Hierarchical Bayes variable selection and microarray experiments

Hierarchical and empirical Bayes approaches to inference are attractive for data arising from microarray gene expression studies because of their ability to borrow strength across genes in making inferences. Here we focus on the simplest case where we have data from replicated two colour arrays which compare two samples and where we wish to decide which genes are differentially expressed and obtain estimates of operating characteristics such as false discovery rates. The purpose of this paper is to examine the frequentist performance of Bayesian variable selection approaches to this problem for different prior specifications and to examine the effect on inference of commonly used empirical Bayes approximations to hierarchical Bayes procedures. The paper makes three main contributions. First, we describe how the log odds of differential expression can usually be computed analytically in the case where a double tailed exponential prior is used for gene effects rather than a normal prior, which gives an alternative to the commonly used B-statistic for ranking genes in simple comparative experiments. The second contribution of the paper is to compare empirical Bayes procedures for detecting differential expression with hierarchical Bayes methods which account for uncertainty in prior hyperparameters to examine how much is lost in using the commonly employed empirical Bayes approximations. Third, we describe an efficient MCMC scheme for carrying out the computations required for the hierarchical Bayes procedures. Comparisons are made via simulation studies where the simulated data are obtained by fitting models to some real microarray data sets. The results have implications for analysis of microarray data using parametric hierarchical and empirical Bayes methods for more complex experimental designs: generally we find that the empirical Bayes methods work well, which supports their use in the analysis of more complex experiments when a full hierarchical Bayes analysis would impose heavy computational demands.     

The likelihood ratio and its applications in sequential analysis

The likelihood ratio is treated as a generalized Radon-Nikodym derivative for measures which need not be absolutely continuous. It is shown that a wide variety of identities and inequalities in sequential analysis follow directly from simple properties of likelihood ratios.     

On approximation of optimal stopping of bayesian sequential test for a normal mean

In this paper, we present a simple and direct approach in which supermartinagles are used to approximate the optimal stopping sets associated with the Bayesian sequential test for normal population means. Several conclusions are given. Project supported by the National Natural Science Foundation of China.     

A preliminary test procedure for the scale parameter of exponential distribution when the selection parameter is unknown

Summary A preliminary test estimator is considered for the scale parameter of the two-parameter exponential distribution with unknown selection parameter, where the distribution does not satisfy the regularity condition of Wilks' theorem—the density is not differentiable. A method of specifying the level of significance of the preliminary test based on is proposed AIC. This work was partly supported by Scientific Research Fund No. 58450058 from the Ministry of Education of Japan. The Institute of Statistical Mathematics     

Ordering and selecting extreme populations by means of entropies and divergences

This paper studies the simultaneous selection of extreme populations from a set of independent populations. Two types of subset selection rules for k populations are proposed and studied. The first type selects one subset of populations that should contain the population with the smallest, and another subset of populations that should contain the population with the largest, φ-entropy. The second type selects analogously, but in terms of the extreme ?-divergences with respect a known control population. Properties of the proposed procedures are stated and studied. Examples are presented in order to illustrate the results.     

On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding

We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu [K. Knight, W. Fu, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356–1378] and Fan and Li [J. Fan, R. Li, Variable selection via non-concave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (2001) 1348–1360]. We show that the distributions are typically highly non-normal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than n^−1/2 in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators’ distribution function is also provided.     

A Bahadur efficiency comparison between one and two sample rank statistics and their sequential rank statistic analogues

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist.9 424–436) and Lombard (1981, South African Statist. J.15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

Strong approximations of the Hoeffding,Blum, Kiefer,Rosenblatt multivariate empirical process

Hoeffding (Ann. Math. Statist. 1948) and Blum, Kiefer and Rosenblatt (Ann. Math. Statist. 1961) constructed distribution free tests of independence based on a multivariate empirical process. We establish strong invariance principles for the latter and also for appropriate functionals of it.     

