discrete series representations for the orbit spaces arising from two involutions of real reductive lie groups leth gbe real reductive lie groups a discrete series representation for a homogeneous spaceg his an irreducible representation ofgrealized as a closedg invariant subspace ofl2 g h the condition for the existence of discrete series representations forg hwas not known in general except for reductive symmetric spaces this paper offers a sufficient condition for the existence of discrete series representations forg hin the setting thatg his a homogeneous submanifold of a symmetric space g hwhereg g h we prove that discrete series representations are non empty for a number of non symmetric homogeneous spaces such assp 2n r sp n0 c gl n1 c gl nk c nj n ando 4m n u 2m j 0 2j n

bayesian hypothesis testing in latent variable models hypothesis testing using bayes factors bfs is known not to be well defined under the improper prior in the context of latent variable models an additional problem with bfs is that they are difficult to compute in this paper a new bayesian method based on the decision theory and the em algorithm is introduced to test a point hypothesis in latent variable models the new statistic is a by product of the bayesian mcmc output and hence easy to compute it is shown that the new statistic is appropriately defined under improper priors because the method employs a continuous loss function in addition it is easy to interpret the method is illustrated using a one factor asset pricing model and a stochastic volatility model with jumps

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

discrete decomposability of the restriction of a_q λ with respect to reductive subgroups ii micro local analysis and asymptotic k support let g c g be real reductive lie groups this paper offers a criterion on the triplet g g ir that the irreducible unitary representation ir of g splits into a discrete sum of irreducible unitary representations of a subgroup g when restricted to g each of finite multiplicity furthermore we shall give an upper estimate of the multiplicity of an irreducible unitary representation of g occurring in wrig

efficient method of moments estimation of a stochastic volatility model a monte carlo study abstract we perform an extensive monte carlo study of efficient method of moments emm estimation of a stochastic volatility model emm uses the expectation under the structural model of the score from an auxiliary model as moment conditions we examine the sensitivity to the choice of auxiliary model using arch garch and egarch models for the score as well as nonparametric extensions emm efficiency approaches that of maximum likelihood for larger sample sizes inference is sensitive to the choice of auxiliary model in small samples but robust in larger samples specification tests and t tests show little size distortion

small sample tests for stock return predictability with possibly non stationary regressors and garch type effects we develop a simulation based procedure to test for stock return predictability with multiple regressors the process governing the regressors is left completely free and the test procedure remains valid in small samples even in the presence of non normalities and garch type effects in the stock returns

spectral multiplicity of derived functor modules abstract let be a discrete torsion free subgroup of a semisimple lie group g such that the volume of g is finite we obtain a cohomological interpretation in the sense of langlands of the multiplicity in the discrete spectrum of l 2 g of an irreducible unitarizable derived functor module r sqc after zuckerman with non singular infinitesimal character

signatures of invariant hermitian forms on irreducible highest weight modules perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of com position factors in a verma module this problem was settled by the proof of the kazhdan lusztig conjecture which states that the multiplicities may be computed via kazhdan lusztig polynomials in this paper we introduce signed kazhdan lusztig polynomials a variation of kazhdan lusztig polyno mials which encodes signature information in addition to composition factor multiplicities and jantzen filtration level careful consideration of gabber and joseph s proof of kazhdan and lusztig s recursive formula for computing kazhdan lusztig polynomials and an application of jantzen s determinant for mula lead to a recursive formula for the signed kazhdan lusztig polynomials we use these polynomials to compute the signature of an invariant hermitian form on an irreducible highest weight module such a formula has applications to unitarity testing 1 1 the unitary dual problem in the 1930s i m gelfand introduced a broad programme in abstract harmonic analysis which would permit the transfer of dif ficult problems in areas as distinct from analysis as topology to more tractable problems in algebra fourier analysis is just one incarnation of this programme an unresolved component in gelfand s programme is the classification of the irre ducible unitary representations of a group known as the unitary dual problem in the case of a real reductive lie group the problem is equivalent to identifying all irreducible harish chandra modules which admit a positive definite invariant hermitian form as harish chandra modules may be constructed via an algebraic method introduced by zuckerman in 1978 known as cohomological induction it is of interest to study signatures of invariant hermitian forms on cohomologically induced modules and to understand how positivity can fail cohomological induction is a two step process in which we compose an induction functor with a zuckerman functor i the intermediate module in cohomological induction is a generalized verma module which admits an invariant hermitian form if the module to which induction was applied admits an invariant hermitian form formulas for the signatures of invariant hermitian forms on these intermediate modules may be used to compute signatures of forms on corresponding cohomo logically induced modules eg wal84 this motivates the study of invariant hermitian forms on verma modules see wal84 yee05 and irreducible highest weight modules

