On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: X_i∼N(θ_i,σ²),i=1,…,p, we consider estimating the vector θ=(θ₁,…,θ_p)^′ with loss ‖d−θ‖² under the constraint , with known τ₁,…,τ_p,σ²,m. In comparing the risk performance of Bayesian estimators δ_α associated with uniform priors on spheres of radius α centered at (τ₁,…,τ_p) with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on α for δ_α to dominate . Large sample determinations of these conditions are provided. Both cases where all such δ_α’s and cases where no such δ_α’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ_m dominates if and only if m≤k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators δ_π with π spherically symmetric and supported on the parameter space dominate whenever m≤c₁(p) with .     

Sieve maximum likelihood estimation for doubly semiparametric zero-inflated Poisson models

For nonnegative measurements such as income or sick days, zero counts often have special status. Furthermore, the incidence of zero counts is often greater than expected for the Poisson model. This article considers a doubly semiparametric zero-inflated Poisson model to fit data of this type, which assumes two partially linear link functions in both the mean of the Poisson component and the probability of zero. We study a sieve maximum likelihood estimator for both the regression parameters and the nonparametric functions. We show, under routine conditions, that the estimators are strongly consistent. Moreover, the parameter estimators are asymptotically normal and first order efficient, while the nonparametric components achieve the optimal convergence rates. Simulation studies suggest that the extra flexibility inherent from the doubly semiparametric model is gained with little loss in statistical efficiency. We also illustrate our approach with a dataset from a public health study.     

Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe

Summary The asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the Malliavin calculus. From this the second order efficiency of the maximum likelihood estimator is proved.The research was supported in part by Grant-in-Aid for Encouragement of Young Scientists from the Ministry of Education, Science and Culture     

Asymptotic properties of particle filter-based maximum likelihood estimators for state space models

We study the asymptotic performance of approximate maximum likelihood estimators for state space models obtained via sequential Monte Carlo methods. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimates are computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the log-likelihood function. Secondly, extensions of this approximation to the whole parameter space are formed by means of piecewise constant functions or B-spline interpolation, and approximate maximum likelihood estimates are obtained through maximization of the resulting functions. In this setting we formulate criteria for how to increase the number of particles and the resolution of the grid in order to produce estimates that are consistent and asymptotically normal.     

Asymptotic expansion of Bayes estimators for small diffusions

Summary Using the Malliavin calculus we derived asymptotic expansion of the distributions of the Bayes estimators for small diffusions. The second order efficiency of the Bayes estimator is proved.     

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.     

Sensitivity of GLS estimators in random effects models

This paper studies the sensitivity of random effects estimators in the one-way error component regression model. Maddala and Mount (1973) [6] give simulation evidence that in random effects models the properties of the feasible GLS estimator are not affected by the choice of the first-step estimator used for the covariance matrix. Taylor (1980) [8] gives a theoretical example of this effect. This paper provides a reason for this in terms of sensitivity. The properties of are transferred via an uncorrelated (and independent under normality) link, called sensitivity. The sensitivity statistic counteracts the improvement in . A Monte Carlo experiment illustrates the theoretical findings.     

Asymptotic Bayesian structure learning using graph supports for Gaussian graphical models

The theory of Gaussian graphical models is a powerful tool for independence analysis between continuous variables. In this framework, various methods have been conceived to infer independence relations from data samples. However, most of them result in stepwise, deterministic, descent algorithms that are inadequate for solving this issue. More recent developments have focused on stochastic procedures, yet they all base their research on strong a priori knowledge and are unable to perform model selection among the set of all possible models. Moreover, convergence of the corresponding algorithms is slow, precluding applications on a large scale. In this paper, we propose a novel Bayesian strategy to deal with structure learning. Relating graphs to their supports, we convert the problem of model selection into that of parameter estimation. Use of non-informative priors and asymptotic results yield a posterior probability for independence graph supports in closed form. Gibbs sampling is then applied to approximate the full joint posterior density. We finally give three examples of structure learning, one from synthetic data, and the two others from real data.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

Moderate deviations of generalized method of moments and empirical likelihood estimators

This paper studies moderate deviation behaviors of the generalized method of moments and generalized empirical likelihood estimators for generalized estimating equations, where the number of equations can be larger than the number of unknown parameters. We consider two cases for the data generating probability measure: the model assumption and local contaminations or deviations from the model assumption. For both cases, we characterize the first-order terms of the moderate deviation error probabilities of these estimators. Our moderate deviation analysis complements the existing literature of the local asymptotic analysis and misspecification analysis for estimating equations, and is useful to evaluate power and robust properties of statistical tests for estimating equations which typically involve some estimators for nuisance parameters.     

