensionality because theirconvergence rate depends on the number of covariates. Hence, we consider additive models and derive
the uniform convergence rate and joint asymptotic normality of kernel estimators for additive functions
based on the backﬁtting method developed by Mammen et al. (1999) and Vogt (2012). Our results
are extensions of the results for time series in Vogt (2012) to random ﬁelds with irregularly spaced
observations, which include irregularly spaced time series as a special case.
To attain the third objective of our study, we discuss examples of locally stationary random ﬁelds on Rd

that satisfy our regularity conditions. For this, we introduce the concept of approximately mn -dependent
locally stationary random ﬁelds (mn → ∞ as n → ∞) and we extend continuous autoregressive andmoving average (CARMA)-type random ﬁelds developed in Brockwell and Matsuda (2017) to locally stationary CARMA-type random ﬁelds. CARMA random ﬁelds are characterized by solutions of (fractional)
stochastic partial diﬀerential equations (cf. Berger (2020)) and are known as a rich class of models for
spatial data (cf. Brockwell and Matsuda (2017)).

References
Berger, D. (2020). Lévy driven CARMA generalized processes and stochastic partial diﬀerential equations.
Stochastic Process. Appl. 130, 5865-5887.
Brockwell, P.J. and Matsuda, Y. (2017). Continuous auto-regressive moving average random ﬁelds on
Rn . J. Roy. Statist. Soc. Ser. B Stat. Methodol. 79 833-857.
Kurisu, D. (2022). Nonparametric regression for locally stationary random ﬁelds under stochastic sampling design. Bernoulli 28, 1250-1275.
Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backﬁtting
projection algorithm under weak conditions. Ann. Statist. 27 1443-1490.
Vogt, M. (2012). Nonparametric regression for locally stationary time series. Ann. Statist. 40 2601-2633.

2

Intergenerational Risk Sharing in a Deﬁned Contribution
Pension System: Analysis with Bayesian Optimization
An Chen1 , Motonobu Kanagawa2 , and Fangyuan Zhang2
1

Institute of Insurance Science, Ulm University
2
Data Science Department, EURECOM

The talk at the 3rd Tohoku-ISM-UULM workshop in Japan
Abstract
We study a fully funded, collective deﬁned-contribution (DC) pension system with multiple
overlapping generations. We investigate whether the welfare of participants can be improved byintergenerational risk sharing (IRS) implemented with a realistic investment strategy (e.g., noborrowing) and without an outside entity (e.g., share holders) that helps ﬁnance the pension fund.
To implement IRS, the pension system uses an automatic adjustment rule for the indexationof individual accounts, which adapts to the notional funding ratio of the pension system. The
pension system has two parameters that determine the investment strategy and the strengthof the adjustment rule, which are optimized by expected utility maximization using Bayesian
optimization. The volatility of the retirement beneﬁts and that of the funding ratio are analyzed,
and it is shown that the trade-oﬀ between them can be controlled by the optimal adjustment
parameter to attain IRS. Compared with the optimal individual DC benchmark using the lifecycle strategy, the studied pension system with IRS is shown to improve the welfare of risk-averse
participants, when the ﬁnancial market is volatile.

1

On Modiﬁed Interdirections
Jana Klicnarová*The high-dimensional data analysis is still more and more required; therefore, simple, robust, and powerful nonparametric tests for high-dimensional data are in demand. One of thepossible concepts of how to handle such data is based on hyperplane signs (see [Randles, 1989]
and ranks (see [Hettmansperger et al., 1999], [Oja and Paindaveine, 2005]). These concepts
became popular due to their simplicity, geometric interpretation, aﬃne invariance, weak moment assumptions, and robustness to both – radial and angular – outliers. Many authors
were interested in such approaches, see [Randles and Peters, 1990], [Jan and Randles, 1996],[Gieser and Randles, 1997], [Hallin and Paindaveine, 2002]. On the other hand, the original
concepts suﬀer from high computational demands, which led to falling out of the fashion ofthese technics. In the talk, we will discuss the possibilities of these concept modiﬁcations,which reduce computational eﬀort, and, if possible, preserve the advantages of these methods.
We focus on statistics based on hyperplanes in more detail. More precisely, on statistics
based on interdirections [Randles, 1989] and lift-interdirections [Hettmansperger et al., 1999],[Oja and Paindaveine, 2005]. Let us suppose p dimensional sample of size n. The classical tests based on interdirections and lift-interdirections are computationally feasible only
for small data dimensions p and small sample sizes n. If we take a closer look at thestatistics, we realize that we handle with complete U-statistics. It means that the computational demand arises exponentially with the dimension p. Therefore, Hudecova et al.
([Hudecová et al., 2020]) discussed the possibility of using incomplete U-statistics instead ofcomplete ones, i.e., they deﬁne so-called incomplete lift-interdirections and incomplete interdirections. They showed, for interdirections, that it is possible, in test statistics given byRandles ([Randles, 1989]), to use incomplete interdirections based on subsets chosen randomly
with or without replacement if the cardinality of the subsets (mn,p−1 ) satisﬁes: mn,p−1 /n → 0
as n → ∞ (n is a size of sample). Also in the test statistics proposed by Oja and Paindaveine [Oja and Paindaveine, 2005], it is possible to replace classical interdirections and liftinterdirections by incomplete ones if design sets are chosen independently and randomly withor without replacement, satisfying that mn,p−1 /n → 0 as n → ∞ and msn,p /n → 0 as n → ∞,where msn,p is a cardinality of design set for lift-interdirections.
Such results allow us to reduce the computational demand of tests. In the case of incomplete interdirections, the number of considered hyperplanes does not grow with the dimensionof observation; hence we can also easily handle samples from p = 100 dimensional spaces.
On the other hand, for lift-interdirections, the number of needed hyperplanes depends on thedimension by the factor 2p . I.e., for p > 10, the computation of the test is already quite
diﬃcult.
*
Faculty of Economics, University of South Bohemia, Studentská 13, CZ-370 05, České Budějovice, Czech
Republic
email: klicnarova@ef.jcu.cz

1Another possibility of allowing a smaller number of hyperplanes is to apply randomized
lift-interdirections. In such a case, the test statistics asymptotics will be aﬀected by randomization; on the other hand, it allows us to use tests based on this concept in higher dimensions.
In the paper [Hudecová and Šiman, 2022], Hudecová and Šiman study such an approach, givetheoretical results and provide a small simulation study. Such randomized lift-interdirections
allow to run of the test for high dimensional data (p > 100) and large samples (n > 1000) ina short time (the computational demand does not depend on p); on the other hand, there issome uncertainty to the statistical inference, given by randomization. Still, it does not aﬀect
the mean p-value too much; moreover, it quickly becomes reasonably small with a growing
number of chosen hyperplanes.

References[Gieser and Randles, 1997] Gieser, P. W., Randles, R. H. (1997). A nonparametric test of
independence between two vectors. Journal of the American Statistical Association,
92(438), 561-567.
[Hallin and Paindaveine, 2002] Hallin, M., Paindaveine, D., 2002. Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Annals of Statistics
30, 1103–1133.
[Hettmansperger et al., 1999] Hettmansperger, TP., Möttönen, J. and Oja, H., 1999. The
geometry of the aﬃne invariant multivariate sign and rank methods. Journal of Nonparametr. Stat. 11, 271–285.
[Hettmansperger and Randles, 2002] Hettmansperger, T. P., Randles, R. H., 2002. A practical aﬃne equivariant multivariate median. Biometrika 89, 851–860.
[Hudecová and Šiman, 2022] Hudecová, Š., Šiman, M. (2022). Multivariate ranks based on
randomized lift-interdirections. Computational Statistics & Data Analysis, 172, 107480.
[Hudecová et al., 2020] Hudecová, Š., Klicnarová, J., and Šiman, M. (2020). Incomplete interdirections and lift-interdirections. Journal of Nonparametric Statistics, 32(1), 93-108.
[Jan and Randles, 1996] Jan, S.-L., Randles, R. H., 1996. Interdirection tests for simple
repeated-measures designs. Journal of the American Statistical Association 91, 1611–
1618.
[Oja and Paindaveine, 2005] Oja, H., Paindaveine, D., 2005. Optimal signed-rank tests basedon hyperplanes. Journal of Statistical Planning and Inference 135, 300–323.
[Randles, 1989] Randles, R. H., 1989. A distribution-free multivariate sign test based on interdirections. Journal of the American Statistical Association 84, 1045–1050.
[Randles and Peters, 1990] Randles, R. H., Peters, D., 1990. Multivariate rank tests for thetwo-sample location problem. Communications in Statistics - Theory and Methods 19,4225–4238.