the langlands parameters of subquotients of certain derived functor modules letgbe a noncompact simple lie group with finite center letkbe a maximal compact subgroup and let g0 k0 p0be the corresponding decomposition of the lie algebra suppose rank g rank k and let t0be a compact cartan subalgebra and b be a borel subalgebra letab be a derived functor module with infinitesimal character which is nondominant with respect to a noncompact simple root suppose that 2 p iskdominant so that thektype with highest weight occurs with multiplicity one inab we develop conditions on the roots of g under which a functor of cohomological induction maps a certain module of parabolic induction to another module of parabolic induction extending a result due to vogan this allows us in many cases to determine the langlands parameters of the subquotient ofab containing via a conjectured method of knapp

an alternative form used to calibrate the heston option pricing model this paper presents an alternative form of the heston model that preserves an essential advantage of the heston model its analytic tractability by imposing the necessary and sufficient conditions for the existence of a solution in affine form while it is in a different form so that it offers certain advantages in parameter determination to demonstrate this we conducted some empirical studies exploring if this new form does have certain advantages over the original version under certain market conditions

markov chain monte carlo methods for stochastic volatility models this paper is concerned with simulation based inference in generalized models of stochastic volatility defined by heavy tailed student t distributions with unknown degrees of freedom and exogenous variables in the observation and volatility equations and a jump component in the observation equation by building on the work of kim shephard and chib rev econom stud 65 1998 361 we develop efficient markov chain monte carlo algorithms for estimating these models the paper also discusses how the likelihood function of these models can be computed by appropriate particle filter methods computation of the marginal likelihood by the method of chib j amer statist assoc 90 1995 1313 is also considered the methodology is extensively tested and validated on simulated data and then applied in detail to daily returns data on the s p 500 index where several stochastic volatility models are formally compared under different priors on the parameters

invariant syzygies and semisimple groups the purpose of this paper is to explore some intriguing connections between lie theory relative homological algebra invariant theory and representations of finite dimensional algebras in the process some results of hilbert 20 cartan and eilenberg 7 hochschild 21 gelfand 171 and others are generalized and a question of zl is partly answered in theorem 1 5 duflo seems to have stronger unpublished results on this problem sections 1 to 3 take place in a fairly general setting whereas section 4 focuses on semisimple lie groups the problem considered here goes back in some sense to hilbert 20 this paper was published 102 years ago to explain this let me start by singling out two out of many fundamental contributions of the great mathematician the first one is invariant theory namely the study of the algebra sv n consisting of those polynomial functions on a finite dimensional vector space v which are invariant under the action of a group h of linear transformations the second one is the so called syzygy theorem which asserts in today s language that the global dimension of the full symmetric algebra sv is dim v this prompts the question is there an algebra d attached to v as naturally as sv is on which h acts by automorphisms such that the global dimension of the algebra b dh of invariants is dim v the most natural candidate is

simulation run length control in the presence of an initial transient this paper studies the estimation of the steady state mean of an output sequence from a discrete event simulation it considers the problem of the automatic generation of a confidence interval of prespecified width when there is an initial transient present it explores a procedure based on schruben s brownian bridge model for the detection of nonstationarity and a spectral method for estimating the variance of the sample mean the procedure is evaluated empirically for a variety of output sequences the performance measures considered are bias confidence interval coverage mean confidence interval width mean run length and mean amount of deleted data if the output sequence contains a strong transient then inclusion of a test for stationarity in the run length control procedure results in point estimates with lower bias narrower confidence intervals and shorter run lengths than when no check for stationarity is performed if the output sequence contains no initial transient then the performance measures of the procedure with a stationarity test are only slightly degraded from those of the procedure without such a test if the run length is short relative to the extent of the initial transient the stationarity tests may not be powerful enough to detect the transient resulting in a procedure with unreliable point and interval estimates

