Estimation of the location parameter under LINEX loss function: multivariate case

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?^r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).     

On the existence of locally mvu estimators

It is proved that there exists an unbiased estimator for some real parameter of a class of distributions, which has minimal variance for some fixed distribution among all corresponding unbiased estimators, if and. only if the corresponding minimal variances for all related unbiased estimation problems concerning finite subsets of the underlying family of distributions are bounded. As an application it is shown that there does not exist some unbiased estimator for θ^k+c(ε≥0) with minimal variance for θ =0 among all corresponding unbiased estimators on the base of k i.i.d. random variables with a Cauchy-distribution, where θ denotes some location parameter.     

A Note on the Estimation of Non-Differentiable Functions of MuItinomial ProbabiIities

Abstract. Consider the problem of estimating f(θ ) where fis a given function and 8 is the unknown parameter of a multinomial distribution. In order to describe the asymptotic behaviour of the frequency substitution estimator, conventional methods typically require differentiability of f , and the error representations depend on the unknown parameter θ.
In this note, a pararneter-free bound for the mean square error is derived which requires continuity of f only.     

A Simulation Approach to Nonparametric Empirical Bayes Analysis

We deal with general mixture of hierarchical models of the form m(x) = f_øf(x |θ) g (θ)dθ , where g(θ) and m(x) are called mixing and mixed or compound densities respectively, and θ is called the mixing parameter. The usual statistical application of these models emerges when we have data x_i, i = 1,…,n with densities f(x_i|θ_i) for given θ_i, and the θ₁ are independent with common density g(θ) . For a certain well known class of densities f(x |θ) , we present a sample-based approach to reconstruct g(θ) . We first provide theoretical results and then we use, in an empirical Bayes spirit, the first four moments of the data to estimate the first four moments of g(θ) . By using sampling techniques we proceed in a fully Bayesian fashion to obtain any posterior summaries of interest. Simulations which investigate the operating characteristics of our proposed methodology are presented. We illustrate our approach using data from mixed Poisson and mixed exponential densities.     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

On an admissibility problem involving the moment generating function of the lognormal distribution

A new location invariant loss function is considered and the best invariant estimator of normal mean is obtained. This estimator is a function of the moment generating function of the lognormal distribution. The admissibility is studied of a class of linear estimators of the form cX+d, where X~N(/,C²), with / unknown and C² known. This yields the admissibility of the best invariant estimator of /.     

Empirical Bayesball Remixed: Empirical Bayes Methods for Longitudinal Data

Empirical Bayes methods for Gaussian and binomial compound decision problems involving longitudinal data are considered. A recent convex optimization reformulation of the nonparametric maximum likelihood estimator of Kiefer and Wolfowitz (Annals of Mathematical Statistics 1956; 27 : 887–906) is employed to construct nonparametric Bayes rules for compound decisions. The methods are illustrated with an application to predict baseball batting averages, and the age profile of batting performance. An important aspect of the empirical application is the general bivariate specification of the distribution of heterogeneous location and scale effects for players that exhibits a weak positive association between location and scale attributes. Prediction of players' batting averages for 2012 based on performance in the prior decade using the proposed methods shows substantially improved performance over more naive methods with more restrictive treatment of unobserved heterogeneity. Comparisons are also made with nonparametric Bayesian methods based on Dirichlet process priors, which can be viewed as a regularized, or smoothed, version of the Kiefer–Wolfowitz method. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating the J function without edge correction

The interaction between points in a spatial point process can be measured by its empty space function F , its nearest-neighbour distance distribution function G , and by combinations such as the J function J = (1 G )/(1 F ). The estimation of these functions is hampered by edge effects: the uncorrected, empirical distributions of distances observed in a bounded sampling window W give severely biased estimates of F and G . However, in this paper we show that the corresponding uncorrected estimator of the function J = (1 G )/(1 F ) is approximately unbiased for the Poisson case, and is useful as a summary statistic. Specifically, consider the estimate _W of J computed from uncorrected estimates of F and G . The function J _W ( r ), estimated by _W , possesses similar properties to the J function, for example J _W ( r ) is identically 1 for Poisson processes. This enables direct interpretation of uncorrected estimates of J , something not possible with uncorrected estimates of either F , G or K . We propose a Monte Carlo test for complete spatial randomness based on testing whether J _W ( r ) 1. Computer simulations suggest this test is at least as powerful as tests based on edge corrected estimators of J .     