2

Extracting Valuable Information based on Human Behavior
Satoshi Shioiri
Advanced Institute of Yotta Informatics & Research Institute of Electrical Communication, Tohoku University

Accumulation of information is essential for human knowledge production, and information technology
has accelerated the speed of data accumulation. The momentum of expansion of its production volume is
growing more and more in this century. Based on a forecast, it will exceed 1 Yottabytes in 2030 [1, 2]. We
introduce an approach for solution of these problems [3, 4] with an example of research along theapproach, in order to solve the problem of quantity and quality of information: that is, which information
deserves to be kept, what information should be left for knowledge production, which information should
be discarded in order to secure storage capacity. This can be recognized as evaluation of data value. It isdifficult to define and evaluate the qualitative aspects of information compared to the quantitative aspects
of information. Our research group proposed a new approach based on information quality and value
concepts as "quali-informatics" [4]. Since evaluation of information is inherently relative to the evaluator's
subjectivity and to the context, the target is a specific data set for individual person, organization or society.
Critical factor of quali-informatics is, therefore, evaluation by human [5].
In this presentation, I introduce three topics related to evaluation of value of information: attention as aselection process, learning effect for selection, and estimation of subjective judgments by machinelearning. First is selecting information by attention. The brain receives a vast amount of input from thesensory organs such as eyes and ears each second and the brain must decide what to process by selecting
the appropriate information. The function of selective attention is indeed what is required in society with
Yottabyte scale data. The human attention process could provide clues for evaluation method of
information values. The visual system, for example, can attend to several aspects of stimulation such as
space, time, features, objects, link to body or action and so on [6,7]. This indicates that the brain selects
information processing a limited number of aspects instead of any possible information from the retina,
such as signals from individual photoreceptors or any possible combination of them. In other words,
aspects that can be attended is along the line of value estimation of the visual system. Value of information
is evaluated based on processing of the information in the brain.
Second is learning effect for selection. It is not surprising that a person can find things in a familiar
place like own office easily than in a place novel to s/he. Either/both explicit or/and implicit learning
generates representation of a place through spending time there, with which s/he can find things there
efficiently. If the value of information from the environment, such as arrangement of things in a room, isconsidered for finding things in a place, objects identified as landmarks and global feature that identify theroom are likely valuable. Since their meaning is only based on previous experience, we can say thatknowledge plays important roles for evaluation of information values. Tsuchiai et al. reported that object
arrangements in a space is memorized implicitly in three dimensional representations throughout repeatedexcursion of visual search [8]. The study compared the transfer of the contextual cueing effect between
cases with and without self-motion by using visual search displays for 3D objects. The contextual cueing
effect is a learning effect of spatial layout in visual search displays. The contextual cueing effect was
obtained with self-motion but disappeared when the display changed without self-motion. This indicatesthat there is an implicit learning effect in spatial coordinates and suggests that the spatial representation of
object layouts or scenes can be obtained and updated implicitly.
Third is the use of image features and facial expressions to evaluate the value of information. To
evaluate value of information, it is fundamental to find features that is relevant to the usage of theinformation. In the condition where the purpose to decide whether to keep or delete pictures in a mobile
device, one method is image features can be used to evaluate human preference to each picture. A
computer can make the decision by finding the link between image features and preference judgements.
Since human judgements are subjective, collection of preference judgements results are required. Similar
to image features, facial expressions during judgements can also be used to evaluate value of the pictures.
Shioiri et al reported that both image features and facial expressions are useful to estimate preference
judgments. In the study, participants was asked to judge whether they like or dislike (prefer/not prefer)
each picture retrieved from Instagram, which is a photo-sharing social network service. Stimulus pictures
were obtained with a hashtag to collect a certain type of pictures (#luchbox and #landscape). The face wasvideo recorded while participants were thinking for the preference, and the facial expressions wereanalyzed later. Pictures used were also analyzed to extract image features using a trained network (Resnet50). Facial features were extracted using open source software (OpenFace). Image and face features wereused to predict participants’ judgments with a machine learning method (LightGBM) to predict theparticipants preference from image and facial features. The results of the analyses showed that facial
expression features are useful for predicting the participants’ judgments of preference. However, it wasalso found that prediction from the facial expression is likely limited for the same individuals sinceprediction accuracy is poor when training data and the test data were from different participants.
Evaluation of information values is critical in Yottabyte scale society. I believe that approaches withconsideration of human sciences will play important roles there as described in this presentation.
[1]
[2]
[3]
[4]

[5]
[6]
[7]
[8]

DC2012 THE DIGITAL UNIVERSE IN 2020: Big Data, Bigger Digital Shadows, and Biggest Growth in the Far East,
https://www.emc.com/collateral/analyst-reports/idc-the-digital-universe-in-2020.pdf
L. Floridi, the 4TH Revolution: How the Infosphere is reshaping Human Reality, Oxford University Press, 2014.
V. Smil, Data World: Racing toward Yotta. IEEE Spectrum 56(7), p. 20, 2019
H. Muraoka, G. Gyoba, Y. Suzuki, S. Shioiri, M. Nakao, M. Nihei & H. Shoji, Gigantic Amount Information and Storage
Technology : Challenge to Yotta-Byte-Scale Informatics (in Japanese), IEICE technical report : 116(441), pp. 27-32, 2017
(Invited)
S. Shioiri, Y. Sato, Y. Horaguchi, H. Muraoka and M. Nihei, Quali-informatics in the society with yotta scale data, IEEE
International Symposium on Circuits and Systems 2021, Pages: 1-4
S. Shioiri, H. Honjyo, Y. Kashiwase, K. Matsumiya and I. Kuriki, Visual attention spreads broadly but selects information
locally, Scientific Reports 2016 Vol. 6 Issue ARTN 35513, Accession Number: WOS:000385953900001
S. Shioiri, T. Sasada and R. Nishikawa, Visual attention around a hand location localized by proprioceptive information, Cereb
Cortex Commun 2022 Vol. 3 Issue 1 Pages tgac005, Accession Number: 35224493
S. Shioiri, H. Honjyo, Y. Kashiwase, K. Matsumiya and I. Kuriki, Visual attention spreads broadly but selects information
locally, Scientific Reports 2016 Vol. 6 Issue ARTN 35513, Accession Number: WOS:000385953900001

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Recommendation Systems with Network Structure and Big Data
Tsukasa ISHIGAKI (Tohoku University)
Recommendation system supports to provide some valued items for each user from huge
number of alternatives and it are used in a variety of industries, including retail, restaurant,
news, trip, music, movie and media. The market of recommendation systems was valued at
USD 1.2 billion in 2020 [Mordor Intelligence 2021] and is forecast to reach USD 16.13
Billion by 2026 [Reports and Data 2021]. Recommendation system has a crucial role intoday’s business.
In this talk, I will talk about some novel recommendation methods using network structuresand big data. First, I will provide some introductory topics about motivations, properties ofdata, problem settings and basic methods on recommendation system. Then, three topics
that we have proposed in [1-3] will be talked. The methods have been realized using deeplearning with implicit feedback or knowledge graph. The results show that the proposed
methods have a high performance in terms of accuracy or novel recommendation in someexperiments using big data.
[1] Linh Nguyen, Tsukasa Ishigaki, D2D-TM: A Cycle VAE-GAN for Multi-Domain
Collaborative Filtering, Proceedings of IEEE International Conference on Big Data (IEEEBig Data 2019), pp.1175-1180, 2019[2] Kachun Lo, Tsukasa Ishigaki, PPNW: Personalized Pairwise Novelty Loss Weighting forNovel Recommendation, Knowledge and Information Systems, Vol.63, No.5, pp.1117-1148,
2021[3] Kachun Lo, Tsukasa Ishigaki, X-2ch: Quad-Channel Collaborative Graph Network over
Knowledge-Embedded Edges, Proceedings of the 44th International ACM SIGIR
Conference on Research and Development in Information Retrieval (SIGIR 2021), pp.20762080, 2021

Non-concave optimization under risk constraints
An Chen∗ , Mitja Stadje∗ and Fangyuan Zhang‡August 20, 2022.
∗

Institute of Insurance Science, Ulm University, Helmholtzstr. 20, 89069 Ulm, Germany.
E–Mails: an.chen@uni-ulm.de, mitja.stadje@uni-ulm.de‡
EURECOM, E–Mail: fangyuan.zhang0069@gmail.com

In the literature, a non-concave maximization problem under a Value-at-Risk (VaR)
or an Average Value-at-Risk (AVaR) constraint has been studied. For example, one
can ﬁnd the optimal solution for a non-concave maximization problem arising from a
non-linear payoﬀ function under a VaR constraint in Nguyen and Stadje (2020), andthe optimal solution with an S-shape utility function under a VaR constraint in Donget al. (2019). In addition, a non-linear payoﬀ function under an AVaR constraint isstudied in De Franco and Tankov (2011), and an S-shape utility function under an AVaR
constraint is discussed in Armstrong and Brigo (2019). However, both of these discussions
under an AVaR constraint are restricted to some special cases. Despite these studies, theclosed form solutions for a non-concave utility maximization under an Expected Shortfall
or an Expected Discounted Shortfall constraint are missing. Moreover, a thoroughly
comparative analysis of the risk constraints on a non-concave maximization problem
based on closed form solutions is also missing in the literature. Our paper ﬁll this gap.
In this work, we consider a non-concave maximization problem under a risk constraint(Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, Average Value-atRisk). We provide closed-form solutions by the Lagrangian approach. Due to the nonconcave utility function and the risk constraint, the constructed Lagrangian for solving
the optimization problem is highly non-concave and discontinuous. We show that bydecomposing the highly non-concave Lagrangian into local concave or aﬃne functions
on disjoint sets, the global maximum is obtained by comparing local maximums of piecewise Lagrangian, which reduces to ﬁnding the zero roots of a sequence of conjugate
1

functions. However, it is non-trivial to determine the global maximum as the conjugate
functions do not necessarily have zero roots, which implies that the existence of the optimal solution is not fully ensured. Note that in traditional concave utility maximization
problems under risk constraints, the existence of the optimal solution is guaranteed by
the existence of the Lagrangian multipliers. But in the non-concave maximization problem under risk constraints, the optimal solution exists if the Lagrangian multipliers aswell as the zero roots of the conjugate functions both exist, which is more challenging
to prove. In this paper, we work out the conditions under which the zero roots of the
conjugate functions exist and obtain full analytical solutions for the non-concave utility
maximization problems under risk constraints.

