splines in statistics abstract this is a survey article that attempts to synthesize a broad variety of work on splines in statistics splines are presented as a nonparametric function estimating technique after a general introduction to the theory of interpolating and smoothing splines splines are treated in the nonparametric regression setting the method of cross validation for choosing the smoothing parameter is discussed and the general multivariate regression surface estimation problem is addressed an extensive discussion of splines as nonparametric density estimators is followed by a discussion of their role in time series analysis a comparison of the spline and isotonic regression methodologies leads to a formulation of a hybrid estimator the closing section provides a brief overall summary and formulates a number of open unsolved problems relating to splines in statistics

partially observable markov decision processes incorporating epistemic uncertainties abstract the use of markov decision processes for inspection maintenance and rehabilitation of civil engineering structures relies on the use of several transition matrices related to the stochastic degradation process maintenance actions and imperfect inspections point estimators for these matrices are usually used and they are evaluated using statistical inference methods and or expert evaluation methods thus considerable epistemic uncertainty often veils the true values of these matrices our contribution through this paper is threefold first we present a methodology for incorporating epistemic uncertainties in dynamic programming algorithms used to solve finite horizon markov decision processes which may be partially observable second we propose a methodology based on the use of dirichlet distributions which answers in our sense much of the controversy found in the literature about estimating markov transition matrices third we show how the complexity resulting from the use of monte carlo simulations for the transition matrices can be greatly overcome in the framework of dynamic programming the proposed model is applied to concrete bridge under degradation in order to provide the optimal strategy for inspection and maintenance the influence of epistemic uncertainties on the optimal solution is underlined through sensitivity analysis regarding the input data

optimal control of hidden markov models with binary observations a hidden markov model hmm consisting of a controlled markov chain and a binary output is considered the binary output reports the occurrence of a certain state versus any other state in the markov chain the transition matrix of the markov chain depends on a control vector which is to be chosen to minimize a discounted infinite horizon cost function the information provided for decision making is the entire history of the binary output which presents an incomplete knowledge of the markov chain and as a consequence leads to a stochastic control problem with partial observations the solution to such a problem is a control policy consisting of two components an estimator to derive the posterior probability distribution of the markov chain from the observation set and a nonlinear control law to map this estimate into the control vector this paper introduces a nonlinear state space representation for the estimator develops analytical expressions for the control law and presents numerical methods for efficient computation of the optimal control application of these results to optimal management of a class of inventory systems is discussed and the performance of optimal control for this application is examined by numerical simulations

on model order estimation for partially observed markov chains abstract model order estimation for the partially observed markov chain pomc also called the hidden markov model is addressed model order is defined as the number of independent model parameters simulation is used to investigate the small sample performance of a number of model order criteria as applied to pomc models with low state dimension and binary observations consistency of these criteria in the context of pomc models with discrete state and observation space is also investigated a criterion due to akaike is shown not to be consistent simulation results indicate that akaike s criterion can provide more accurate estimation in the small sample case

control limits for two state partially observable markov decision processes abstract a production process can be in either a good or a bad state the true state is unknown and can only be inferred from observations if the state is good during one period it may deteriorate and become bad during the next period two actions are available continue or replace for a fixed cost the objective is to maximize the expected discounted value of the total future profits we prove that dominance in expectation the expected profit is larger in the good state than in the bad state suffices for the optimal policy to be of a control limit clt type continue if and only if the good state probability exceeds the clt this condition is weaker than stochastic dominance which has been prevailing we also show that the expected profit function is convex strictly increasing

arakelov self intersection numbers of minimal regular models of modular curves x_0 p 2 x0 p2 we compute an asymptotic expression for the arakelov self intersection number of the relative dualizing sheaf of edixhoven s minimal regular model for the modular curve x 0 p 2 over mathbb q the computation of the self intersection numbers are used to prove an effective version of the bogolomov conjecture for the semi stable models of modular curves x 0 p 2 and obtain a bound on the stable faltings height for those curves in a companion article banerjee and chaudhuri in isr j math 2020

optimal stopping under partial observation near value iteration we propose a new approximate value iteration method namely near value iteration nvi to solve continuous state optimal stopping problems under partial observation which in general cannot be solved analytically and also pose a great challenge to numerical solutions nvi is motivated by the expression of the value function as the supremum over an uncountable set of linear functions in the belief state after a smart manipulation of the operations in the updating equation for the value function we reduce the set to only two functions at every time step so as to achieve significant computational savings nvi yields a value function approximation bounded by the tightest lower and upper bounds that can be achieved by existing algorithms in the same class so the nvi approximation is closer to the true value function than at least one of these bounds we demonstrate the effectiveness of our approach on an example of pricing american options under stochastic volatility

