This article gives a theoretical treatment of the asymptotics of the L₁ error of a model-based estimate of a density f(x|θ) on a finite dimensional Euclidean space ℛ^k.

The dimension p of the parameter vector θ is considered arbitrary but fixed in Section 2. Two theorems in Section 2 lay out the weak limits of a suitably scaled L₁ error, with a general estimating sequence θ̂ and a general family of smooth densities f(x|θ) dominated by some σ-finite measure, the discrete case included. We show that the L₁ error converges at the coarsest rate corresponding to the different coordinates of the parameter vector θ. Four applications are detailed in Section 3, a special one being a new confidence interval for a Poisson mean.

Section 4 considers the high and the ultra high dimensional case, where p grows with n. The exact critical growth rate for p when maximum likelihood starts to falter is derived. Maximum likelihood is shown to exhibit a trichotomy of behavior; the desired behavior below the threshold, problematic behavior at the threshold, and disastrous behavior above the threshold.

It is then shown that regularization, if coupled with the right amount of sparsity, can return consistent density estimation, even at the best possible n^−1/2 rate. We give a complete description of the limiting behavior of the regularized density estimate under different sparsity conditions. Section 4 is specialized to the Gaussian case due to its special importance and well known links to function estimation.

The estimation of a linear combination of several restricted location parameters is addressed from a decision-theoretic point of view. A bench-mark estimator of the linear combination is an unbiased estimator, which is minimax, but inadmissible relative to the mean squared error. An interesting issue is what is a prior distribution which results in the generalized Bayes and minimax estimator. Although it seems plausible that the generalized Bayes estimator against the uniform prior over the restricted space should be minimax, it is shown to be not minimax when the number of the location parameters, k, is more than or equal to three, while it is minimax for k=1. In the case of k=2, a necessary and sufficient condition for the minimaxity is given, namely, the minimaxity depends on signs of coefficients of the linear combination. When the underlying distributions are normal, we can obtain a prior distribution which results in the generalized Bayes estimator satisfying minimaxity and admissibility. Finally, it is demonstrated that the estimation of ratio of normal variances converges to the estimation of difference of the normal positive means, which gives a motivation of the issue studied here.

This paper considers estimation of the predictive density for a normal linear model with unknown variance under α-divergence loss for −1≤α≤1. We first give a general canonical form for the problem, and then give general expressions for the generalized Bayes solution under the above loss for each α. For a particular class of hierarchical generalized priors studied in Maruyama and Strawderman (2005, 2006) for the problems of estimating the mean vector and the variance respectively, we give the generalized Bayes predictive density. Additionally, we show that, for a subclass of these priors, the resulting estimator dominates the generalized Bayes estimator with respect to the right invariant prior, i.e., the best (fully) equivariant minimax estimator when α=1.

Consider a balanced one way layout without a normality assumption. That is, assume each population has an unknown translation parameter and has a continuous distribution. We wish to test the C^k₂ pairwise differences in translation parameters assuming there are k populations.We propose a multiple testing method based on ranks that is analogous to the method developed for testing pairwise contrasts among means by Cohen, Sackrowitz and Chen (2010) [3] (CSC). This latter method has an intuitive and important practical property not shared by most multiple testing methods. Namely, for each individual pairwise hypothesis relevant acceptance sections are intervals. Furthermore, the CSC method is shown to do well in terms of power compared to competitive methods. In the nonparametric setting the analogous procedure has the desirable interval property and also stacks up well in a comparative simulation study.

Procedures to reduce nonconformity in interlaboratory studies by shrinking multivariate data toward a consensus matrix-weighted mean are discussed. Some of them are shown to have a smaller quadratic risk than the ordinary least squares rule. Bayes procedures and shrinkage estimators in random effects models are also considered. The results are illustrated by an example of collaborative studies.

This paper provides an extension of earlier results on minimax estimation of a high-dimensional sparse vector to even more sparse vectors. Specifically, an approximation of the minimax ℓ_q risk is obtained and threshold estimators are proved to achieve the minimax risk within an infinitesimal fraction in all small ℓ_p balls.

For the problem of estimating a general loss of the form c(‖x−θ‖²), Stein’s identity is particularly relevant in deriving unbiased estimators of loss when x is used as an estimate of θ and is distributed as $\mathcal{N}_{p}(\theta,I)$, and when c is the identity function. In [3], Fourdrinier and Lepelletier show that extensions to other distributions (actually, to spherically symmetric distributions) and to general functions c are conceivable, but through another approach involving a Green’s formula. Somewhat surprisingly, the statistical context induces an unusual weak functional framework. The main goal of this paper is to present such an analytic context.

We consider estimation problem of a normal quantile μ+ησ. For the scale invariant squared error loss and unrestricted values of the population mean and standard deviation μ and σ, [13] established the inadmissibility of the MRE estimator for η≠0. In this paper, we explore: (i) the impact of the loss with the study of scale invariant absolute value loss, and (ii) situations where there is a parameter space restriction of a lower bounded mean μ. We establish

(i) the inadmissibility of the MRE estimator of μ+ησ; η≠0; under scale invariant absolute value loss;

(ii) the inadmissibility of the Generalized Bayes estimator of μ+ησ; η>0; under scale invariant squared error loss, associated with the prior measure 1_(0,∞)(μ)1_(0,∞)(σ) which represents the truncation of the usual non-informative prior measure onto the restricted parameter space.