References
Armstrong, J. and Brigo, D. (2019). Risk managing tail-risk seekers: Var and expectedshortfall vs s-shaped utility. Journal of Banking and Finance, 101:122–135.
De Franco, C. and Tankov, P. (2011). Portfolio insurance under a risk-measure constraint.
Insurance: Mathematics and Economics, 49(3):361–370.
Dong, Y., Wu, S., Lv, W., and Wang, G. (2019). Optimal asset allocation for participating contracts under the var and pi constraint. Scandinavian Actuarial Journal, pages
1–26.
Nguyen, T. and Stadje, M. (2020). Nonconcave optimal investment with value-at-risk
constraint: An application to life insurance contracts. SIAM Journal on Control andOptimization, 58(2):895–936.

2

Statistical modeling for the nonlinear structure ofquantiles in hierarchical data via regularization
Yuko Araki
Graduate School of Information Sciences
Tohoku University
The purpose of this study was to develop a model that can estimate thetransition of quantile points of a response variable from longitudinal datawith a nonlinear structure when the nonlinear function cannot be identiﬁed
in advance. To achieve this goal, we consider a basis expansion method
for both ﬁxed- and random-eﬀects terms in quantile mixed-eﬀects models.
Since models with basis expansions can be expressed in the same form asa linear model, they can be treated as a special case of the linear quantile
mixed-eﬀects model, such as those proposed by Geraci and Bottai (2007)
and Geraci and Bottai (2014).
To overcome the expensive computational cost of this approach, we proposed the Bayesian nonparametric quantile mixed-eﬀects models (BNQMs,
Tanabe, Araki et al. 2022). In BNQMs, Bayesian regularization can be
performed by assuming a speciﬁc distribution of the prior for the coeﬃcientparameters of the basis function. The priors or hyperpriors are assumed
hierarchically for the parameters of the priors for the coeﬃcient parameters, and regularization parameters can be estimated simultaneously by the
MCMC method. We also proposed the use of a Gaussian process (GP) priorfor the coeﬃcient parameter vectors of the basis functions. We showed thatthe smoothness can be appropriately adjusted when the number of basis
functions is excessive by using a GP prior for Bayesian regularization.
We take the observation of the i-th individual at the j-th measurement
time tij , {(tij , yij ); i = 1, ..., N, j = 1, ..., ni }. Then, the τ -th quantile of yij
at tij can be modeled as
Qyij (τ |tij ) =pβτ k φk (tij ) +

k=1

q


bτ il ψl (tij )l=1= βτ φ(tij ) + b
τ i ψ(tij ),1(1)

where τ ∈ (0, 1), Qyij (·) ≡ F −1 (·), and φ(t) = (φ1 (t), · · · , φp (t)) andψ(t) = (ψ1 (t), · · · , ψq (t)) are vectors of the basis functions in the ﬁxedand random-eﬀects terms, respectively, βτ = (βτ 1 , · · · , βτ p ) is a p × 1
coeﬃcient parameter vector of φ(t), and bτ i = (bτ i1 , · · · , bτ iq ) is a q × 1
coeﬃcient parameter vector of ψ(t), where bτ i ∼ N (0, Γτ ) is assumed. Here,Γτ is a q × q positive-deﬁnite covariance matrix.
We assume that the conditional distribution of yij is an asymmetric
Laplace distribution. Here, the conditional distribution of yij is written as
yij |β, bi , σ ∼ AL(μij , σ, τ ); therefore, its probability density function can be
written as
yij − μij
τ (1 − τ )p(yij |β, bi , σ) =exp −ρτ,(2)
σσwhere μij = β  φ(tij ) + bi ψ(tij ) is the location parameter, −∞ < μij <∞, σ > 0 is the scale parameter, 0 < τ < 1 is the skewness parameter,and ρτ (u) = u(τ − I(u < 0)) is the loss function with indicator function
I(·). For further details of the model and estimation, including GP priorregularization for basis expansion method and prior setting and posterior of
BNQM, please refer to Tanabe, Araki et al. (2022).
The performance of BNQMs was evaluated by a Monte Carlo simulation.
The proposed BNQMs showed the highest estimation accuracy for each data
structure and were shown to be useful as a quantile regression technique inhierarchical data with a nonlinear structure. Then, a BNQM was applied
to longitudinal data of cortisol in infants. The results suggested that thecortisol secretion rhythm in infancy is bimodal, and the magnitude of theamplitude increases as the cortisol level itself increases.

References[1] Geraci M, Bottai M. (2007). Quantile regression for longitudinal datausing the asymmetric Laplace distribution. Biostatistics. 8(1), 140-54.
[2] Geraci, M., Bottai, M. (2014). Linear quantile mixed models. Stat Comput . 24, 461–479.
[3] Tanabe, Y., Araki, Y., Kinoshita, M. et al. (2022). Bayesian nonparametric quantile mixed-eﬀects models via regularization using Gaussian
process priors. Jpn J Stat Data Sci., 5, 241–267.

2

Prediction of Heavy-Tailed Random Functions
Vitalii Makogin∗, Evgeny Spodarev†
September 9, 2022
This talk is based on a recent paper [3]. The R extrapolation code for heavy tailed time series can befound in [2].
Let Y : Ω → R be a square integrable random variable deﬁned on a probability space (Ω, F, P), and let
G ⊂ F be an observable sub–σ–algebra.The classical L2 –theory of prediction of random variables states thatthe best unbiased predictor of Y with respect to G is given by the conditional expectation E(Y |G). But as
far as Y has no ﬁnite moments, no uniﬁed widely accepted prediction theory exists. Our contribution createssuch theory which also applies to the ﬁnite variance case. Its main idea is the following. Let u ∈ R be an
excursion level chosen according to a ﬁnite measure m(·) on R. For any two random variables Y1 , Y2 : Ω → R
introduce the excursion pseudo-metricP({Y1 > u}  {Y2 > u}) m(du),Em (Y1 , Y2 ) :=R

which is an m-weighted average probability of symmetric diﬀerence of excursions of Y1 and Y2 over u ∈ R.
Then, we say that Z is a prediction of a random variable Y onto the σ–algebra G introduced above if
Z = argminY0 Em (Y, Y0 ),whenever this minimum (taken over all G-measurable random variables Y0 ) exists and is unique. Sometimes
we add more constraints to the geometry of our projection space saying that, additionally to G-measurability,dY0 = Y or that Y0 is a linear combination of observables. Apparently, the above solution Z, its existence
and uniqueness may heavily depend on the choice of measure m(·). A natural candidate for this would be
the distribution of Y . We will show that Em is a metric on the space of random variables whenever the
distribution function of m(·) is strictly increasing.
The intuition behind the use of the new metric is the following. Assume that a stationary heavy–tailed
time series {Yt , t ∈ R} is observed at locations t1 , . . . , tn in a compact window W ⊂ R. As proposed in [1],nthe linear predictor Yt =λj Yt , t ∈ {t1 , . . . , tn }, is a minimizer of the functional
j

j=1


R

 

E v1 AY (u)ΔAY (u) m(du) =Em (Yt , Yt ) dt

Wwith respect to the choice of weights λ1 , . . . , λn subject to the constraint Yt = Yt . The left hand side
term is the mean length of the symmetric diﬀerence of excursion sets AY (u) := {t ∈ W : Yt > u} andAY (u) := {t ∈ W : Yt > u} averaged over the levels u ∈ R picked up according to the measure m(·). Herev1 (·) is the Lebesgue measure on R.
Let U be a random variable with probability law m which is independent of Y1 , Y2 . It can be interpreted
as a random excursion level which we choose to build the metric Em . We show that Em coincides with theso–called separation (pseudo) metric [4] whenever m is a probability measure.
dLemma 1. Let m be a probability measure on R with c.d.f. FU (x) =xm(dy), x ∈ R. Then−∞

Em (Y1 , Y2 ) = E|FU (Y2 −) − FU (Y1 −)|.
∗ Institute of Stochastics, Ulm University, Germany; vitalii.makogin@uni-ulm.de† Institute of Stochastics, Ulm University, Germany; evgeny.spodarev@uni-ulm.de1

Theorem 1. Let XS be the space of random variables with support S ⊆ R. If FU is strictly increasing on
S, then Em is a metric on XS × XS .
Restricted to the space of random variables Y with the same absolutely continuous distribution F , themetric EF turns to be distribution–free depending only on bivariate copulas. We call this metric a Gini
metric. It properties are investigated.
Consider the prediction of a value of random variable X with continuous distribution function FX basedon the set Xn := (X1 , . . . , Xn ) of realizations of X via the excursion metric metric EFX . Namely, we propose
λ := g(λ, Xn ), where λ = (λ1 , . . . , λn ) ∈ Λ ⊂ Rn is deterministic and g : Rn × Λ → R, n ∈ N,a predictor X
is a continuous measurable function such that the excursion metric is minimal:
λ ) = arg min 2EFX (X ∨ Xλ ) − EFX (Xλ ) .
λ̂ := arg min EFX (X, X
λ∈Λ