On Stein’s lemma, dependent covariates and functional monotonicity in multi-dimensional modeling

Tracking the correct directions of monotonicity in multi-dimensional modeling plays an important role in interpreting functional associations. In the presence of multiple predictors, we provide empirical evidence that the observed monotone directions via parametric, nonparametric or semiparametric fit of commonly used multi-dimensional models may entirely violate the actual directions of monotonicity. This breakdown is caused primarily by the dependence structure of covariates, with negligible influence from the bias of function estimation. To examine the linkage between the dependent covariates and monotone directions, we first generalize Stein’s Lemma for random variables which are mutually independent Gaussian to two important cases: dependent Gaussian, and independent non-Gaussian. We show that in both two cases, there is an explicit one-to-one correspondence between the monotone directions of a multi-dimensional function and the signs of a deterministic surrogate vector. Moreover, we demonstrate that the second case can be extended to accommodate a class of dependent covariates. This generalization further enables us to develop a de-correlation transform for arbitrarily dependent covariates. The transformed covariates preserve modeling interpretability with little loss in modeling efficiency. The simplicity and effectiveness of the proposed method are illustrated via simulation studies and real data application.     

On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives

Based on a random sample from a population with (unknown) probability density f, this note exhibits a class of statistics f^(p) for each fixed integer $\text{p ≧ 0}$. It is shown that f^(p) are uniformly strongly consistent estimators of f^(p), the pth order derivative of f, if and only iff^(p)is bounded and uniformly continuous.     

On the Maximum Likelihood estimation of a linear structural relationship when the intercept is known

This paper considers the Maximum Likelihood (ML) estimation of the five parameters of a linear structural relationship y = α + βx when α is known. The parameters are β, the two variances of observation errors on x and y, the mean and variance of x. When the ML estimates of the parameters cannot be obtained by solving a simple simultaneous system of five equations, they are found by maximizing the likelihood function directly. Some asymptotic properties of the estimates are also obtained.     

Addendum to “a third-order optimum property of the maximum likelihood estimator”

A third-order optimum property of the maximum likelihood estimator is extended to not necessarily symmetric loss functions under an appropriate restriction on the class of competing estimators.     

On the ruin probability in a dependent discrete time risk model with insurance and financial risks

This paper considers the discrete-time risk model with insurance risk and financial risk in some dependence structures. Under assumptions that the insurance risks are heavy tailed (belong to the intersection of the long-tailed class and the dominatedly varying-tailed class) and the financial risks satisfy some moment conditions, the asymptotic and uniformly asymptotic relations for the finite-time and ultimate ruin probabilities are derived.     

Admissibility and minimaxity of Bayes estimators for a normal mean matrix

In some invariant estimation problems under a group, the Bayes estimator against an invariant prior has equivariance as well. This is useful notably for evaluating the frequentist risk of the Bayes estimator. This paper addresses the problem of estimating a matrix of means in normal distributions relative to quadratic loss. It is shown that a matricial shrinkage Bayes estimator against an orthogonally invariant hierarchical prior is admissible and minimax by means of equivariance. The analytical improvement upon every over-shrinkage equivariant estimator is also considered and this paper justifies the corresponding positive-part estimator preserving the order of the sample singular values.     

On the maximum total sample size of a group sequential test about bivariate binomial proportions

For testing “univariate” binomial proportions, it has been proven that, under mild conditions, there exist group sequential designs which satisfy the pre-specified Type I error and power of the single-stage design while the sample size is bounded above by that of the single-stage design (Kepner and Chang, 2003). In this article, we extend this result and prove the existence of such group sequential designs for various decision rules in the space of bivariate binomial variables. We also demonstrate how to obtain the actual group sequential designs for detecting changes in bivariate binomial variables.     

On the eigenvalues of a class of saddle point matrices

We study spectral properties of a class of block 2 × 2 matrices that arise in the solution of saddle point problems. These matrices are obtained by a sign change in the second block equation of the symmetric saddle point linear system. We give conditions for having a (positive) real spectrum and for ensuring diagonalizability of the matrix. In particular, we show that these properties hold for the discrete Stokes operator, and we discuss the implications of our characterization for augmented Lagrangian formulations, for Krylov subspace solvers and for certain types of preconditioners. The work of this author was supported in part by the National Science Foundation grant DMS-0207599 Revision dated 5 December 2005.     

Completeness of location families,translated moments,and uniqueness of charges

Summary A sufficient condition for statistical completeness of location families generated by a probability density in euclidean space is given. As an application, completeness of families generated by a symmetric stable law is proved. Our criterion, complementing a classical result of Wiener and recent work of Isenbeck and Rüschendorf, is in terms of regularity of the generating density and zerofreeness of its characteristic function. Its proof rests on a local version of the convolution theorem for Fourier transforms of tempered distributions. A more general version of the criterion is applicable to apparently different problems, as is illustrated by giving a simultaneous proof of a theorem on translated moments by P. Hall and a uniqueness result of M. Riesz in potential theory.