Robust linear static panel data models using ε-contamination

The paper develops a general Bayesian framework for robust linear static panel data models using$\epsilon$-contamination. A two-step approach is employed to derive the conditional type-II maximum likelihood (ML-II) posterior distribution of the coefficients and individual effects. The ML-II posterior means are weighted averages of the Bayes estimator under a base prior and the data-dependent empirical Bayes estimator. Two-stage and three stage hierarchy estimators are developed and their finite sample performance is investigated through a series of Monte Carlo experiments. These include standard random effects as well as Mundlak-type, Chamberlain-type and Hausman–Taylor-type models. The simulation results underscore the relatively good performance of the three-stage hierarchy estimator. Within a single theoretical framework, our Bayesian approach encompasses a variety of specifications while conventional methods require separate estimators for each case.     

Estimation of multivariate models for time series of possibly different lengths

We consider the problem of estimating parametric multivariate density models when unequal amounts of data are available on each variable. We focus in particular on the case that the unknown parameter vector may be partitioned into elements relating only to a marginal distribution and elements relating to the copula. In such a case we propose using a multi‐stage maximum likelihood estimator (MSMLE) based on all available data rather than the usual one‐stage maximum likelihood estimator (1SMLE) based only on the overlapping data. We provide conditions under which the MSMLE is not less asymptotically efficient than the 1SMLE, and we examine the small sample efficiency of the estimators via simulations. The analysis in this paper is motivated by a model of the joint distribution of daily Japanese yen–US dollar and euro–US dollar exchange rates. We find significant evidence of time variation in the conditional copula of these exchange rates, and evidence of greater dependence during extreme events than under the normal distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Empiric bayes estimation of hypergeometric probability

A Bayes-empiric Bayes estimator of a parameter of the hypergeometric distribution, based on orthogonal polynomials on non-negative integers, is introduced. It is shown that this estimator is asymptotically optimal; and the resulting estimator of the prior probability function is mean square consistent.     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.     

Bias in estimation from type I censored samples

Summary
Let be a family of probability distributions on R¹. This paper raises the question whether a parameter θ=θ (P), Pt, is estimable on the basis of a type I censored sample (i.e. censored on a fixed set C). Two theorems are given that state conditions on θ and C that ensure that θ is not estimable. The results are applied to estimation problems for the normal and POISSON distributions; it turns out that unbiased estimation is impossible in the majority of practical cases.     

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

On efficient estimation of the ordered response model

In this paper, we derive efficiency bounds for the ordered response model when the distribution of the errors is unknown. Furthermore, we develop an estimator that is efficient under suitable conditions. Interestingly, neither the bounds nor the estimator are trivial extensions of what has been proposed in the literature for the binary response model. The estimator is composed of quadratic B-splines, and estimation is performed by the method of sieves. In addition, the estimator of the distribution function is restricted to be a proper distribution function. An empirical example on the effect of fees on attendance rates at universities and community colleges is also included; we get substantively different results by relaxing the assumption that the distribution of the errors is normal.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

Estimation of quantities determined as implicit functions of unknown parameters

Consider a family of distribution functions ${\{F(x, \theta),\,\theta \in \Theta\}}Consider a family of distribution functions {F(x, q), q ? Q}{\{F(x, \theta),\,\theta \in \Theta\}} . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The results are then applied to estimate change point of failure rate function, and change point of mean residual life function.     

Estimation of Fisher information using model selection

In the paper the problem of estimation of Fisher information I _f for a univariate density supported on [0, 1] is discussed. A starting point is an observation that when the density belongs to an exponential family of a known dimension, an explicit formula for I _f there allows for its simple estimation. In a general case, for a given random sample, a dimension of an exponential family which approximates it best is sought and then estimator of I _f is constructed for the chosen family. As a measure of quality of fit a modified Bayes Information Criterion is used. The estimator, which is an instance of Post Model Selection Estimation method is proved to be consistent and asymptotically normal when the density belongs to the exponential family. Its consistency is also proved under misspecification when the number of exponential models under consideration increases in a suitable way. Moreover we provide evidence that in most of considered parametric cases the small sample performance of proposed estimator is superior to that of kernel estimators.     

Estimation of inefficiency in stochastic frontier models: a Bayesian kernel approach

We propose a kernel-based Bayesian framework for the analysis of stochastic frontiers and efficiency measurement. The primary feature of this framework is that the unknown distribution of inefficiency is approximated by a transformed Rosenblatt-Parzen kernel density estimator. To justify the kernel-based model, we conduct a Monte Carlo study and also apply the model to a panel of U.S. large banks. Simulation results show that the kernel-based model is capable of providing more precise estimation and prediction results than the commonly-used exponential stochastic frontier model. The Bayes factor also favors the kernel-based model over the exponential model in the empirical application.

     

Model misspecification effects for biased samples

The model misspecification effects on the maximum likelihood estimator are studied when a biased sample is treated as a random one as well as when a random sample is treated as a biased one. The relation between the existence of a consistent estimator under model misspecification and the completeness of the distribution is also considered. The cases of the weight invariant distribution and the scale parameter distribution are examined and finally an example is presented to illustrate the results.