λ∈Λ

Here, the set of 
admissible parameters Λ as well as the analytic form of g(λ, Xn ) depend on the law PX ,ne.g., g(λ, Xn ) = j=1 λj Xj for inﬁnitely divisible laws of X or g(λ, Xn ) = maxj=1,...,n λj Xj for max–stableddλ =λ ) =
X. Since g and FX are continuous, the constraint X
X is equivalent to FX (XU ∼ U (0, 1). Denoteλ ) = U }. Should our prediction be law-preserving, the above optimization problem
Λg := {λ ∈ Rn : FX (Xrewritesλ ) .
λ̂ := arg min EFX (X ∨ Xdλ∈ΛgIf the analytic form of Λ is given explicitly but Λg is hardly available, we modify the minimization
λ ) and U (0, 1):functional by adding a term which penalizes a diﬀerence between the law of FX (Xλ ) − EFX (Xλ ) + γ EF 2 (Xλ ) − E[FX (Xλ ) ∨ Y ]λ̂ := arg min 2EFX (X ∨ X
X
λ∈Λ

,

λ ), which allows for an equivalent reformulation
where γ > 0 and Y is an independent copy of FX (X 1λ ) − EFX (Xλ ) + γ
FY1 (y) [FY1 (y) − 2y] dy .
λ̂ := arg min 2EFX (X ∨ Xλ∈Λ0Existence of the solution and consistency of the above predictors are discussed as well. A special caseof extrapolating heavy-tailed time series is considered in detail. Numerical examples predicting Gaussian,
α–stable and autoregressive heavy–tailed stationary time series round up the talk. We explore several
advantages of our excursion predictors using theoretical results and computational studies. Namely, they arecomputationally fast, weakly consistent for stochastically continuous random ﬁelds, and work for randomﬁelds without ﬁnite moments. In addition, they are robust with respect to the statistical estimation of thec.d.f. FX . These results reveal a great potential for many real world applications.

References[1] A. Das, V. Makogin, and E. Spodarev. Extrapolation of stationary random ﬁelds via level sets. Theoryof Probability and Mathematical Statistics, 106:85–103, 2022.
[2] V. Makogin.
Prediction of random time series via the excursion metric.
R code, August 2022.
https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.110/forschung/
Software/ExtrapolationGradient3.R.
[3] V. Makogin and E. Spodarev. Prediction of random variables by excursion metric projections. Preprint,
arXiv:2207.00447v2, September 2022. https://doi.org/10.48550/arXiv.2207.00447.
[4] M. D. Taylor. Separation metrics for real-valued random variables. Internat. J. Math. Math. Sci.,
7(2):407–408, 1984.

2

Expected Euler characteristic heuristic for smooth Gaussian
random ﬁelds with inhomogeneous marginals
Satoshi Kuriki
The Institute of Statistical Mathematics
10-3 Midoricho, Tachikawa, Tokyo 190-8562, Japankuriki@ism.ac.jp
Abstract
Expected Euler characteristic (EC) heuristic is a method for approximating the tailprobability of the maximum of a Gaussian random ﬁeld. In this talk, we provide an expected Euler characteristic formula for the approximate tail probability and its relative
approximation error when the index set M is a closed manifold and the mean and varianceof the marginal distribution are not necessarily constant. When the variance is constant,[TTA05] proved that the relative approximation error is exponentially small in a general
setting where the index set M is a stratiﬁed manifold. When the variance is not constant,
it is shown that only the subset Msupp of M , referred to as the supporting index set,
contributes to the maximum tail probability. The proposed tail probability formula isan integral of the Euler characteristic density over Msupp , and its relative approximation
error is proven to be exponentially small as in the case of constant variance. These results
are generalizations of [KTT22], who addressed a restricted case of ﬁnite Karhunen-Loève
expansion by the volume-of-tube method. As an example, the tail probability formulafor the largest eigenvalues of noncentral Wishart matrices Wp (ν, Σ; Φ) and its relative approximation error are obtained. Numerical experience supports the high accuracy of the
expected Euler characteristic formulas regardless of whether the marginals are homogeneous or inhomogeneous.
Keywords: Borel’s inequality, Kac-Rice formula, noncentral Wishart distribution, volumeof-tube method, Weyl’s tube formula.

References[KTT22] Satoshi Kuriki, Akimichi Takemura, and Jonathan E. Taylor, The volume-oftube method for gaussian random ﬁelds with inhomogeneous variance, Journal ofMultivariate Analysis 188 (2022), 104819.
[TTA05] Jonathan E. Taylor, Akimichi Takemura, and Robert J. Adler, Validity of the
expected Euler characteristic heuristic, Ann. Probab. 33 (2005), no. 4, 1362–
1396.

Counterfactual Mean Embeddings
Motonobu Kanagawa
EURECOM, Sophia Antipolis, France
Joint work with
Krikamol Muandet (MPI Tuebingen), Sorawit Saengkyongam (U. Copenhagen),
Sanparith Marukatat (NECTEC)
To make a rational decision, a decision maker must be able to anticipate the eﬀects of a decision tothe outcomes of interest, before committing to that decision. For instance, before building a certain
facility in a city, e.g., a dam, policymakers and citizens must seek to understand its environmental
eﬀects. In medicine, a doctor has some prior knowledge about the eﬀects a certain drug will have on apatient’s health, before actually prescribing it. In business, a company needs to understand the eﬀects
of a certain strategy of advertisement to its revenue. One approach to addressing these questions iscounterfactual inference.
Counterfactual inference we consider in this work consists of the following three main ingredients.
Suppose that there exists a hypothetical subject (e.g., a patient in medical treatment), and let X
be covariates representing the features of the subject (e.g., age, weight, medical record, etc.), T be atreatment indicator representing the treatment assigned to the subject (a drug of interest or a placebo),
and Y be the observed outcome representing the post-treatment quantity of interest (e.g., whether thepatient is recovered or not). Given certain realizations of these variables {(xi , ti , yi )}ni=1 , in which eachindex i represents the identity of a subject, an analyst wishes to know how the treatment aﬀects theoutcome.
This problem is called counterfactual since for each subject i, we only observe the outcome yi
resulting from the assigned treatment ti and can never observe the outcome (say yi ) that would
have been realized under an alternative treatment ti = ti . For example, if a patient receives anactive treatment (e.g., a drug of interest), we can never observe the outcome from the same patientunder a control treatment (e.g., a placebo). This is known as the fundamental problem of causal
inference [9] and also as bandit feedback in the bandit literature [5]. One way to partially address
this issue is a randomized experiment [6], in which treatments are randomly assigned to subjects.
Although considered a gold standard, in practice randomization can be too expensive, time-consuming,or unethical. In most cases, therefore, analysis about treatment eﬀects needs to be done on the basisof observational data {(xi , ti , yi )}ni=1 in which the treatment assignment ti may depend on covariates
xi and possibly on some hidden confounders; this setting is commonly known as observational studies[14, 17].
A fundamental framework for observational studies is the potential outcome framework [13, 16].
It provides a clear notation for potential outcomes, i.e., the outcomes that would have been observed
under diﬀerent treatments, and elucidates the conditions required for making a valid inference about
treatment eﬀects. The framework has been studied extensively in statistics, and has a wide range
of applications in biomedical and social sciences; see, e.g., [10]. Moreover, important applications ofmachine learning such as oﬀ-policy evaluation for online advertisement and recommendation systems
can be reformulated under this framework [18, 11].
In this work, we propose a novel approach to counterfactual inference that addresses the above
challenges, which we term counterfactual mean embedding (CME). Our approach is built on kernelmean embedding [1, 19, 12], a framework for representing probability distributions as elements in a
reproducing kernel Hilbert space (RKHS), so that each element representing a distribution maintains
all of its information. We deﬁne an element representing a counterfactual distribution, for which we
propose a nonparametric estimator. Notable advantages of the proposed approach are summarized asfollows:
1. The proposed estimator can be computed based only on linear algebraic operations involving
kernel matrices. Being a kernel method, it can be applied to not only standard domains (suchas the Euclidean space), but also more complex and structured covariates and/or outcomes such1

as images, sequences, and graphs, by using oﬀ-the-shelf kernels designed for such data [7]; thiswidens possible applications of counterfactual inference in general. Thus our work oﬀers more
ﬂexibility than the existing approaches by [15] and [4], who focused on estimating the cumulative
distribution functions of counterfactual distributions by assuming real-valued outcomes.
2. The proposed estimator can be used for computing a distance between the counterfactual andcontrolled distributions, thereby providing a way of quantifying the eﬀect of a treatment to the
distribution of outcomes; we deﬁne this distance as the maximum mean discrepancy (MMD)[2, 8] between the counterfactual and controlled distributions. It also provides a way to sample
points from a counterfactual distribution based on kernel herding [3], a kernel-based deterministic
sampling method.
3. The proposed estimator is nonparametric, and has theoretical guarantees. Speciﬁcally, we provethe consistency of the proposed estimator under a very mild condition, and derive its convergence
rates under certain regularity assumptions involving kernels and underlying distributions. Bothresults hold without assuming any parametric assumption.