Who's afraid of reduced-rank parameterizations of multivariate models? Theory and example

Reduced-rank restrictions can add useful parsimony to coefficient matrices of multivariate models, but their use is limited by the daunting complexity of the methods and their theory. The present work takes the easy road, focusing on unifying themes and simplified methods. For Gaussian and non-Gaussian (GLM, GAM, mixed normal, etc.) multivariate models, the present work gives a unified, explicit theory for the general asymptotic (normal) distribution of maximum likelihood estimators (MLE). MLE can be complex and computationally hard, but we show a strong asymptotic equivalence between MLE and a relatively simple minimum (Mahalanobis) distance estimator. The latter method yields particularly simple tests of rank, and we describe its asymptotic behavior in detail. We also examine the method's performance in simulation and via analytical and empirical examples.     

Efficient estimators and LAN in canonical bivariate POT models

Bivariate generalized Pareto distributions (GPs) with uniform margins are introduced and elementary properties such as peaks-over-threshold (POT) stability are discussed. A unified parameterization with parameter ?∈[0,1] of the GPs is provided by their canonical parameterization. We derive efficient estimators of ? and of the dependence function of the GP in various models and establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of the observations. From this result we can deduce that the estimator of ? suggested by Falk and Reiss (2001, Statist. Probab. Lett. 52, 233-242) is not efficient, whereas a modification actually is.     

Power of edge exclusion tests for graphical log-linear models

Asymptotic multivariate normal approximations to the joint distributions of edge exclusion test statistics for saturated graphical log-linear models, with all variables binary, are derived. Non-signed and signed square-root versions of the likelihood ratio, Wald and score test statistics are considered. Non-central chi-squared approximations are also considered for the non-signed versions of the test statistics. Simulation results are used to assess the quality of the proposed approximations. These approximations are used to estimate the overall power of edge exclusion tests. Power calculations are illustrated using data on university admissions.     

Depth estimators and tests based on the likelihood principle with application to regression

We investigate depth notions for general models which are derived via the likelihood principle. We show that the so-called likelihood depth for regression in generalized linear models coincides with the regression depth of Rousseeuw and Hubert (J. Amer. Statist. Assoc. 94 (1999) 388) if the dependent observations are appropriately transformed. For deriving tests, the likelihood depth is extended to simplicial likelihood depth. The simplicial likelihood depth is always a U-statistic which is in some cases not degenerated. Since the U-statistic is degenerated in the most cases, we demonstrate that nevertheless the asymptotic distribution of the simplicial likelihood depth and thus asymptotic α-level tests for general types of hypotheses can be derived. The tests are distribution-free. We work out the method for linear and quadratic regression.     

Admissibility of linear estimators with respect to inequality constraints under matrix loss function

In this paper we investigate the admissibility of linear estimators in the multivariate linear model with respect to inequality constraints under matrix loss function. The necessary and sufficient conditions for a linear estimator to be admissible in the class of homogeneous linear estimators and the class of inhomogeneous linear estimators are obtained, respectively.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

Efficiencies of tests and estimators forp-order autoregressive processes when the error distribution is nonnormal

Summary We considerpth order autoregressive time series where the shocks need not be normal. By employing the concept of contiguity, we obtain the sysmptotic power for tests of hypothesis concerning the autoregressive parameters. Our approach allows consideration of the double exponential and other thicker-tailed distributions for the shocks. We derive a new result in the contiguity framework that leads directly to an expression for the Pitman efficiencies of tests as well as estimators. The numerical values of the efficiencies suggest a lack of robustness for the normal theory least squares estimators when the shock distribution is thick tailed or an outlier prone mixed normal. An important alternative test statistic is proposed that competes with the normal theory tests. This research was supported by the Office of Naval Research under Grant No. N00014-78-C-0722 and by the Army Research Office.     

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.     

Modifying estimators of ordered positive parameters under the Stein loss

This paper treats the problem of estimating positive parameters restricted to a polyhedral convex cone which includes typical order restrictions, such as simple order, tree order and umbrella order restrictions. In this paper, two methods are used to show the improvement of order-preserving estimators over crude non-order-preserving estimators without any assumption on underlying distributions. One is to use Fenchel’s duality theorem, and then the superiority of the isotonic regression estimator is established under the general restriction to polyhedral convex cones. The use of the Abel identity is the other method, and we can derive a class of improved estimators which includes order-statistics-based estimators in the typical order restrictions. When the underlying distributions are scale families, the unbiased estimators and their order-restricted estimators are shown to be minimax. The minimaxity of the restrictedly generalized Bayes estimator against the prior over the restricted space is also demonstrated in the two dimensional case. Finally, some examples and multivariate extensions are given.