Characterizing priors by posterior expectations in multiparameter exponential families

Summary Exploiting the notion of identifiability of mixtures of exponential families with respect to a vector parameter θ, it is shown that the posterior expectation of θ characterizes the prior distribution of θ. The result is applied to normal and negative multinomial distributions.     

ψ-correct decision for selection and elimination

Summary The selection oft out ofk populations with parameters θ _i (i=1, ...,k) is said to result in an ψ-correct decision provided ψ (minimum selected θ)>maximum non-selected θ where ψ(θ) (>θ) is an increasing function. For the cases of location or scale parameters the minimum probability of ψ-correct decision over the entire parameter space is shown to be no less than the minimum probability of correct selection over a preference zone determined by ψ(θ). For other types of parameters this result is shown to be true under certain conditions linking the distribution function and the ψ function.     

Large sample properties of the mle and mcle for the natural parameter of a truncated exponential family

Summary Consider a truncated exponential family of absolutely continuous distributions with natural parameter θ and truncation parameter γ. Strong consistency and asymptotic normality are shown to hold for the maximum likelihood and maximum conditional likelihood estimates of θ with γ unknown. Moreover, these two estimates are also shown to have the same limiting distribution, coinciding with that of the maximum likelihood estimate for θ when γ is assumed to be known.     

Computation of exact confidence limits from discrete data

Suppose that the data have a discrete distribution determined by (∞, ψ) where θ is the scalar parameter of interest and ψ is a nuisance parameter vector. The Buehler 1 - α upper confidence limit for θ is as small as possible, subject to the constraints that (a) its coverage probability is at least 1 - α and (b) it is a nondecreasing function of a pre-specified statisticT. This confidence limit has important biostatistical and reliability applications. The main result of the paper is that for a wide class of models (including binomial and Poisson), parameters of interest 9 and statisticsT (which possess what we call the “logical ordering” property) there is a dramatic increase in the ease with which this upper confidence limit can be computed. This result is illustrated numerically for θ a difference of binomial probabilities. Kabaila & Lloyd (2002) also show that ifT is poorly chosen then an assumption required for the validity of the formula for this confidence limit may not be satisfied. We show that for binomial data this assumption must be satisfied whenT possesses the “logical ordering” property.     

Improved confidence set estimators of a multivariate normal mean and generalizations

Summary LetX be the observed vector of thep-variate (p≧3) normal distribution with mean θ and covariance matrix equal to the identity matrix. Denotey ₊=max{0,y} for any real numbery. We consider the confidence set estimator of θ of the formC _δa,φ={θ:|θ−δ^a,φ(X)}≦c}, whereδ ^a,φ=[1−aφ({X})/{X}²]₊X is the positive part of the Baranchik (1970,Ann. Math. Statist.,41, 642–645) estimator. We provide conditions on ϕ(•) anda which guarantee thatC _δa.φ has higher coverage probability than the usual one, {θ:|θ−X|≦c}. This dominance result will be shown to hold for spherically symmetric distributions, which include the normal distribution,t-distribution and double exponential distribution. The latter result generalizes that of Hwang and Chen (1983,Technical Report, Dept. of Math., Cornell University).     

On a selection and ranking procedure for gamma populations

Summary The problem of selecting a subset of k gamma populations which includes the “best” population, i.e. the one with the largest value of the scale parameter, is studied as a multiple decision problem. The shape parameters of the gamma distributions are assumed to be known and equal for all the k populations. Based on a common number of observations from each population, a procedure R is defined which selects a subset which is never empty, small in size and yet large enough to guarantee with preassigned probability that it includes the best population regardless of the true unknown values of the scale parameters θ_i. Expression for the probability of a correct selection using R are derived and it is shown that for the case of a common number of observations the infimum of this probability is identical with the probability integral of the ratio of the maximum of k-1 independent gamma chance variables to another independent gamma chance variable, all with the same value of the other parameter. Formulas are obtained for the expected number of populations retained in the selected subset and it is shown that this function attains its maximum when the parameters θ_i are equal. Some other properties of the procedure are proved. Tables of constants b which are necessary to carry out the procedure are appended. These constants are reciprocals of the upper percentage points of F_max, the largest of several correlated F statistics. The distribution of this statistic is obtained. This work was supported in part by Office of Naval Research Contract Nonr-225 (53) at Stanford University. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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Two-parameter family of infinite-dimensional diffusions on the Kingman simplex