References[1] Alain Berlinet and Christine Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer
Academic Publishers, 2004.
[2] Karsten Borgwardt, Arthur Gretton, Malte Rasch, Hans-Peter Kriegel, Bernhard Schölkopf, and Alex Smola. Integrating
structured biological data by kernel maximum mean discrepancy. Bioinformatics, 22(14):49–57, 2006.
[3] Y. Chen, M. Welling, and A. Smola. Super samples from kernel-herding. In Proceedings of the 26th Conference on
Uncertainty in Artiﬁcial Intelligence, pages 109–116. AUAI Press, 2010.
[4] Victor Chernozhukov, Iván Fernández-Val, and Blaise Melly. Inference on counterfactual distributions. Econometrica,
81(6):2205–2268, 2013.
[5] Miroslav Dudík, John Langford, and Lihong Li. Doubly Robust Policy Evaluation and Learning. In Proceedings of the
28th International Conference on Machine Learning, pages 1097–1104. Omnipress, 2011.
[6] R.A. Fisher. The Design of Experiments. Oliver and Boyd, 1935.
[7] Thomas Gärtner. A survey of kernels for structured data. SIGKDD Explor. Newsl., 5(1):49–58, 2003.
[8] Arthur Gretton, Karsten Borgwardt, Malte Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test.
Journal of Machine Learning Research, 13:723–773, 2012.
[9] Paul Holland. Statistics and causal inference. Journal of the American Statistical Association, 81(396):945–960, 1986.
[10] Guido W. Imbens and Donald B. Rubin. Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction. Cambridge University Press, New York, NY, USA, 2015.
[11] Nathan Kallus and Angela Zhou. Policy evaluation and optimization with continuous treatments. In International Conference on Artiﬁcial Intelligence and Statistics (AISTATS), pages 1243–1251, 2018.
[12] Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, and Bernhard Schölkopf. Kernel mean embedding of distributions: A review and beyond. Foundations and Trends in Machine Learning, 10(1-2):1–141, 2017.
[13] Jerzy Neyman. Sur les applications de la theorie des probabilites aux experiences agricoles: Essai des principes. Master’s
thesis, 7 1923. Excerpts reprinted in English, Statistical Science, Vol. 5, pp. 463–472. (D. M. Dabrowska, and T. P. Speed,
Translators.).
[14] Paul Rosenbaum. Observational Studies. Springer Series in Statistics. Springer-Verlag, New York, 2nd edition, 2002.
[15] C. Rothe. Nonparametric estimation of distributional policy eﬀects. Journal of Econometrics, 155:56–70, 2010.
[16] D.B. Rubin. Estimating causal eﬀects of treatments in randomized and nonrandomized studies. Journal of Educational
Psychology, 66(5):688–701, 1974.
[17] Donald Rubin. Causal inference using potential outcomes. Journal of the American Statistical Association, 100(469):322–
331, 2005.
[18] Tobias Schnabel, Adith Swaminathan, Ashudeep Singh, Navin Chandak, and Thorsten Joachims. Recommendations as
treatments: Debiasing learning and evaluation. In Proceedings of The 33rd International Conference on Machine Learning, volume 48, pages 1670–1679, 2016.
[19] Alexander J. Smola, Arthur Gretton, Le Song, and Bernhard Schölkopf. A Hilbert space embedding for distributions. In
Proceedings of the 18th International Conference on Algorithmic Learning Theory (ALT), pages 13–31. Springer-Verlag,
2007.

2

Central Limit Theorems in High-Dimensions:
Recent Developments
Yuta Koike
The University of Tokyo and CREST JST

Let X1 , . . . , Xn be independent random vectors in Rd with mean 0 and covariance matrix Σ. Set
1 
Sn = √
Xi .
n
n

i=1

It is well-known by the central limit theorem (CLT) that the law of Sn can be approximated by N(0, Σ),the d-dimensional normal distribution with mean 0 and covariance matrix Σ, under some regularity assumptions, particularly when d is fixed. This approximation is ubiquitous in statistics because it gives a basisof asymptotically valid construction of confidence intervals and hypothesis testing. Recently, motivatedby the growing interest in high-dimensional data analysis, it is of great interest to develop CLTs for Sn inhigh-dimensional settings such as d = dn tends to infinity as n → ∞.
For statistical application, we need the following convergence for a sufficiently rich class A of Borel sets
in Rd :
ρn (A) := sup |P(Sn ∈ A) − P(Z ∈ A)| → 0,A∈A

where Z ∼ N(0, Σ). If A = Cd is the class of convex Borel sets in Rd , it is known that such convergence isgenerally impossible unless d = o(n). In the pathbreaking work of Chernozhukov et al. (2013, 2017), they
showed that one can prove ρn (A) → 0 under mild moment assumptions even when d  n if A = Rd is
the class of rectangles in Rd . Moreover, the class Rd is sufficiently rich in the sense that the convergenceρn (Rd ) → 0 justifies many uniform inference procedures; see Chernozhukov et al. (2013), Belloni et al.
(2018) and Chernozhukov et al. (2022a).
In the original work of Chernozhukov et al. (2017), the authors have shown that
1/6 
log7 (dn)ρn (Rd ) = Onunder some moment assumptions, and this convergence rate was conjectured to be optimal with respect tothe dependence on n. However, the recent work of Chernozhukov et al. (2022b) has showed that this rate
can be improved toρn (Rd ) = O
1

log5 (dn)n1/4 
(1)

under the same assumptions. This improvement poses the question of whether further improvement of thisrate is possible, especially whether we can achieve the Berry–Esseen rate n−1/2 while keeping the poly-log
dependence on d. In this talk, we will review the recent progress in this problem.
The convergence rate (1) is obtained without assuming the invertibility of Σ, and it is known that a
Berry–Esseen type bound is generally not achievable when Σ is degenerate (cf. Senatov (1986)). Therefore,
it is natural to assume the invertibility of Σ in order to improve (1). In this direction, Chernozhukov et al.
(2021) have shown that⎛ρn (Rd ) = O ⎝

3

⎞

log d
log n⎠nwhen the smallest eigenvalue of Σ is lower bounded and coordinates of Xi are uniformly bounded. It is
known that the rate

log3 dn gives a lower bound for the minimax optimal convergence rate for ρn (Rd ). In

the meantime, the recent work of Fang & Koike (2022) found that, if X1 , . . . , Xn are i.i.d. and log-concave,the rate n−1/2 is achievable while keeping the poly-log dependence on d without any restriction on Σ. To
be precise, they have shown that⎛ρn (Rd ) = O ⎝ψd⎞log3 (dn) ⎠,nwhere ψd is the so-called KLS constant, which is known to be (at most) of order O(log3.3336 d) and conjectured to be bounded by a universal constant (see Jambulapati et al. (2022)).
References
Belloni, A., Chernozhukov, V., Chetverikov, D., Hansen, C. & Kato, K. (2018). High-dimensional econometrics andregularized GMM. Working paper. arXiv: 1806.01888.
Chernozhukov, V., Chetverikov, D. & Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima
of sums of high-dimensional random vectors. Ann. Statist. 41, 2786–2819.
Chernozhukov, V., Chetverikov, D. & Kato, K. (2017). Central limit theorems and bootstrap in high dimensions. Ann.
Probab. 45, 2309–2353.
Chernozhukov, V., Chetverikov, D., Kato, K. & Koike, Y. (2022a). High-dimensional data bootstrap. Annu. Rev. Stat.
Appl. (forthcoming) .
Chernozhukov, V., Chetverikov, D., Kato, K. & Koike, Y. (2022b). Improved central limit theorem and bootstrap
approximation in high dimensions. Ann. Statist. (forthcoming) .
Chernozhukov, V., Chetverikov, D. & Koike, Y. (2021). Nearly optimal central limit theorem and bootstrap approximations in high dimensions. Ann. Appl. Probab. (forthcoming) .
Fang, X. & Koike, Y. (2022). Sharp high-dimensional central limit theorems for log-concave distributions. Working
paper. Available at arXiv: ?iiTb,ff`tBpXQ`;f#bfkkydXR98je.
Jambulapati, A., Lee, Y. T. & Vempala, S. S. (2022). A slightly improved bound for the KLS constant. Working paper.
Available at arXiv: ?iiTb,ff`tBpXQ`;f#bfkky3XRRe99.
Senatov, V. V. (1986). Four examples of lower estimates in the multi-dimensional central limit theorem. Theory Probab.
Appl. 30, 797–805.