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

hauteur de faltings de quotients de j 0 n discriminants d algebres de hecke et congruences entre formes modulaires in this paper we compute the faltings height of the jacobian of the modular curve j 0 n we explain how it is related to the discriminant of the hecke algebra this discriminant measures congruences between modular forms as a corollary we obtain a lower bound for this discriminant the faltings height of a quotient a of j 0 n is also computed

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

state of the art a survey of partially observable markov decision processes theory models and algorithms this paper surveys models and algorithms dealing with partially observable markov decision processes a partially observable markov decision process pomdp is a generalization of a markov decision process which permits uncertainty regarding the state of a markov process and allows for state information acquisition a general framework for finite state and action pomdp s is presented next there is a brief discussion of the development of pomdp s and their relationship with other decision processes a wide range of models in such areas as quality control machine maintenance internal auditing learning and optimal stopping are discussed within the pomdp framework lastly algorithms for computing optimal solutions to pomdp s are presented

lower bounds on the arithmetic self intersection number of the relative dualizing sheaf on arithmetic surfaces we give an explicitly computable lower bound for the arithmetic self intersection number of the dualizing sheaf on a large class of arithmetic surfaces if some technical conditions are satisfied then this lower bound is positive in particular these technical conditions are always satisfied for minimal arithmetic surfaces with simple multiplicities and at least one reducible fiber but we have also used our techniques to obtain lower bounds for some arithmetic surfaces with non reduced fibers

convergence of reinforcement learning algorithms and acceleration of learning the techniques of reinforcement learning have been gaining increasing popularity recently however the question of their convergence rate is still open we consider the problem of choosing the learning steps ensuremath alpha n and their relation with discount ensuremath gamma and exploration degree ensuremath epsilon appropriate choices of these parameters may drastically influence the convergence rate of the techniques from analytical examples we conjecture optimal values of ensuremath alpha n and then use numerical examples to verify our conjectures

stochastic control theory and operational research abstract the paper gives an overview of stochastic optimal control theory and its applications to operational research stochastic control theory deals with the intertemporal optimization of dynamic systems under uncertainty after a short review of deterministic optimal control theory and the theory of stochastic dynamic systems a general stochastic optimal control problem is formulated both in discrete and continuous time among the method for solving such problems particularly stochastic dynamic programming and the stochastic maximum principle are introduced examples from the areas of finance advertising production inventory models and information policy illustrate the applicability of stochastic control theory to operational research problems moreover its actual and potential impact on or is discussed from a methodological point of view

a simple derivation of deletion diagnostic results for the general linear model with correlated errors a general simple and intuitive derivation is provided for diagnostics associated with the deletion of arbitrary subsets for the linear model with general covariance structure these are seen to be most simply expressed even for the well studied case of independent and identically distributed data in terms of a residual known variously as the conditional residual the deletion prediction residual and the cross validation residual particularly simple specializations arise when the subsets are of size 1 and of size 2 but the method is easy to apply for all subsets and to conditional deletions

void probabilities and cauchy schwarz divergence for generalized labeled multi bernoulli models the generalized labeled multi bernoulli glmb is a family of tractable models that alleviates the limitations of the poisson family in dynamic bayesian inference of point processes in this paper we derive closed form expressions for the void probability functional and the cauchy schwarz divergence for glmbs the proposed analytic void probability functional is a necessary and sufficient statistic that uniquely characterizes a glmb while the proposed analytic cauchy schwarz divergence provides a tractable measure of similarity between glmbs we demonstrate the use of both results on a partially observed markov decision process for glmbs with cauchy schwarz divergence based reward and void probability constraint

