Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

A Berry-Esseen theorem for the kernel quantile estimator with application to studying the deficiency of quantile estimators

A Berry-Esseen bound is established for the kernel quantile estimator under various conditions. The results improve an earlier result of Falk (1985,Ann. Statist.,13, 428–433) and rely on the local smoothness of the quantile function. This new Berry-Esseen bound is applied to studying the deficiency of the sample quantile estimator with respect to the kernel quantile estimator. A new result is obtained which is an extension of that in Falk (1985).     

A projection method of estimation for a subfamily of exponential families

Summary This paper is concerned with estimation for a subfamily of exponential-type, which is a parametric model with sufficient statistics. The family is associated with a surface in the domain of a sufficient statistic. A new estimator, termed a projection estimator, is introduced. The key idea of its derivation is to look for a one-to-one transformation of the sufficient statistic so that the subfamily can be associated with a flat subset in the transformed domain. The estimator is defined by the orthogonal projection of the transformed statistic onto the flat surface. Here the orthogonality is introduced by the inverse of the estimated variance matrix of the statistic on the analogy of Mahalanobis's notion (1936,Proc. Nat. Inst. Sci. Ind.,2, 49–55). Thus the projection estimator has an explicit representation with no iterations. On the other hand, the MLE and classical estimators have to be sought as numerical solutions by some algorithm with a choice of an initial value and a stopping rule. It is shown that the projection estimator is first-order efficient. The second-order property is also discussed. Some examples are presented to show the utility of the estimator.     

Estimation of the ratio of the scale parameters of two exponential distributions with unknown location parameters

We consider the estimation of the ratio of the scale parameters of two independent two-parameter exponential distributions with unknown location parameters. It is shown that the best affine equivariant estimator (BAEE) is inadmissible under any loss function from a large class of bowl-shaped loss functions. Two new classes of improved estimators are obtained. Some values of the risk functions of the BAEE and two improved estimators are evaluated for two particular loss functions. Our results are parallel to those of Zidek (1973, Ann. Statist., 1, 264–278), who derived a class of estimators that dominate the BAEE of the scale parameter of a two-parameter exponential distribution.     

柯西分布的参数估计

研究了柯西分布的参数估计问题,给出了位置参数的最小一乘估计和尺度参数的低阶矩估计.证明了柯西分布位置参数的最小一乘估计具有渐近无偏性与强相合性;尺度参数的低阶矩估计具有强相合性.     

Iterative Estimation of the Extreme Value Index

Let {X_n, n ≥ 1} be a sequence of independent random variables with common continuous distribution function F having finite and unknown upper endpoint. A new iterative estimation procedure for the extreme value index γ is proposed and one implemented iterative estimator is investigated in detail, which is asymptotically as good as the uniform minimum varianced unbiased estimator in an ideal model. Moreover, the superiority of the iterative estimator over its non iterated counterpart in the non asymptotic case is shown in a simulation study.AMS 2000 Subject Classification: 62G32Supported by Swiss National Science foundation.     

The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chi-squared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw's minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chi-squared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.     

Nonparametric Estimation for Semi-Markov Processes Based on its Hazard Rate Functions

The problem of estimating the Markov renewal matrix and the semi-Markov transition matrix based on a history of a finite semi-Markov process censored at time T (fixed) is addressed for the first time. Their asymptotic properties are studied. We begin by the definition of the transition rate of this process and propose a maximum likelihood estimator for the hazard rate functions and then we show that this estimator is uniformly strongly consistent and converges weakly to a normal random variable. We construct a new estimator for an absolute continous semi-Markov kernel and give detailed derivation of uniform strong consistency and weak convergence of this estimator as the censored time tends to infinity. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Dirichlet process functional approach to heteroscedastic-consistent covariance estimation

The mixture of Dirichlet process (MDP) defines a flexible prior distribution on the space of probability measures. This study shows that ordinary least-squares (OLS) estimator, as a functional of the MDP posterior distribution, has posterior mean given by weighted least-squares (WLS), and has posterior covariance matrix given by the (weighted) heteroscedastic-consistent sandwich estimator. This is according to a pairs bootstrap distribution approximation of the posterior, using a Pólya urn scheme. Also, when the MDP prior baseline distribution is specified as a product of independent probability measures, this WLS solution provides a new type of generalized ridge regression estimator. Such an estimator can handle multicollinear or singular design matrices even when the number of covariates exceeds the sample size, and can shrink the coefficient estimates of irrelevant covariates towards zero, which makes it useful for nonlinear regressions via basis expansions. Also, this MDP/OLS functional methodology can be extended to methods for analyzing the sensitivity of the heteroscedasticity-consistent causal effect size over a range of hidden biases, due to missing covariates omitted from the regression; and more generally, can be extended to a Vibration of Effects analysis. The methodology is illustrated through the analysis of simulated and real data sets. Overall, this study establishes new connections between Dirichlet process functional inference, the bootstrap, consistent sandwich covariance estimation, ridge shrinkage regression, WLS, and sensitivity analysis, to provide regression methodology useful for inferences of the mean dependent response.     