2

A Modelling Framework for Regression with Collinearity
Takeaki KARIYA , Nagoya University of Commerce and Business,
The corresponding author: thekariya70@gmail.com
Hiroshi KURATA2 University of Tokyo, Graduate School of Arts and Sciences,
Takaki HAYASHI3 Keio University, Graduate School of Business Administration,
This study addresses a fundamental, yet overlooked, gap between the standard theory
and empirical practices in the OLS regression . To fill it, introducing a new concept“accommodation”, this paper formulates a novel conceptual framework for developing
our own model selection process in empirical modelling for given (X) with collinearity
in X. With no use of , the new process enables us to find a class of effective andcollinearity-resilient models. In fact, it directly controls not only the sampling varianceof each OLSE, which includes Variance Inflation Factor, but also the individual powerproperty of each t-test on regression coefficient, which includes what we call “Power
Deflation Factor” as a collinearity factor. And to materialize our model selection
process, two computational algorithms are proposed.
Consequently, it will provide an advance model-screening process and serve as anempirical platform for pre-selecting a class of effective models that well accommodate
with both collinearity and inefficiency controlled in advance. In such a class of models
we can freely use such statistical measures and procedures with use of as OLS
estimation, t-value, coefficient of determination, stepwise model selection, etc.
More specifically, within the traditional OLS (Ordinary Least Squares) framework in
the linear regression model,(1.1)
with and ,we aim to formulate a conceptual framework for developing our model selection
process, called XMOSEP, by connecting the OLS sampling theory with methodology of
empirically effective modelling. Here, is assumed to be given, contain all possible
variables for analysis and to be used as they are. As its vast literature shows, itself does
not have a capacity to identify the collinearity structure of and avoid the ill-effect even
if is really generated from it, though hybrid remedies have been provided. Our
XMOSEP stands for X-based model selection process and with no use of , it enables us
to obtain a class of empirically effective and collinearity-resilient models, which iscalled “ -accommodating class of models”, where y here simply denotes “dependent
variable” symbolically. In the class y will come into its own capacity to enableempirically effective and reliable modelling via frequently used variable and model
selection procedures (VMOSEPs) that use .
These VMOSEPs include such measures and procedures as those using t-value, Fvalue, adjusted coefficient of determination and so on. In fact, without controllingstrong collinearity in advance, those procedures will not lead us to empirically effectivemodels in practice within the OLS framework. While, whether or not is used in themodel selection, VMOSEPs and XMOSEP, which are called commonly MOSEPs, areregarded as a process of replacing in (1.1) by a “better” submatrix where data (X) is
given. Our XMOSEP connects the sampling variances of individual OLSEs inestimation and the power performances of individual t-tests with a practical process of1

making empirically effective modelling, so that standard errors of estimates and powerof t-tests are controlled against collinearity and inefficiency, where inefficiency isimplied by large individual sampling variances (IndSVs).
In the OLS framework, as is well known, if the initial model in (1.1) is supposedly“true” with rank(X) =K,
is the best linear unbiased estimator (BLUE) in thenonnegative definite ordering, in which no shrinkage-type estimator will beat the OLS
estimator in risk matrix. This optimality holds no matter how strong the collinearity in X
may be, implying no collinearity effect on this basic optimality of . However, in
empirical modelling, it is often the case that a final model is selected via a MOSEP
together with various diagnostics including t- or F- tests and variance analysis. Then is
not only different from X (1.1) but also it may depend on analyst’s view. The final
OLS model thus selected faces a conflict in claiming its BLUE-ness and effectivenessof the diagnostic results, because is not X. In Section 3, this problem will be overcome
by defining the model in (1.1) as a “bundle model”. After all, a finally selected model
via a MOSEP will have to be regarded as the model having generated y in any empiricalanalysis, so long as y is regressed on .
Key words: OLS, model selection process, collinearity effect, empirically modelling, ttest
Kariya and Hayashi’ s portion of this work was supported by JSPS KAKENHI Grant
Number 21K01431. Kurata’s portion is partially supported by JSPS KAKENHI Grant
Number 19K11853.

2

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SPATIAL REGRESSION DISCONTINUITY DESIGNS12

Takuya Ishiharaa , Daisuke Kurisub , Yasumasa Matsudaa and Masayuki Sawadac
We propose a mean squared error optimal estimator for regression discontinuity (RD) designs
with a vector of running variables. For the estimator, we show the asymptotic normality of amulti-dimensional local polynomial estimator with dimension-speciﬁc bandwidths. We illustrate
our results for two-dimensional cases. Consider a non-parametric regression model
Yi = m(Xi ) + i , E[ i |Xi ] = 0, i = 1, . . . , n, Yi ∈ R and Xi = (Xi,1 , Xi,2 ) ∈ R2
for a sequence of i.i.d. random vectors {(Yi , Xi )}ni=1 . A pair of running variables Xi = (Xi1 , Xi2 )
is a geographical location on a map. The location Xi determines treatment Di . Speciﬁcally, Di =1{Xi ∈ T } where T is a subset of the support of Xi .
Let c be a particular point on the boundary of T , our target parameter is θ := limx→c,x∈T m(x)−limx→c,x∈T C m(x). Under appropriate conditions (Hahn et al., 2001, Keele and Titiunik, 2015, forexample), θ is the average treatment eﬀect for the units at the boundary point c.
Local linear estimation such as Calonico et al. (2014) is dominant for RD designs because of its
intuitive construction from the identiﬁcation strategy that compares units around the boundary.
However, existing local linear estimators are limited to a uni-variate running variable, and multidimensional case is available only in a non-kernel procedure such as Imbens and Wager (2019).
Consequently, empirical practices often imposes that m(·) is a function of the Euclidean distancefrom a point or the boundary. We provide a local linear estimator without such a strong restriction.
Consider the local linear estimator β̂ + = (β̂0+ , β̂1+ , β̂2+ ) solving the following problemβ̂ + = arg min(β0 ,...,β2 ) ∈R3

where Kh (Xi − c) = Kn(Yi − β0 − β1 (Xi,1 − c1 ) − β2 (Xi,2 − c2 ))2 Kh (Xi − c) 1{Xi ∈ T }i=1
Xi,1 −c1 Xi,2 −c2
, h2
h1


and each hj is a sequence of positive constants (band-

widths) such that hj → 0 as n → ∞. Similarly, let β̂ − be the estimator using 1{Xi ∈ T c }1
2

Prepared for 3rd TIU workshop on 2022 October 12 through 14.
Graduate School of Economics and Management, Tohoku Universityb
Graduate School of International Social Sciences, Yokohama National UniversitycInstitute of Economic Research, Hitotsubashi Universitya2
subsample. Our spatial RD estimator is β̂0+ − β̂0− .
Unlike Masry (1996) who study the asymptotic property of the local polynomial estimator witha common bandwidth h1 = h2 , we show the asymptotic normality of the local polynomial estimator with heterogeneous bandwidth. Furthermore, following Imbens and Kalyanaraman (2012), we
propose a procedure to choose the pair of bandwidths h1 = h2 by minimizing the asymptotic meansquared error at the cutoﬀ normalized to c = 0 without loss of generality.
+−+−− ∂11 m− (0)K11) + h22 (∂22 m+ (0)K22− ∂22 m− (0)K22){h21 (∂11 m+ (0)K11+−− ∂12 m− (0)K12)}2 ++ h1 h2 (∂12 m+ (0)K12

1 σ 2 (0)  −1 −1e S KS e1nh1 h2 f (0) 1

where e1 = (1, 0, 0) , ∂lk m+ (0) = limx→0,x∈T ∂lk m(x), ∂lk m− (0) = limx→0,x∈T C ∂lk m(x), S, K and
+
−
+
−
+
−
, K11
, K12
, K12
, K22
and K22are constants determined by the kernel K(·), σ 2 (0) is the conditional
K11

variance of the error term at the cutoﬀ, and f (0) is the density at the cutoﬀ.
However, for a particular shape of m, there exists a pair (h1 , h2 ) such that the asymptotic
bias is zero, and consequently the bandwidths cannot be determined. For a product kernel K(x) =K1 (x1 )K2 (x2 ), we resolve this issue by rotating the axis so that K2 becomes the one-sided triangularkernel K2 (x) = 2(1 − x)1{0 ≤ x ≤ 1}. In Monte Carlo simulation, we demonstrate that our spatial
RD estimator outperforms the conventional RD estimator with the shape restriction. We apply ourprocedure to Ehrlich and Seidel (2018) data for a spatial RD evaluation of a place-based policy.
REFERENCES
Calonico, S., M. D. Cattaneo, and R. Titiunik (2014): “Robust Nonparametric Conﬁdence Intervals for
Regression-Discontinuity Designs,” Econometrica, 82, 2295–2326.
Ehrlich, M. v. and T. Seidel (2018): “The Persistent Eﬀects of Place-Based Policy: Evidence from the WestGerman Zonenrandgebiet,” American Economic Journal: Economic Policy, 10, 344–374.
Hahn, J., P. Todd, and W. V. d. Klaauw (2001): “Identiﬁcation and Estimation of Treatment Eﬀects with aRegression-Discontinuity Design,” Econometrica, 69, 201–209.
Imbens, G. and K. Kalyanaraman (2012): “Optimal Bandwidth Choice for the Regression Discontinuity Estimator,” The Review of Economic Studies, 79, 933–959.
Imbens, G. and S. Wager (2019): “Optimized Regression Discontinuity Designs,” The Review of Economics andStatistics, 101, 264–278.
Keele, L. J. and R. Titiunik (2015): “Geographic Boundaries as Regression Discontinuities,” Political Analysis,
23, 127–155.
Masry, E. (1996): “Multivariate regression estimation local polynomial ﬁtting for time series,” Stochastic Processes
and their Applications, 65, 81–101.