Both of these results are obtained through a conditional risk analysis and may be viewed as extensions of [13]. Finally, we provide further applications to two-sample problems under the presence of the additional information of ordered means.

We evaluate priors by the second order asymptotic behaviour of the corresponding estimators. Under certain regularity conditions, the risk differences between efficient estimators of parameters taking values in a domain D, an open connected subset of R^d, are asymptotically expressed as elliptic differential forms depending on the asymptotic covariance matrix V. Each efficient estimator has the same asymptotic risk as a “local Bayes” estimate corresponding to a prior density p. The asymptotic decision theory of the estimators identifies the smooth prior densities as admissible or inadmissible, according to the existence of solutions to certain elliptic differential equations. The prior p is admissible if the quantity pV is sufficiently small near the boundary of D. We exhibit the unique admissible invariant prior for V=I, D=R^d−{0}. A detailed example is given for a normal mixture model.

Some Bayesian agents must produce a joint estimator of the mean of an exponentially distributed random variable S from a sample of realizations S. Their priors may differ but they have the same utility function. For the case of two agents, the Pareto efficient boundary of the utility set generated by the class of all non-randomized linear estimation rules is explored in this paper. Conditions are given that make those rules G-complete within the class of non-randomized linear estimators, meaning that optimum non-random estimators can be found on the Pareto boundary thereby providing a basis for a meaningful consensus.

Using a Bayesian model with a class of hierarchically specified scale-mixture-of-normals priors as motivation, we consider a generalization of the grouped LASSO in which an additional penalty is placed on the penalty parameter of the L₂ norm. We show that the resulting MAP estimator obtained by jointly minimizing the corresponding objective function in both the mean and penalty parameter is a thresholding estimator that generalizes (i) the grouped lasso estimator of Yuan and Lin (2006) and (ii) the univariate minimax concave penalization procedure of Zhang (2010) to the setting of a vector of parameters. An exact formula for the risk and a corresponding SURE formula are obtained for the proposed class of estimators.

A new universal threshold is proposed under appropriate sparsity assumptions; in combination with the proposed class of estimators, we subsequently obtain a new and interesting motivation for the class of positive part estimators. In particular, we establish that the original positive part estimator corresponds to a suboptimal choice of this thresholding parameter. Numerical comparisons between the proposed class of estimators and the positive part estimator show that the former can achieve further, significant reductions in risk near the origin.

This manuscript examines this task of approximating significance levels for a test of symmetry in square contingency tables. The null sampling distribution of this test statistic is the same as that of the sum of squared independent centered binomial random variables, weighted by their separate sample size; each of these variables may be taken to have success probability half. This manuscript applies an existing asymptotic correction to the standard chi-squared approximation to the distribution of the quadratic form of a random vector confined to a multivariate lattice, when the quadratic form is formed from the inverse variance matrix of the random vector. This manuscript also investigates non-asymptotic corrections to approximations to this distribution, when the separate binomial sample sizes are small.

Consider the problem of estimating a response function which depends upon a non-stochastic independent variable under our control. The data are independent Bernoulli random variables where the probabilities of success are given by the response function at the chosen values of the independent variable. Here we present a nonparametric Bayesian method for estimating the response function. The only prior information assumed is that the response function can be well approximated by a mixture of step functions.

This article proposes an alternative approach to incorporate information from observed data with its corresponding prior information using a recipe developed for combining confidence distributions. The outcome function is called a CD posterior, an alternative to Bayes posterior, which is shown here to have the same coverage property as the Bayes posterior. This approach to incorporating a prior distribution has a great advantage that it does not require any prior on nuisance parameters. It also can ease the computational burden which a typical Bayesian analysis endures. An error bound is established on the CD-posterior when there is an error in prior specification.

The Bayesian linear model framework has become an increasingly popular building block in regression problems. It has been shown to produce models with good predictive power and can be used with basis functions that are nonlinear in the data to provide flexible estimated regression functions. Further, model uncertainty can be accounted for by Bayesian model averaging. We propose a simpler way to account for model uncertainty that is based on generalized ridge regression estimators. This is shown to predict well and to be much more computationally efficient than standard model averaging methods. Further, we demonstrate how to efficiently mix over different sets of basis functions, letting the data determine which are most appropriate for the problem at hand.

Ensemble classification methods like Random Forest are powerful and versatile classifiers. We explore variations in the ensemble approach and demonstrate the strong performance of ensemble versions of Linear Discriminant Analysis (LDA) variants such as LDA-PCA (LDA after a Principal Components Analysis step to reduce dimensionality) and LASSO in situations characterized by a huge number of features and a small number of samples such as DNA microarray data. We also demonstrate the value of enriching the ensembles with features that are most likely to be informative in situations where only a very small percentage of the features actually carries classification information. Notably, in the case studies we analyzed, the enriched ensemble procedure with LDA-PCA as base classifier had a misclassification rate that was essentially half that observed with Random Forest.

A hunting program was instituted in a previously un-hunted 836-ha county park in New Jersey for white-tailed deer (Odocoileus virginianus) in 1994. Harvested deer were aged and weighed. The reproductive tracts of female deer were examined and the number of embryos counted. Hierarchical Bayes modeling was used to examine whether or not compensatory reproduction was occurring in the park. Compensatory reproduction is an issue which is frequently raised by anti-hunting groups.