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

estimating a semiparametric asymmetric stochastic volatility model with a dirichlet process mixture in this paper we extend the parametric asymmetric stochastic volatility model asv where returns are correlated with volatility by flexibly modeling the bivariate distribution of the return and volatility innovations nonparametrically its novelty is in modeling the joint conditional return volatility distribution with an infinite mixture of bivariate normal distributions with mean zero vectors but having unknown mixture weights and covariance matrices this semiparametric asv model nests stochastic volatility models whose innovations are distributed as either normal or student t distributions plus the response in volatility to unexpected return shocks is more general than the fixed asymmetric response with the asv model the unknown mixture parameters are modeled with a dirichlet process prior this prior ensures a parsimonious finite posterior mixture that best represents the distribution of the innovations and a straightforward sampler of the conditional posteriors we develop a bayesian markov chain monte carlo sampler to fully characterize the parametric and distributional uncertainty nested model comparisons and out of sample predictions with the cumulative marginal likelihoods and one day ahead predictive log bayes factors between the semiparametric and parametric versions of the asv model shows the semiparametric model projecting more accurate empirical market returns a major reason is how volatility responds to an unexpected market movement when the market is tranquil expected volatility reacts to a negative positive price shock by rising initially declining but then rising when the positive shock is large however when the market is volatile the degree of asymmetry and the size of the response in expected volatility is muted in other words when times are good no news is good news but when times are bad neither good nor bad news matters with regards to volatility

gift wrapping based preimage computation algorithm based on a classical convex hull algorithm called gift wrapping the purpose of the paper is to provide a new algorithm for computing the vertices of a polytope called preimage roughly the set of naive digital planes containing a finite subset s of z 3 the vertices of the upper hemisphere the ones of the lower hemisphere and at last the equatorial vertices are computed independently the principle of the algorithm is based on duality and especially on the fact that the vertices of the preimage correspond to faces of the input set s or of its chords set email protected email protected 0 0 1 it allows to go from one vertex to another by gift wrapping until the whole region of interest has been explored

a bayesian robust chi squared test for testing simple hypotheses abstract in this paper we introduce a new bayesian chi squared test based on an adjusted quadratic loss function for testing a simple null hypothesis we show that the asymptotic null distribution of our suggested test is a central chi squared distribution under some assumptions required for the bayesian large sample theory we refer to our test as the bayesian robust chi squared test since it is robust to parametric misspecification in the alternative model that is the limiting null distribution of our test is a central chi squared distribution irrespective of parametric misspecification in the alternative model in addition to being robust to parametric misspecification our test also shares properties of the test suggested by li et al 2015 based on a quadratic loss function we provide four examples to illustrate the implementation of our suggested bayesian test statistic

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

marginal likelihood from the gibbs output abstract in the context of bayes estimation via gibbs sampling with or without data augmentation a simple approach is developed for computing the marginal density of the sample data marginal likelihood given parameter draws from the posterior distribution consequently bayes factors for model comparisons can be routinely computed as a by product of the simulation hitherto this calculation has proved extremely challenging our approach exploits the fact that the marginal density can be expressed as the prior times the likelihood function over the posterior density this simple identity holds for any parameter value an estimate of the posterior density is shown to be available if all complete conditional densities used in the gibbs sampler have closed form expressions to improve accuracy the posterior density is estimated at a high density point and the numerical standard error of resulting estimate is derived the ideas are applied to probit regression and finite mixture models

unitary highest weight modules the unitary highest weight modules of a semisimple lie group g have been classified s 93 unfortunately this classification does not fit well with a more general picture of the unitary dual of g we give such a classification for g a classical group using derived functors and the philosophy of unipotent representations in particular these representations are associated to coadjoint orbits of g let 8 lie g 0 a cartan involution of 6 8 c a derived functor module is defined by the data q i u a o stable parabolic subalgebra of 0 and 17 an irreducible unitary representation of l the stabilizer of l in g considering z7 as a representation of i write z7 a rc where a e c c center of i denotes dual and rt z7l r we define a rc i rc cf definition 6 1 now fix n and vary 1 under certain restrictions on i which amount to a positivity condition on the corresponding bundle over g l a i rr is known to be irreducible and unitary as ii varies within this range the langlands parameters of a a rr vary smoothly however one cannot obtain all unitary highest weight modules under these restrictions the orbit method suggests how to obtain the remaining unitary highest weight modules the modules a rr are associated to coadjoint orbits as follows if rc is trivial we say a rr is associated to the elliptic orbit 0 g ii not all unitary highest weight modules may be obtained with rc trivial an example of which is the oscillator representation of 2n r the metapletic group we define 2n irreducible unitary highest weight modules rc 1 2 n k r is 113 oool 8708 87 7 50

a new approach to bayesian hypothesis testing abstract in this paper a new bayesian approach is proposed to test a point null hypothesis based on the deviance in a decision theoretical framework the proposed test statistic may be regarded as the bayesian version of the likelihood ratio test and appeals in practical applications with three desirable properties first it is immune to jeffreys concern about the use of improper priors second it avoids jeffreys lindley s paradox third it is easy to compute and its threshold value is easily derived facilitating the implementation in practice the method is illustrated using some real examples in economics and finance it is found that the leverage effect is insignificant in an exchange time series and that the fama french three factor model is rejected