We construct a two-parameter family of diffusion processes X _α,θ on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers. For α = 0, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981) in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present paper, we extend Ethier and Kurtz’s main results to the two-parameter case. Namely, we show that the (two-parameter) Poisson-Dirichlet distribution PD(α,θ) is the unique stationary distribution for the process X _α,θ and that the process is ergodic and reversible with respect to PD(α, θ). We also compute the spectrum of the generator of X _α,θ . The Wright-Fisher diffusions on finite-dimensional simplices turn out to be special cases of X _α,θ for certain degenerate parameter values.     

On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Monotone maximum likelihood estimate for hazard function

A wide class of reliability theory models or lifetime data can be described as follows. Assume that the lifetime distribution function is F(t, θ)=F₀(λ(θ)t), where θ is the parameter characterizing some inner properties of a product and λ(θ) is an unknown increasing function. The paper deals with methods of estimation of λ(θ) from the sample (t _i ,θ _i ),i = 1, ...,n, for the case of exponentialF ₀. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 46–51, Perm, 1991.     

On the lower bounds for mean square error of empirical bayes estimators

The usual empirical Bayes setting is considered with θ being a shift or a scale parameter. A class of empirical Bayes estimators of a function b(θ) is proposed. The properties of the estimates are studied and mean square errors are calculated. The lower bounds are constructed for mean square errors of the empirical Bayes estimators over the class of all empirical Bayes estimators of b(θ). The results are applied to the case b(θ)=θ. The examples of the upper and lower bounds for mean square error are presented for the most popular families of conditional distributions. Added to the English translaion.     

Frequentist and Bayesian measures of confidence via multiscale bootstrap for testing three regions

A new computation method of frequentist p values and Bayesian posterior probabilities based on the bootstrap probability is discussed for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region of the parameter vector. We introduce new functional forms for the scaling-law of bootstrap probability so that the multiscale bootstrap method, which was designed for a one-sided test, can also compute confidence measures of a two-sided test, extending applicability to a wider class of hypotheses. Parameter estimation for the scaling-law is improved by the two-step multiscale bootstrap and also by including higher order terms. Model selection is important not only as a motivating application of our method, but also as an essential ingredient in the method. A compromise between frequentist and Bayesian is attempted by showing that the Bayesian posterior probability with a noninformative prior is interpreted as a frequentist p value of “zero-sided” test.     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

不同先验分布下的后验分布确定土力学参数

岩土工程中各土层参数的取值是根据现场及室内试验数据,采用经典统计学方法进行确定的,但这往往忽略了先验信息的作用.与经典统计学方法不同的是,Bayes法能从考虑先验分布的角度结合样本分布去推导后验分布,为岩土参数的取值提供一种新的分析方法.岩土工程勘察可视为对总体地层的随机抽样,当抽样完成时,样本分布密度函数是确定的,故Bayes法中的后验分布取决于先验分布,因此推导出两套不同的先验分布:利用先验信息确定先验分布及共轭先验分布.通过对先验及后验分布中超参数的计算,当样本总体符合N(μ,σ²)正态分布时,对所要研究的未知参数μ和σ展开分析,综合对比不同先验分布下后验分布的区间长度,给出岩土参数Bayes推断中最佳后验分布所要选择的先验分布.结果表明:共轭情况下的后验分布总是比无信息情况下的后验区间短,概率密度函数分布更集中,取值更方便.在正态总体情形下,根据未知参数μ和σ的联合后验分布求极值方法,确定样本总体中最大概率均值μ_max和方差σ_max作为工程设计采用值,为岩土参数取值方法提供了一条新的路径,有较好的工程意义.     

Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions

In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.     