a multiperiod newsvendor problem with partially observed demand we consider a newsvendor problem with partially observed markovian demand demand is observed if it is less than the inventory otherwise only the event that it is larger than or equal to the inventory is observed these observations are used to update the demand distribution from one period to the next the state of the resulting dynamic programming equation is the current demand distribution which is generally infinite dimensional we use unnormalized probabilities to convert the nonlinear state transition equation to a linear one this helps in proving the existence of an optimal feedback ordering policy so as to learn more about the demand the optimal order is set to exceed the myopic optimal order the optimal cost decreases as the demand distribution decreases in the hazard rate order in a special case with finitely many demand values we characterize a near optimal solution by establishing that the value function is piecewise linear

a two state partially observable markov decision process with three actions abstract a process can be in either a stable or an unstable state interchangeably the true state is unobservable and can only be inferred from observations three actions are available continue with the process con repair the process for a certain fee bring the process to the stable state rep and obtain the state of the process for a cost ins the objective is to maximize the expected discounted value of the total future profits we formulate the problem as a discrete time partially observable markov decision process pomdp we show that the expected profit function is convex and strictly increasing and that the optimal policy has either one or two control limits also we show that dominance in expectation the expected revenue is larger in the stable state than in the unstable state suffices for a control limit structure

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

partially observed inventory systems the case of zero balance walk in many inventory control contexts inventory levels are only partially i e not fully observed this may be due to nonobservation of demand spoilage misplacement or theft of inventory we study a partially observed inventory system where the demand is not observed inventory level is noticed when it reaches zero the unmet demand is lost and replenishment orders must be decided so as to minimize the total discounted costs over an infinite horizon this problem has an infinite dimensional state space and for it we establish the existence of a feedback policy when single period costs are bounded or when the discount factor is sufficiently small we also provide an approximately optimal feedback policy that uses a finite state representation

bounds on faltings s delta function through covers let x be a compact riemann surface of genus gx 1 in 1984 g faltings introduced a new invariant fal x associated to x in this paper we give explicit bounds for fal x in terms of fundamental dierential geometric invariants arising from x when gx 1 as an application we are able to give bounds for faltings s delta function for the family of modular curves x0 n in terms of the genus only in combination with work of a abbes p michel and e ullmo this leads to an asymptotic formula for the faltings height of the jacobian j0 n associated to x0 n

residuals for the linear model with general covariance structure a general theory is presented for residuals from the general linear model with correlated errors it is demonstrated that there are two fundamental types of residual associated with this model referred to here as the marginal and the conditional residual these measure respectively the distance to the global aspects of the model as represented by the expected value and the local aspects as represented by the conditional expected value these residuals may be multivariate some important dualities are developed which have simple implications for diagnostics the results are illustrated by reference to model diagnostics in time series and in classical multivariate analysis with independent cases

linear equality constraints in the general linear mixed model scientists may wish to analyze correlated outcome data with constraints among the responses for example piecewise linear regression in a longitudinal data analysis can require use of a general linear mixed model combined with linear parameter constraints although well developed for standard univariate models there are no general results that allow a data analyst to specify a mixed model equation in conjunction with a set of constraints on the parameters we resolve the difficulty by precisely describing conditions that allow specifying linear parameter constraints that insure the validity of estimates and tests in a general linear mixed model the recommended approach requires only straightforward and noniterative calculations to implement we illustrate the convenience and advantages of the methods with a comparison of cognitive developmental patterns in a study of individuals from infancy to early adulthood for children from low income families

a method for fitting regression splines with varying polynomial order in the linear mixed model the linear mixed model has become a widely used tool for longitudinal analysis of continuous variables the use of regression splines in these models offers the analyst additional flexibility in the formulation of descriptive analyses exploratory analyses and hypothesis driven confirmatory analyses we propose a method for fitting piecewise polynomial regression splines with varying polynomial order in the fixed effects and or random effects of the linear mixed model the polynomial segments are explicity constrained by side conditions for continuity and some smoothness at the points where they join by using a reparameterization of this explicitly constrained linear mixed model an implicitly constrained linear mixed model is constructed that simplifies implementation of fixed knot regression splines the proposed approach is relatively simple handles splines in one variable or multiple variables and can be easily programmed using existing commercial software such as sas or s plus the method is illustrated using two examples an analysis of longitudinal viral load data from a study of subjects with acute hiv 1 infection and an analysis of 24 hour ambulatory blood pressure profiles copyright 2005 john wiley sons ltd