Improper priors for translation families

We consider the problem of inference when sampling from a translation family with an improper prior. Properties of the formal Bayes inference will be studied. We give conditions (on the prior and/or the family) guaranteeing the HS-coherence (see Heath and Sudderth, Ann. Statist. 6 (1978), 333–345) of the formal Bayes posterior. Since HS-coherence is equivalent to being a posterior of a finitely additive prior, all coherence results imply the existence of a finitely additive prior which has the formal Bayes inference as a posterior.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

Nonparametric linear bayes estimation of survival curves from incomplete observations

A linear Bayes estimator of a survival curve is derived.The estimator has a relatively simple interpretation as a Kaplan-Meier estimator based on an augemented data base - prior information plus sampling information.It is Bayes if the prior is a Dirichlet process, and otherwise an approximation to the Bayes rule against any prior.     

Improved estimation of the mean in one-parameter exponential families with known coefficient of variation

The value for which the mean square error of a biased estimatoraT for the mean μ is less than the variance of an unbiased estimatorT is derived by minimizingMSE(aT). The resulting optimal value is 1/[1+c(n)v ²], wherev=σ/μ, is the coefficient of variation. WhenT is the UMVUE , thenc(n)=1/n, and the optimal value becomes 1/(n+v ²) (Searls, 1964). Whenever prior information about the size ofv is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic function, it becomes easy to determinev.     

Heteroskedasticity-Robust Inference in Linear Regressions

The assumption that all errors share the same variance (homoskedasticity) is commonly violated in empirical analyses carried out using the linear regression model. A widely adopted modeling strategy is to perform point estimation by ordinary least squares and then perform testing inference based on these point estimators and heteroskedasticity-consistent standard errors. These tests, however, tend to be size-distorted when the sample size is small and the data contain atypical observations. Furno (1996 Furno , M. ( 1996 ). Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator . Journal of Statistical Computation and Simulation 54 : 115 – 128 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested performing point estimation using a weighted least squares mechanism in order to attenuate the effect of leverage points on the associated inference. In this article, we follow up on her proposal and define heteroskedasticity-consistent covariance matrix estimators based on residuals obtained using robust estimation methods. We report Monte Carlo simulation results (size and power) on the finite sample performance of different heteroskedasticity-robust tests. Overall, the results favor inference based on HC0 tests constructed using robust residuals.     

Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.     

Simultaneous marginal survival estimators when doubly censored data is present

A doubly censoring scheme occurs when the lifetimes T being measured, from a well-known time origin, are exactly observed within a window [L, R] of observational time and are otherwise censored either from above (right-censored observations) or below (left-censored observations). Sample data consists on the pairs (U, δ) where U = min{R, max{T, L}} and δ indicates whether T is exactly observed (δ = 0), right-censored (δ = 1) or left-censored (δ = −1). We are interested in the estimation of the marginal behaviour of the three random variables T, L and R based on the observed pairs (U, δ). We propose new nonparametric simultaneous marginal estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} for the survival functions of T, L and R, respectively, by means of an inverse-probability-of-censoring approach. The proposed estimators [^(S)]_T, [^(S)]_L{\hat S_{T}, \hat S_{L}} and [^(S)]_R{\hat S_{R}} are not computationally intensive, generalize the empirical survival estimator and reduce to the Kaplan-Meier estimator in the absence of left-censored data. Furthermore, [^(S)]_T{\hat S_{T}} is equivalent to a self-consistent estimator, is uniformly strongly consistent and asymptotically normal. The method is illustrated with data from a cohort of drug users recruited in a detoxification program in Badalona (Spain). For these data we estimate the survival function for the elapsed time from starting IV-drugs to AIDS diagnosis, as well as the potential follow-up time. A simulation study is discussed to assess the performance of the three survival estimators for moderate sample sizes and different censoring levels.     

Nonparametric empirical Bayes for the Dirichlet process mixture model

The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G ₀. Most previous applications have either fixed G ₀ as a member of a parametric family or treated G ₀ in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G ₀. We combine this method with a technique for estimating α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G ₀ are of intrinsic interest, or where the structure of G ₀ cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.     