Shrinkage Methods for Treatment Choice
Takuya Ishihara

Daisuke Kurisu

This paper studies the problem of determining whether or not to treat individuals based on
observed covariates. In this problem, the most common decision rule is the conditional empirical
success (CES) rule proposed by Manski (2004), which is a rule assigning individuals to treatmentsthat yield the best experimental outcomes conditional on observed covariates. The CES rule uses
only the average treatment effect (ATE) estimate conditional on each covariate value. In contrast,
a common method in statistical estimation problem is to shrink unbiased but noisy preliminary
estimates toward the average of these estimates and it is well known that shrinkage estimators
have smaller mean squared error than unshrunk estimators. In this study, we assume that thedispersion of the conditional ATEs (CATEs) is bounded and propose the shrinkage rule that is arule assigning individuals to treatments based on shrinkage estimators. We select the shrinkage
factor by minimizing an upper bound of the maximum regret. By considering the treatment rule
for individuals that uses not only each CATE but also the CATEs of other individuals, it is possible
to incorporate information across individuals. This allows the proposed shrinkage rule to perform
as well as or better than existing treatment rules in the sense of regret and to be more flexible tothe heterogeneity of the individuals. In addition, we compare the shrinkage rule with other rules
when the parameter space is correctly specified and misspecified.
The contributions of this paper are threefold. First, our approach is attractive from acomputational point of view. Computation of the exact minimax regret rule often becomeschallenging in the context of statistical treatment choice. Indeed, when the parameter space isrestricted and the number of the possible covariate values is large, it is difficult to obtain theshrinkage rule that minimizes the maximum regret. To overcome this problem, we propose theshrinkage rule that minimizes tractable upper bound of the maximum regret. Because in thisapproach, each shrinkage factor is obtained by optimizing over a single parameter, the proposed
shrinkage rule is easy to compute.
Second, we compare the maximum regrets of the shrinkage, CES, and pooling rules when theparameter space is correctly specified. As an alternative to the CES and shrinkage rules, one could
consider using the pooling rule that determines whether or not to treat the individuals based on
the average of the CATE estimates. Because the CES and pooling rules are special cases ofshrinkage rules, the proposed shrinkage rule is expected to outperform these two rules. However,
because the proposed shrinkage rule does not minimize the exact maximum regret, its maximum

regret may be larger than that of the CES and pooling rules. Hence, it is important to compare
these maximum regrets. If the dispersion of the standard errors of estimated CATEs is small
compared to the dispersion of the CATEs, then the proposed shrinkage rule has a smaller
maximum regret than the CES rule. We also show that the maximum regret of the shrinkage ruleis always less than twice that of the pooling rule. Furthermore, when the dispersion of CATEs issufficiently small or large, the shrinkage rule is no worse than the pooling rule.
Third, we evaluate the maximum regret of the shrinkage rule when the parameter space is
misspecified. Because the minimax decision rule depends on the parameter space, the choice ofthe parameter space is important in practice. For example, Armstrong and Kolesar (2018) andArmstrong and Kolesar (2021) consider the minimax estimation and inference problem for thetreatment effects and show that it is not possible to choose the parameter space automatically ina data-driven way. Hence, it is important to analyze the decision rule under misspecification of
the parameter space. We investigate the performance of the shrinkage rule and show that our
results are robust against misspecification of the parameter space. To the best of our knowledge,
this is the first study that considers misspecification of the parameter space in the treatment choiceproblem.

Referencesz

Armstrong, Timothy B., and Michal Kolesár. "Optimal inference in a class of regression
models." Econometrica 86.2 (2018): 655-683.

z

Armstrong, Timothy B., and Michal Kolesár. "Finite‐Sample Optimal Estimation and Inference on
Average Treatment Effects Under Unconfoundedness." Econometrica 89.3 (2021): 1141-1177.

z

Manski, Charles F. "Statistical treatment rules for heterogeneous populations.”(2004): 1221-1246.

Econometrica 72.4

Forecasting Stock Returns with Conditional
Quantile-level Dependence
Stanley Iat-Meng Ko∗

Sung Y. Park†

Finance practitioners and researchers have long attempted to forecast the equity premium, which is essential for asset pricing, portfolio construction and so on. Rozeﬀ (1984),
Fama and French (1988) and Campbell and Shiller (1988a) formally study the predictabilityof the dividend-price ratio for stock returns. Since then, various predictors have been developed and studied; for example, the dividend-earnings ratio (Campbell and Shiller, 1988b;
Lamont, 1998), yields on treasury and corporate bonds (Keim and Stambaugh, 1986; Campbell, 1987; Fama and French, 1989), and book-to-market ratio (Kothari and Shanken, 1997;
Pontiﬀ and Schall, 1998), among many others. Welch and Goyal (2008) comprehensively
summarizes various stock return predictors and extensively examines their forecasting performance. They conclude that most predictors forecast poorly both in-sample and out-ofsample. Moreover, the prevailing mean model (average historical return) simply provides
better forecasts; see also Butler, Grullon and Weston (2005). Recent studies have utilizedﬁnancial economic theory to improve the forecastability of predictors. Using the same setof predictors, Campbell and Thompson (2008) and Pettenuzzo, Timmermann and Valkanov (2014) propose a constrained forecasting method in which the restrictions on mean andvariance suggested by investment theory are imposed.
In this paper, we propose a novel approach to forecasting stock excess returns using
conditional quantile levels. We deﬁne the “optimal forecasting quantile level” as the corresponding conditional quantile value that is the closest to the true yt . To illustrate theidea, we consider a hypothetical time series and its conditional quantiles at t = 1, . . . , 8 in
Figure 1. For illustrative purposes, we only consider ﬁve quantiles, at 0.05, 0.25, 0.5, 0.75,
and 0.95. The red circles denote the quantiles of returns at diﬀerent times. The realized
returns are plotted as triangles at each time. The quantile levels with blue dots are theclosest to the true yt , which we call the “optimal forecasting quantile level”, α̃t . In thisexample, the optimal forecasting quantile-level sequence or trajectory is thus 0.5 → 0.95 →0.25 → 0.75 → 0.75 → 0.05 → 0.5 → 0.75.
∗
Graduate School of Economics and Management, Tohoku University, Sendai, Japan. Tel: +81 022-7956288. Email: stanleyko@tohoku.ac.jp.
†
School of Economics, Chung-Ang University, Seoul, Korea.
Tel: +82-2-820-5622.
Email:
sungpark@cau.ac.kr.

1

1.0

Figure 1: Hypothetical optimal forecasting quantile levels.

q_0.95q_0.95q_0.95q_0.95

q_0.75
q_0.500.5

q_0.75
q_0.75Returnq_0.25q_0.95q_0.95q_0.95q_0.25

q_0.50q_0.50q_0.75q_0.05
q_0.75q_0.05q_0.25
q_0.75

q_0.50q_0.95

q_0.75

q_0.50q_0.75

q_0.500.0

q_0.50q_0.05

q_0.50q_0.25q_0.25q_0.25q_0.25q_0.05q_0.25q_0.05q_0.05q_0.05−0.5

q_0.05

1

2

3

4

5

6

7

8

Time

We discover an interesting phenomenon that the α̃t sequence forms a stationary timeseries with stable autocorrelation. Speciﬁcally, there is signiﬁcant lag-5 autocorrelation inα̃t of the monthly equity premium of S&P 500, and signiﬁcant lag-7 autocorrelation in thequarterly data. However, the autocorrelation of α̃t disappears in the annual data. This
provides us a way to forecast excess return by exploring the time series property of the
optimal forecasting quantile levels. Typically, we ﬁrst forecast the future optimal forecasting quantile level, based on the past optimal levels; for example, for the monthly return,ˆ t = γ̂0 + γ̂1 α̃t−5 . Then we forecast the excess return using quantile regression at α̃ˆ t ; that is,α̃ŷt|t−1 ≡ q̂α̃ˆ t ,t+1|t . We show that a time varying risk premium adjustment is implied by our
method. In the empirical study, we show that the proposed two-step approach outperforms
other methods in forecasting the equity premium. Our empirical results demonstrate thesuperiority of our method for forecasting the monthly and quarterly equity premium. Using
the conventional out-of-sample R2 performance measure, we show extraordinary robust predictive power for our method in diﬀerent sample periods. Our method also achieves better
prediction performance under the alternative absolute loss measure.

2

3rd Tohoku-ISM-UUlm workshop

Issues in Spatial Processes with Long Range DependencePeter M Robinson

London School of Economics

Abstract

We discuss issues of statistical inference for spatial processes with 'long range dependence'. Long
range dependence, or strong dependence, in time series is a topic that has been quite extensively
studied in recent years. After a number of probabilistic contributions and empirical studies, serious
treatment of issues of statistical inference could be said to have begun in the mid-1980s, withactivity then increasing through the 1990s and this century. Predominately this literature has
focussed on observations that are regularly-spaced over time, and the bulk of the theoreticaldevelopment has been in terms of asymptotic statistical theory, with the number of observations
regarded as diverging, finite-sample theory proving mathematically intractable, even under the
precise distributional assumptions that are typically not required in a large-sample treatment.
Parametric, semiparametric and nonparametric models have all featured. The major
characteristic feature of a long range dependent covariance stationary time series process is that its
autocovariance function decays so slowly with increase in lag length as not to be summable, or,nearly equivalently, its spectral density diverges, typically at zero frequency, while some
nonstationary processes, such as ones with a unit root can, a fortiori, be regarded as having evenlonger memory. By contrast, 'short range dependent' time series typically have autocovariances thatare summable and spectral density that is more or less smooth (for example a stationary
autoregressive moving average (ARMA) has exponentially decaying autocovariance and analytic
spectral density), though for some relevant purposes a short range dependent process is sometimes
defined as merely having spectral density that is positive and finite at zero frequency.
Spatial data have long attracted the attention of statisticians, and the configurations of some suchdata, especially some arising in such fields as meteorology, cosmology and agriculture, can beviewed as generalisations of the typical regularly-spaced time series one mentioned above. In
particular, they constitute observations in M≥2 dimensions, (where M=1 in the time series case).
This brings to mind regularly spaced agricultural plots on a field. In fact, `spatial long range
dependence' goes back at least to the agriculturally-motivated paper of Smith (1938), which is also avery early reference relative to the literature on time series long range dependence. It is interestingthat Smith (1938) thought of a power law decay: this might seem natural to one coming to thesubject unschooled in time series modelling which, after World War II, stressed exponential decay,
as in ARMA modelling.

