Mean square error matrix comparisons of estimators in linear regression

The aim of this paper is to provide criteria which allow to compare two estimators of the parameter vector in the linear regression model with respect to their mean square error matrices, where the main interest is focussed on the case when the difference of the covariance matrices is singular. The results obtained are applied to equality restricted and pretest estimators.     

Improved estimators for the common mean and ordered means of two normal distributions with ordered variances

We first consider the problem of estimating the common mean of two normal distributions with unknown ordered variances. We give a broad class of estimators which includes the estimators proposed by Nair (1982) and Elfessi et al. (1992) and show that the estimators stochastically dominate the estimators which do not take into account the order restriction on variances, including the one given by Graybill and Deal (1959). Then we propose a broad class of individual estimators of two ordered means when unknown variances are ordered. We show that in estimating the mean with larger variance, estimators which do not take into account the order restriction on variances are stochastically dominated by the proposed class of estimators which take into account both order restrictions. However, in estimating the mean with smaller variance, similar improvement is not possible even in terms of mean squared error. We also show a domination result in the simultaneous estimation problem of two ordered means. Further, improving upon the unbiased estimators of the two means is discussed.     

A monte carlo study on the robustness of four MANOVA criterion tests

Four MANOVA tests (Wilk's Lambda, Roy's Largest Root Test, the Hotelling-Lawley Trace and the Pillai-Bartlett Trace) were studied when restricted sample data were drawn from normal populations. Robustness was compared by examining bias at critical points and fluctuations in the standard error of the empirical distributions. The Wilk's Lambda statistic was found to be the least affected by the restricted sampling.     

ONE-SIDED ESTIMATING AND TESTING PROBLEMS FOR LOCATION MODELS FROM GROUPED SAMPLES

This article deals with one-sided problems for location models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjointed intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. This article considers one-sided estimating and testing problems for location models. Some methods for computing the maximum likelihood estimates of the parameters subject to order restrictions are proposed and a numerical example by the method is given.     

Maximum and Minimum Prediction Variance for Spherical and Cuboidal Split Plot Designs

The impact of restricted randomization on the information matrix has created challenges for the computation of design optimality criteria. This article focuses on the computation of the maximum and minimum prediction variance for Central Composite (CCD) and Box–Behnken (BBD) split plot designs (SPD). The approach is to analytically determine the exact maximum and minimum prediction variance for both spherical and cuboidal second-order SPD. A particular feature of these analytical functions is that they are functions of the design parameters. Finally, the application of these analytical functions is demonstrated for a CCD SPD.     

QUALITY IMPROVEMENT FROM DISASSEMBLY-REASSEMBLY EXPERIMENTS

The goal of achieving high quality products has led to an emphasis on reducing variation in performance characteristics. It may often happen that one of the product's components is responsible for much of the observed variation. This research is stimulated by the problem of detecting a component that impairs quality by systematically inflating the variance in a product that is assembled from “interchangeable components.” We consider the class of “disassembly-reassembly” experiments, in which components are swapped among assemblies. The specific units used in the experiment are sampled from a large population of units, so it is natural to measure the influence of each factor by its variance component. We present the model for these experiments as a special case of the mixed linear model, compare several estimators for the variance components and consider the problem of testing hypotheses to identify troublesome components.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Estimation of a distribution function by extrapolating upper and lower bounds

Constrained optimization is proposed as a practical solution to the problem of estimating a distribution function at each point in a given set from monotone sequences of upper and lower bounds. The proposed solution employs least absolute value estimation and, hence, has a linear programming formulation. The special structure inherent in this formulation is exploited and an efficient computational method is discussed. The procedure is illustrated by two examples.     

Parameter estimation under orthant restrictions

The authors consider the estimation of linear functions of a multivariate parameter under orthant restrictions. These restrictions are considered both for location models and for the Poisson distribution. For these models, situations are characterized for which the restricted maximum likelihood estimator dominates the unrestricted one for the estimation of any linear function of the parameter. The results obtained point directly to the importance of the dimension of the parameter space, the central direction of the cone and its vertex in these cases. Special attention is given to examples, such as the one‐way analysis of variance, where the estimation of individual interesting linear functions of the parameter, as the coordinates and the differences between them, is also treated.     

Inference for multivariate normal hierarchical models

This paper provides a new method and algorithm for making inferences about the parameters of a two-level multivariate normal hierarchical model. One has observed J p -dimensional vector outcomes, distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices. At level 2, the unknown mean vectors also have normal distributions, with common unknown covariance matrix A and with means depending on known covariates and on unknown regression coefficients. The algorithm samples independently from the marginal posterior distribution of A by using rejection procedures. Functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of A drawn. This estimation accounts for the uncertainty in A , unlike standard restricted maximum likelihood empirical Bayes procedures. It is based on independent draws from the exact posterior distributions, unlike Gibbs sampling. The procedure is demonstrated for profiling hospitals based on patients' responses concerning p =2 types of problems (non-surgical and surgical). The frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting.