Proper Bayes minimax estimators of the normal mean matrix with common unknown variances

This paper addresses the problem of estimating a matrix of the normal means, where the variances are unknown but common. The approach to this problem is provided by a hierarchical Bayes modeling for which the first stage prior for the means is matrix-variate normal distribution with mean zero matrix and a covariance structure and the second stage prior for the covariance is similar to Jeffreys’ rule. The resulting hierarchical Bayes estimators relative to the quadratic loss function belong to a class of matricial shrinkage estimators. Certain conditions are obtained for admissibility and minimaxity of the hierarchical Bayes estimators.     

Selection of error probability laws by generalized modified profile likelihood

Although error probability law selection of models of location-scale forms is of importance in some sense, the commonly used model selection procedures, such as AIC and BIC, do not apply to it. By treating error probability law as a “parameter” of interest, location and scale as nuisance parameters, this paper proposes that generalized modified profile likelihood (GMPL), considered as a quasi-likelihood function of error probability law, be used to select the error probability laws. The GMPL method achieves minimax rate optimality and proves to be consistent. Simulations show its good performance for finite and even small samples. Note that it is straightforward to generalize the GMPL of location-scale models to various models of location-scale forms particularly including the various linear regression models and their variations, to select their error probability laws. The author believes that GMPL and its variations would be quite promising for various model selection problems.     

On multivariate quantile regression

To detect the dependence on the covariates in the lower and upper tails of the response distribution, regression quantiles are very useful tools in linear model problems with univariate response. We consider here a notion of regression quantiles for problems with multivariate responses. The approach is based on minimizing a loss function equivalent to that in the case of univariate response. To construct an affine equivariant notion of multivariate regression quantiles, we have considered a transformation retransformation procedure based on ‘data-driven coordinate systems’. We indicate some algorithm to compute the proposed estimates and establish asymptotic normality for them. We also, suggest an adaptive procedure to select the optimal data-driven coordinate system. We discuss the performance of our estimates with the help of a finite sample simulation study and to illustrate our methodology, we analyzed an interesting data-set on blood pressures of a group of women and another one on the dependence of sales performances on creative test scores.     

On an adaptive transformation–retransformation estimate of multivariate location

An affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied. The work is primarily motivated by earlier work on different versions of the multivariate median and their properties. We explore an issue related to efficiency and equivariance that was originally raised by Bickel and subsequently investigated by Brown and Hettmansperger. Our estimate has better asymptotic performance than the vector of co-ordinatewise medians when the variables are substantially correlated. The finite sample performance of the estimate is investigated by using Monte Carlo simulations. Some examples are presented to demonstrate the effect of the adaptive transformation–retransformation strategy in the construction of multivariate location estimates for real data.     

Practical Point Estimation from Higher-Order Pivots

Point estimators for a scalar parameter of interest in the presence of nuisance parameters can be defined as zero-level confidence intervals as explained in Skovgaard (1989). A natural implementation of this approach is based on estimating equations obtained from higher-order pivots for the parameter of interest. In this paper, generalising the results in Pace and Salvan (1999) outside exponential families, we take as an estimating function the modified directed likelihood. This is a higher-order pivotal quantity that can be easily computed in practice for a wide range of models, using recent advances in higher-order asymptotics (HOA, 2000). The estimators obtained from these estimating equations are a refinement of the maximum likelihood estimators, improving their small sample properties and keeping equivariance under reparameterisation. Simple explicit approximate versions of these estimators are also derived and have the form of the maximum likelihood estimator plus a function of derivatives of the loglikelihood function. Some examples and simulation studies are discussed for widely-used model classes.     

An efficient algorithm for estimating the parameters of superimposed exponential signals

An efficient computational algorithm is proposed for estimating the parameters of undamped exponential signals, when the parameters are complex valued. Such data arise in several areas of applications including telecommunications, radio location of objects, seismic signal processing and computer assisted medical diagnostics. It is observed that the proposed estimators are consistent and the dispersion matrix of these estimators is asymptotically the same as that of the least squares estimators. Moreover, the asymptotic variances of the proposed estimators attain the Cramer–Rao lower bounds, when the errors are Gaussian.