discrete decomposability of the restriction of aq λ with respect to reductive subgroupsiii restriction of harish chandra modules and associated varieties let h g be real reductive lie groups and an irreducible unitary representation of g we introduce an algebraic formulation discretely decomposable restriction to single out the nice class of the branching problem breaking symmetry in physics in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction h this paper offers basic algebraic properties of discretely decomposable restrictions especially for a reductive symmetric pair g h and for the zuckerman vogan derived functor module and proves that the sufficient condition invent math 94 is in fact necessary a finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum an application to the restriction h of discrete series for a symmetric space g h is also given

on leverage in a stochastic volatility model this paper is concerned with the specification for modelling financial leverage effect in the context of stochastic volatility sv models two alternative specifications co exist in the literature one is the euler approximation to the well known continuous time sv model with leverage effect and the other is the discrete time sv model of jacquier et al j econometrics 122 2004 185 using a gaussian nonlinear state space form with uncorrelated measurement and transition errors i show that it is easy to interpret the leverage effect in the conventional model whereas it is not clear how to obtain and interpret the leverage effect in the model of jacquier et al empirical comparisons of these two models via bayesian markov chain monte carlo mcmc methods further reveal that the specification of jacquier et al is inferior simulation experiments are conducted to study the sampling properties of bayes mcmc for the conventional model

signature characters of invariant hermitian forms on irreducible verma modules and hall littlewood polynomials the unitary dual problem is one of mathematics most important open problems classify the irreducible unitary representations of a group the general approach has been to classify all representations admitting non degenerate invariant hermitian forms compute the signatures of those forms and then determine which forms are positive definite signature character algorithms and formulas arising from deforming representations and analysing changes at reducibility points as in adams et al unitary representations of real reductive groups arxiv e prints 2012 and yee represent theory 9 638 677 2005 produce very complicated formulas or algorithms from the resulting recursion this paper shows that in the case of irreducible verma modules all of the complexity can be encapsulated by the affine hecke algebra for compact real forms and for alcoves corresponding to translations of the fundamental alcove by a regular weight signature characters of irreducible verma modules are in fact negatives of hall littlewood polynomial summands evaluated at q 1 times a version of the weyl denominator establishing a simple signature character formula and drawing an important connection between signature characters and the affine hecke algebra signature characters of irreducible highest weight modules are shown to be related to kazhdan lusztig basis elements this paper also handles noncompact real forms the current state of the art for the unitary dual is a computer algorithm for determining if a given representation is unitary these results suggest the potential to move the state of the art to a closed form classification for the entire unitary dual

monte carlo estimation for nonlinear non gaussian state space models we develop a proposal or importance density for state space models with a nonlinear non gaussian observation vector y p y and an unobserved linear gaussian signal vector p the proposal density is obtained from the laplace approximation of the smoothing density p y we present efficient algorithms to calculate the mode of p y and to sample from the proposal density the samples can be used for importance sampling and markov chain monte carlo methods the new results allow the application of these methods to state space models where the observation density p y is not log concave additional results are presented that lead to computationally efficient implementations we illustrate the methods for the stochastic volatility model with leverage

bayesian analysis of stochastic volatility models with fat tails and correlated errors abstract the basic univariate stochastic volatility model specifies that conditional volatility follows a log normal auto regressive model with innovations assumed to be independent of the innovations in the conditional mean equation since the introduction of practical methods for inference in the basic volatility model j business econom statist 12 4 371 it has been observed that the basic model is too restrictive for many financial series we extend the basic svol to allow for a so called leverage effect via correlation between the volatility and mean innovations and for fat tails in the mean equation innovation a bayesian markov chain monte carlo algorithm is developed for the extended volatility model thus far likelihood based inference for the correlated svol model has not appeared in the literature we develop bayes factors to assess the importance of the leverage and fat tail extensions sampling experiments reveal little loss in precision from adding the model extensions but a large loss from using the basic model in the presence of mis specification there is overwhelming evidence of a leverage effect for weekly and daily equity indices the evidence in favor of fat tails is very strong for daily exchange rate and equity indices but less so for weekly data we also find that volatility estimates from the extended model are markedly different from those produced by the basic svol

brst cohomology and highest weight vectors i we initiate a program to study certain recent problems in non compact coset cft by the brst approach we derive a reduction formula for the brst cohomology by making use of a twisting by highest weight modules as illustrations we apply the formula to the bosonic string model and a rank one non compact coset model dpl our formula provides a completely new approach to non compact coset construction