Cramér-type conditions and quadratic mean differentiability

Summary Let (Ω,A) be a measurable space, let Θ be an open set inR ^k , and let {P _θ; θ∈Θ} be a family of probability measures defined onA. Let μ be a σ-finite measure onA, and assume thatP _θ≪μ for each θ∈Θ. Let us denote a specified version ofdP _θ /d _μ byf(ω; θ). In many large sample problems in statistics, where a study of the log-likelihood is important, it has been convenient to impose conditions onf(ω; θ) similar to those used by Cramér [2] to establish the consistency and asymptotic normality of maximum likelihood estimates. These are of a purely analytical nature, involving two or three pointwise derivatives of lnf(ω; θ) with respect to θ. Assumptions of this nature do not have any clear probabilistic or statistical interpretation. In [10], LeCam introduced the concept of differentially asymptotically normal (DAN) families of distributions. One of the basic properties of such a family is the form of the asymptotic expansion, in the probability sense, of the log-likelihoods. Roussas [14] and LeCam [11] give conditions under which certain Markov Processes, and sequences of independent identically distributed random variables, respectively, form DAN families of distributions. In both of these papers one of the basic assumptions is the differentiability in quadratic mean of a certain random function. This seems to be a more appealing type of assumption because of its probabilistic nature. In this paper, we shall prove a theorem involving differentiability in quadratic mean of random functions. This is done in Section 2. Then, by confining attention to the special case when the random function is that considered by LeCam and Roussas, we will be able to show that the standard conditions of Cramér type are actually stronger than the conditions of LeCam and Roussas in that they imply the existence of the necessary quadratic mean derivative. The relevant discussion is found in Section 3. This research was supported by the National Science Foundation, Grant GP-20036.     

Uniformly robust tests in errors-in-variables models

Consider an ordinary errors-in-variables model. The true level α _n (θ*) of a test at nominal level α and sample size n is said to be pointwise robust if α _n (θ*) → α as n → ∞ for each parameter θ*. Let Ω* be a set of values of θ*. Define α _n = sup _{θ* ∈Ω*}α _n (θ*). The test is said to be uniformly robust over Ω* if α _n → α as n → ∞. Corresponding definitions apply to the coverage probabilities of confidence sets. It is known that all existing large-sample tests for the parameters of the errors-in-variables model are pointwise robust. However, they might not be uniformly robust over certain null parameter spaces. In this paper, we construct uniformly robust tests for testing the vector coefficient parameter and vector slope parameter in the functional errors-in-variables model. These tests are established through constructing the confidence sets for the same parameters in the model with similar desirable property. Power comparisons based on simulation studies between the proposed tests and some existing tests in finite samples are also presented.     

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions. Research supported by NSERC of Canada.     

Asymptotic Behaviour of Trajectory Fitting Estimators for Certain Non-ergodic SDE

Suppose one observes a path of a stochastic processX = (X_t)_t≥0 driven by the equation dX_t=θ a(X_t)dt + dW_t, t≥0, θ ≥ 0 with a(x) = x or a(x) = |x|^α for some α ∈ [0,1) and given initial condition X ₀. If the true but unknown parameter θ₀ is positive then X is non-ergodic. It is shown that in this situation a trajectory fitting estimator for θ₀ is strongly consistent and has the same limiting distribution as the maximum likelihood estimator, but converges of minor order. This revised version was published online in August 2006 with corrections to the Cover Date.     

Iterative Approximation of Statistical Distributions and Relation to Information Geometry

The optimal control of stochastic processes through sensor estimation of probability density functions is given a geometric setting via information theory and the information metric. Information theory identifies the exponential distribution as the maximum entropy distribution if only the mean is known and the Γ distribution if also the mean logarithm is known. The surface representing Γ models has a natural Riemannian information metric. The exponential distributions form a one-dimensional subspace of the two-dimensional space of all Γ distributions, so we have an isometric embedding of the random model as a subspace of the Γ models. This geometry provides an appropriate structure on which to represent the dynamics of a process and algorithms to control it. This short paper presents a comparative study on the parameter estimation performance between the geodesic equation and the B-spline function approximations when they are used to optimize the parameters of the Γ family distributions. In this case, the B-spline functions are first used to approximate the Γ probability density function on a fixed length interval; then the coefficients of the approximation are related, through mean and variance calculations, to the two parameters (i.e. μ and β) in Γ distributions. A gradient based parameter tuning method has been used to produce the trajectories for (μ, β) when B-spline functions are used, and desired results have been obtained which are comparable to the trajectories obtained from the geodesic equation. This revised version was published online in August 2006 with corrections to the Cover Date.