Inference Under Heteroskedasticity and Leveraged Data

We evaluate the finite-sample behavior of different heteros-ke-das-ticity-consistent covariance matrix estimators, under both constant and unequal error variances. We consider the estimator proposed by Halbert White (HC0), and also its variants known as HC2, HC3, and HC4; the latter was recently proposed by Cribari-Neto (2004 Cribari-Neto , F. ( 2004 ). Asymptotic inference under heteroskedasticity of unknown form . Computat. Statist. Data Anal. 45 : 215 – 233 .[Crossref], [Web of Science ®] , [Google Scholar]). We propose a new covariance matrix estimator: HC5. It is the first consistent estimator to explicitly take into account the effect that the maximal leverage has on the associated inference. Our numerical results show that quasi-t inference based on HC5 is typically more reliable than inference based on other covariance matrix estimators.     

Bounds for the Bayes Error in Classification: A Bayesian Approach Using Discriminant Analysis

We study two of the classical bounds for the Bayes error P _e , Lissack and Fu’s separability bounds and Bhattacharyya’s bounds, in the classification of an observation into one of the two determined distributions, under the hypothesis that the prior probability χ itself has a probability distribution. The effectiveness of this distribution can be measured in terms of the ratio of two mean values. On the other hand, a discriminant analysis-based optimal classification rule allows us to derive the posterior distribution of χ, together with the related posterior bounds of P _e . Research partially supported by NSERC grant A 9249 (Canada). The authors wish to thank two referees, for their very pertinent comments and suggestions, that have helped to improve the quality and the presentation of the paper, and we have, whenever possible, addressed their concerns.     

Quadratic subspaces and construction of Bayes invariant quadratic estimators of variance components in mixed linear models

Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic estimators of variance components in normal mixed linear models of the form where the matrices V _i , 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V _k is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV ₁ M,...,MV _k-1 M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms. We also consider a class of matrices sharing some of the symmetries common to MV ₁ M,...,MV _k-1 M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices MV ₁ M,...,MV _k-1 M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods. Dedicated to the memory of Professor Stanisław Gnot.     

Investigating the sensitivity of Gaussian processes to the choice of their correlation function and prior specifications

A Gaussian process (GP) can be thought of as an infinite collection of random variables with the property that any subset, say of dimension n, of these variables have a multivariate normal distribution of dimension n, mean vector β and covariance matrix Σ [O'Hagan, A., 1994, Kendall's Advanced Theory of Statistics, Vol. 2B, Bayesian Inference (John Wiley & Sons, Inc.)]. The elements of the covariance matrix are routinely specified through the multiplication of a common variance by a correlation function. It is important to use a correlation function that provides a valid covariance matrix (positive definite). Further, it is well known that the smoothness of a GP is directly related to the specification of its correlation function. Also, from a Bayesian point of view, a prior distribution must be assigned to the unknowns of the model. Therefore, when using a GP to model a phenomenon, the researcher faces two challenges: the need of specifying a correlation function and a prior distribution for its parameters. In the literature there are many classes of correlation functions which provide a valid covariance structure. Also, there are many suggestions of prior distributions to be used for the parameters involved in these functions. We aim to investigate how sensitive the GPs are to the (sometimes arbitrary) choices of their correlation functions. For this, we have simulated 25 sets of data each of size 64 over the square [0, 5]×[0, 5] with a specific correlation function and fixed values of the GP's parameters. We then fit different correlation structures to these data, with different prior specifications and check the performance of the adjusted models using different model comparison criteria.     

Pooling multivariate data under W, LR and LM tests

Two independent random samples are drawn from two multivariate normal populations with mean vectors μ1 and μ2 and a common variance-covariance matrix Σ. Ahmed and Saleh (1990) considered preliminary test maximum likelihood estimator (PMLTE) for estimating μ1 based on the Hotelling's T _N ², when it is suspected that μ1=μ2. In this paper, the PTMLE based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. Using the quadratic risk function, the conditions of superiority of the proposed estimator for departure parameter are derived. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the PTMLE based on W test produces the highest minimum guaranteed efficiencies compared to UMLE among the three test procedures.     

A note on simplicial depth function

We investigate the behaviour of simplicial depth under the perturbation (1−ε)F+ε δ _z , where F is a p-dimensional probability distribution and δ _z is the point-mass distribution concentrated at the point z. The influence function of simplicial depth at the point x, up to a scalar multiplier, turns out to be the difference between the conditional depth, given that one of the vertices of the random simplex is fixed at the position z, and the unconditional depth. The scalar multiplier is p+1, which suggests that simplicial depth can be more sensitive to perturbations as the dimensionality grows higher. The geometrical properties of the influence function give new insight into the observed behaviour of simplicial depth and its relation with halfspace depth. The behaviour of the perturbed simplicial median is also investigated.     

Near Optimal Prediction from Relevant Components

Abstract. The random x regression model is approached through the group of rotations of the eigenvectors for the x ‐covariance matrix together with scale transformations for each of the corresponding regression coefficients. The partial least squares model can be constructed from the orbits of this group. A generalization of Pitman's Theorem says that the best equivariant estimator under a group is given by the Bayes estimator with the group's invariant measure as the prior. A straightforward application of this theorem turns out to be impossible since the relevant invariant prior leads to a non‐defined posterior. Nevertheless we can devise an approximate scale group with a proper invariant prior leading to a well‐defined posterior distribution with a finite mean. This Bayes estimator is explored using Markov chain Monte Carlo technique. The estimator seems to require heavy computations, but can be argued to have several nice properties. It is also a valid estimator when p>n.     

Posterior Bayes factor analysis for an exponential regression model

In the exponential regression model, Bayesian inference concerning the non-linear regression parameter has proved extremely difficult. In particular, standard improper diffuse priors for the usual parameters lead to an improper posterior for the non-linear regression parameter. In a recent paper Ye and Berger (1991) applied the reference prior approach of Bernardo (1979) and Berger and Bernardo (1989) yielding a proper informative prior for . This prior depends on the values of the explanatory variable, goes to 0 as goes to 1, and depends on the specification of a hierarchical ordering of importance of the parameters.This paper explains the failure of the uniform prior to give a proper posterior: the reason is the appearance of the determinant of the information matrix in the posterior density for . We apply the posterior Bayes factor approach of Aitkin (1991) to this problem; in this approach we integrate out nuisance parameters with respect to their conditional posterior density given the parameter of interest. The resulting integrated likelihood for requires only the standard diffuse prior for all the parameters, and is unaffected by orderings of importance of the parameters. Computation of the likelihood for is extremely simple. The approach is applied to the three examples discussed by Berger and Ye and the likelihoods compared with their posterior densities.     

A family of admissible minimax estimators of the mean of a multivariate,normal distribution

Let X has a p-dimensional normal distribution with mean vector θ and identity covariance matrix I. In a compound decision problem consisting of squared-error estimation of θ, Strawderman (1971) placed a Beta (α, 1) prior distribution on a normal class of priors to produce a family of Bayes minimax estimators. We propose an incomplete Gamma(α, β) prior distribution on the same normal class of priors to produce a larger family of Bayes minimax estimators. We present the results of a Monte Carlo study to demonstrate the reduced risk of our estimators in comparison with the Strawderman estimators when θ is away from the zero vector.     

Bayesian Estimation of Change Point in Inverse Weibull Sequence

A sequence of independent lifetimes X ₁,…, X _m , X _m+1,…, X _n were observed from inverse Weibull distribution with mean stress θ₁ and reliability R ₁(t ₀) at time t ₀ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in mean stress θ₁ and in reliability R ₂(t ₀) at time t ₀. The Bayes estimators of m, R ₁(t ₀) and R ₂(t ₀) are derived when a poor and a more detailed prior information is introduced into the inferential procedure. The effects of correct and wrong prior information on the Bayes estimators are studied.