Bayesian variable sampling plan for the weibull distribution with type I censoring

In this article, we study a model of a single variable sampling plan with Type I censoring. Assume that the quality of an item in a batch is measured by a random variable which follows a Weibull distributionW (λ,m), with scale parameter λ and shape parameterm having a gamma-discrete prior distribution or σ=1/λ andm having an inverse gamma-uniform prior distribution. The decision function is based on the Kaplan-Meier estimator. Then, the explicit expressions of the Bayes risk are derived. In addition, an algorithm is suggested so that an optimal sampling plan can be determined approximately after a finite number of searching steps.     

Asymptotic Improvement of the Usual Confidence Set in a Multivariate Normal Distribution with Unknown Variance

We consider confidence sets for the mean of a multivariate normal distribution with unknown covariance matrix of the formσ²I. The coverage probability of the usual confidence set is shown to be improved asymptotically by centering at a shrinkage estimator.     

多元t分布下相依回归模型参数的两步估计

本文把文献中关于正态分布下相依回归模型参数Zellner估计的有限样本均方误差结果和效率结果以及两步协方差改进估计的一般均方误差结果推广到多元t分布情况,在该分布下两种估计的统计优效性质均不变.     

Simultaneous estimation of means of classified normal observations

Simultaneous estimation of normal means is considered for observations which are classified into several groups. In a one-way classification case, it is shown that an adaptive shrinkage estimator dominates a Stein-type estimator which shrinks observations towards individual class averages as Stein's (1966,Festschrift for J. Neyman, (ed. F. N. David), 351–366, Wiley, New York) does, and is minimax even if class sizes are small. Simulation results under quadratic loss show that it is slightly better than Stein's (1966) if between variances are larger than within ones. Further this estimator is shown to improve on Stein's (1966) with respect to the Bayes risk. Our estimator is derived by assuming the means to have a one-way classification structure, consisting of three random terms of grand mean, class mean and residual. This technique can be applied to the case where observations are classified into a two-stage hierarchy.     

Kullback-Leibler Information Consistent Estimation for Censored Data

This paper is intended as an investigation of parametric estimation for the randomly right censored data. In parametric estimation, the Kullback-Leibler information is used as a measure of the divergence of a true distribution generating a data relative to a distribution in an assumed parametric model M. When the data is uncensored, maximum likelihood estimator (MLE) is a consistent estimator of minimizing the Kullback-Leibler information, even if the assumed model M does not contain the true distribution. We call this property minimum Kullback-Leibler information consistency (MKLI-consistency). However, the MLE obtained by maximizing the likelihood function based on the censored data is not MKLI-consistent. As an alternative to the MLE, Oakes (1986, Biometrics, 42, 177–182) proposed an estimator termed approximate maximum likelihood estimator (AMLE) due to its computational advantage and potential for robustness. We show MKLI-consistency and asymptotic normality of the AMLE under the misspecification of the parametric model. In a simulation study, we investigate mean square errors of these two estimators and an estimator which is obtained by treating a jackknife corrected Kaplan-Meier integral as the log-likelihood. On the basis of the simulation results and the asymptotic results, we discuss comparison among these estimators. We also derive information criteria for the MLE and the AMLE under censorship, and which can be used not only for selecting models but also for selecting estimation procedures.     

r-k Class estimation in regression model with concomitant variables

We treat with the r-k class estimation in a regression model, which includes the ordinary least squares estimator, the ordinary ridge regression estimator and the principal component regression estimator as special cases of the r-k class estimator. Many papers compared total mean square error of these estimators. Sarkar (1989, Ann. Inst. Statist. Math., 41, 717–724) asserts that the results of this comparison are still valid in a misspecified linear model. We point out some confusions of Sarkar and show additional conditions under which his assertion holds.     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

Estimating the Scale Parameter of a Lévy-stable Distribution via the Extreme Value Approach

The characteristic exponent α of a Lévy-stable law S _α (σ, β, μ) was thoroughly studied as the extreme value index of a heavy tailed distribution. For 1 < α < 2, Peng (Statist. Probab. Lett. 52: 255–264, 2001) has proposed, via the extreme value approach, an asymptotically normal estimator for the location parameter μ. In this paper, we derive by the same approach, an estimator for the scale parameter σ and we discuss its limiting behavior.      

Admissibility of estimators of the probability of unobserved outcomes

The problem of estimating the probability of unobserved outcomes or, as it is sometimes called, the conditional probability of a new species, is studied. Good's estimator, which is essentially the same as Robbins' estimator, namely the number of singleton species observed divided by the sample size, is studied from a decision theory point of view. The results obtained are as follows: (1) When the total number of different species is assumed bounded by some known number, Good's and Robbins' estimators are inadmissible for squared error loss. (2) If the number of different species can be infinite, Good's and Robbins' estimators are admissible for squared error loss. (3) Whereas Robbins' estimator is a UMVUE for theunconditional probability of a new species obtained in one extra sample point, Robbins' estimator is not a uniformly minimum mean squared error unbiased estimator of the conditional probability of a new species. This answers a question raised by Robbins. (4) It is shown that for Robbins' model and squared error loss, there are admissible Bayes estimators which do not depend only on a minimal sufficient statistic. A discussion of interpretations and significance of the results is offered. Research supported by NSF Grant DMS-88-22622.