Since then, many papers on 'spatial long range dependence' have appeared, but the topic has notbeen developed as systematically or comprehensively as 'long range dependent time series'. Somedistinctive issues arising in the 'spatial' case, all of which have been studied far more under shortrange dependence than long range dependence, are:
1. Is there isotropy or not? If not, we might model each dimension separately, or with someinteraction, and possibly with a different memory parameter for each dimension.
2. Is sampling regularly or irregularly spaced? Whereas in time series regular spacing has been
predominately studied, irregular spacing is perhaps more likely to be found with spatial data.
3. Should modelling be unilateral or multilateral? For time series unilateral modelling,reflecting one-sided transition from past to future, is usually natural, but this is not the case withspatial data, where, for example, the dimensions might be latitude and longitude.
4. The edge effect. In estimating lagged quantities there is loss of information at the boundaryof the observation region, which has negligible effect when M=1, but increasing, and damaging,effect as M increases unless corrected for.
We discuss the following topics:
1. Inference on location and regression with long range dependent errors.
2. Inference on second-order properties of long range dependent stationary processes.
3. Miscellaneous topics: nonstationary processes, irregular spacing, adaptive estimation,
nonparametric regression.
We do not consider 'spatial autoregressive'-type ('SAR') models (which depend on a user-chosenweight matrix of geographic or economic inverse distances); these do not fit into our framework andtypically possess a kind of short range dependence.

On estimation of
fractional Browninan ﬁelds and sheets
Yoshihiro Yajima
University of Tokyo(Emeritus Professor)
at Tohoku University Oct. 12-14
Abstract
Originally a Gaussian semiparametric estimator is an approximate likelihood estimator in a frequency domain
for long memory models of stationary and nonstationary time series(Robinson(1995); Velasco(1999)). We
apply it to fractional Brownian ﬁelds(FBF) and Brownian sheet(FBS) observed on a regular grid and derive
its asymptotic properties. They are Gaussian random ﬁelds and used to model many physical processes inspace. The estimator is consistent and has the limiting normal distribution as the sample size goes to inﬁnity.
Keywords:spatio-temporal models; fractional Brownian ﬁeld; fractional Brownian sheet.
1. Introduction FBF is a popular model for intrinsic stationary random ﬁelds(ISRFs). Let {X(s) : s ∈ Rd }be a random ﬁeld. If {X(s)} satisﬁes that for any ﬁxed h(∈ Rd ), the increment Zh (s) = X(s + h) − X(s)is a stationary random ﬁeld, {X(s)} is called an ISRF. Then {X(s)} is characterized by
E(X(s + h) − X(s))=

0,

Var(X(s + h) − X(s))=2γ(h),

where 2γ(h) is the variogram function(see Chilès & Delﬁner (2012); Cressie (1993)). Hereafter we also assume
that X(0) = 0.
For λ and h, let (λ, h) be the inner product and  λ  be the norm.
Then if 2γ(h) is a continuous function on Rd satisfying γ(0) = 0, it has the spectral representation1 − cos((λ, h))G(dλ) + Q(h),(1)2γ(h) =d(2π)dR
where Q(h)(≥ 0) is a quadratic form and G(λ) is a positive, symmetric measure such that  λ 2 G(λ) is
continuous at the origin and λ 2G(dλ) < ∞.
(2)
d 1+  λ 2R(See Chilès& Delﬁner (2012); Cressie(1993); Solo (1992); Yaglom (1957).)
Hereafter we assume that Q(h) ≡ 0 and G(λ) is absolutely continuous with density g(λ). Then (1) and (2)reduce to
1 − cos((λ, h))g(λ)dλ,2γ(h) =
d
(2π)d
R
and



 λ 2g(λ)dλ < ∞,2
Rd 1+  λ 

respectively.
An interesting special class of ISRF’s that is often applied to empirical data analysis in space is a fractional
Brownian ﬁeld(FBF)(see Adler (1981); Mandelbrot& Van Ness (1968);Yaglom (1957); Zhu& Stein (2002) andthe references therein). FBF has 2γ(h) = C  h 2H , which is equivalent tog(λ) =
where

CHKH
,
 λ d+2H

KH = π d/2 22H+d Γ((d + 2H)/2)/Γ(1 − H), 0 < H < 1,1and C is a scale parameter and H is a smoothness parameter with larger values corresponding to smoothersurfaces.
The other popular random ﬁeld is FBS. FBS is a Gaussian random ﬁeld and its covariance is deﬁned by
Cov(X(s), X(t)) = C

d


Hi
Hi
i
(sH
i + ti − |si − ti | ), 0 < Hi < 1.

i=1




where s = (s1 , · · · , sd ) and t = (t1 , · · · , td ) .
2.Theoretical results Hereafter for simplicity of calculation and notation we assume that d = 2. For FBF,
we also denote λ by (λ1 , λ2 ) and g(λ) by g(λ1 , λ2 ) respectively.
First we consider FBF.
Assumption 1 g(λ1 , λ2 ) is expressed byg(λ1 , λ2 ) = λ −2H−2 go (λ1 , λ2 ),0 < H < 1,

where go (λ1 , λ2 ) is a nonnegative with go (0, 0) > 0, symmetric , go (λ1 , λ2 ) = go (−λ1 , −λ2 ), twice continuously
diﬀerentiable function for −∞ < λ1 , λ2 < ∞ and is bounded with bounded ﬁrst and second order partial
derivatives.
Theorem 1 Under Assumption 1 and additional conditions, GSE Ĥn converges to H0 (the true parameter)in probability as n → ∞ and for p ≥ 2, m1/2 (Ĥn − H0 ) converges to N (0, 1) in distribution as n → ∞ wherem is the number of the discrete Fourier transforms used to construct the estimator and p is the order of thedata taper..
Next we have the following result for FBS.
Theorem 2 Assume p ≥ 3. Under some assumptions, GSE Ĥin (i = 1, 2)converges to Hi0 (the true parameters) in probability as n → ∞ and m(Ĥ1n − H10 , Ĥ2n − H20 ) converges to N (0, I2 ) in distribution as n → ∞where m2 is the number of the discrete Fourier transforms used to construct the estimator.
References
Adler,R.J. (1981). The Geometry of Random Fields.Wiley, New York.
Chilès, J.-P. and Delﬁner,P. (2012). Geostatistics :Modeling Spatial Uncertainty. 2nd ed. Wiley, New York.
Cressie, N.A.C. (1993). Statistics for Spatial Data. Revised ed. Wiley, New York.
Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motion, fractal noises and applications.
SIAM. Rev. 10 422-437.
Robinson,P.M. (1995). Gaussian semiparametric estimation of long range dependence. Ann. Statist. 23
1630-1661.
Stein,M.L. (1999). Interpolation of Spatial Data:Some Theory for Kriging. Springer, New York.
Solo,V. (1992). Intrinsic stationary random functions and the paradox of 1/f noise. SIAM J. Appl. Math.
52 270-291.
Velasco,C. (1999). Gaussian semiparametric estimation of non-stationary time series. J. Time Ser. Anal.
20 87-127.
Yaglom,A.M. (1957). Some classes of random ﬁelds in n-dimensional space, related to stationary random
processes. Theory Probab. Appl. 2 273-320.
Zhu,Z. and Stein,M.L. (2002). Parametric estimation for fractional Brownian surfaces. Statist. Sinica12
863-883.

2

Joint circular distributions in view of higher order spectra of time series and copula
Masanobu Taniguchi* Waseda University,Tokyo,
Shogo Kato, Institute of Statistical Mathematics,Tokyo,
Hiroaki Ogata, Tokyo Metropolitan University,
Arthur Pewsey, University of Extremadura, Spain
Abstract
Circular data analysis is emerging as an important component of statistics. For thishalf century, various circular distributions have been proposed, e.g., von Mises
distribution, wrapped Cauchy distribution, among other things.
Also, regarding the joint distribution, Wehrly and Johnson(1980) proposed a bivariate
circular distribution which is related to a family of Markov processes on the circle.
Because the sample space is on a circle, various new statistical methods have beendeveloped. In this talk we provide a new look at circular distributions in view ofspectral distributions of time series because the typical circular distributions
correspond to spectral densities of time series models. For example, autoregressive
AR(1) spectral density corresponds to wrapped Cauchy distribution, and von Mises
distribution corresponds to exponential spectral density (Bloomfield(1973)), etc.
Furthermore we introduce a class of joint circular distributions from the higher order
spectra of time series, which can describe very general joint circular distributions.
Hence we can develop the statistical inference for dependent observations on thecircle. We present a family of distributions on the circle derived from the ARMA
spectral density. It is seen that the proposed family includes some existing circularfamilies as special cases. For these special cases, the normalizing constant andtrigonometric moments are shown to have simple and closed form. We develop theasymptotic optimal inference theory based on the local asymptotic normality (LAN)on the circle. Because the observations are permitted to be dependent, the theory
opens a new paradigm in the estimation for joint circular distributions. Because weintroduced very general joint circular distributions, we can discuss the problem of
copula for them.