Shrinkage Estimation Under Multivariate Elliptic Models

The estimation of the location vector of a p-variate elliptically contoured distribution (ECD) is considered using independent random samples from two multivariate elliptically contoured populations when it is apriori suspected that the location vectors of the two populations are equal. For the setting where the covariance structure of the populations is the same, we define the maximum likelihood, Stein-type shrinkage and positive-rule shrinkage estimators. The exact expressions for the bias and quadratic risk functions of the estimators are derived. The comparison of the quadratic risk functions reveals the dominance of the Stein-type estimators if p ≥ 3. A graphical illustration of the risk functions under a “typical” member of the elliptically contoured family of distributions is provided to confirm the analytical results.     

On the efficiencies of some common quick estimators

The quick estimators of location and scale have broad applications and are widely used. For a variety of symmetric populations we obtain the quantiles and the weights for which the asymptotic variances of the quick estimators are minimum. These optimal quick estimators are then used to obtain the asymptotic relative efficiencies of the commonly used estimators such as trimean. gastwirth. median, midrange. and interquartile range with respect to the optimal quick estimators in order to determine a choice among them and to check whether they are unacceptably poor. In the process it is seen that the interquartile range is the optimal quick estimator of scale for Cauchy populations; but the interdecile range is in general preferable. Also the optimal estimator of the location for the logistic distribution puts weights 0.3 on each of the two quartiles and 0.4 on the median. It is shown that for the symmetric distributions, such as the beta and Tukey- lambda with [d] > 0, which have finite support and short tails, i.e. the tail exponents (Parzen, 1979) satisfy [d] < 1, the midrange and the range are the optimal quick estimators of location and scale respectively if [d] < 1/2. The class of such distributions Include the distributions with high discontinuous tails, e.g. Tukey-lambda with [d] > 1, as well as some distributions with p.d.f.'s going to zero at the ends of the finite support, such as Tukey-lambda with 1/2 < [d] < 1. As a byproduct an interesting tail correspondence between beta and Tukey-lambda distributions is seen.     

Likelihood-Based Inference for Weak Exogeneity in I(2) Cointegrated VAR Models

This article develops limit theory for likelihood analysis of weak exogeneity in I(2) cointegrated vector autoregressive (VAR) models incorporating deterministic terms. Conditions for weak exogeneity in I(2) VAR models are reviewed, and the asymptotic properties of conditional maximum likelihood estimators and a likelihood-based weak exogeneity test are then investigated. It is demonstrated that weak exogeneity in I(2) VAR models allows us to conduct asymptotic conditional inference based on mixed Gaussian distributions. It is then proved that a log-likelihood ratio test statistic for weak exogeneity in I(2) VAR models is asymptotically χ² distributed. The article also presents an empirical illustration of the proposed test for weak exogeneity using Japan's macroeconomic data.     

On Erlang(2) Risk Process Perturbed by Diffusion

ABSTRACT

In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.     

Small Sample Confidence Intervals for the Odds Ratio

Abstract

Numerous methods—based on exact and asymptotic distributions—can be used to obtain confidence intervals for the odds ratio in 2 × 2 tables. We examine ten methods for generating these intervals based on coverage probability, closeness of coverage probability to target, and length of confidence intervals. Based on these criteria, Cornfield’s method, without the continuity correction, performed the best of the methods examined here. A drawback to use of this method is the significant possibility that the attained coverage probability will not meet the nominal confidence level. Use of a mid-P value greatly improves methods based on the “exact” distribution. When combined with the Wilson rule for selection of a rejection set, the resulting method is a procedure that performed very well. Crow’s method, with use of a mid-P, performed well, although it was only a slight improvement over the Wilson mid-P method. Its cumbersome calculations preclude its general acceptance. Woolf's (logit) method—with the Haldane–Anscombe correction— performed well, especially with regard to length of confidence intervals, and is recommended based on ease of computation.     

ON THE ECONOMIC DESIGN OF MULTIVARIATE CONTROL CHARTS

ABSTRACT

It is an increasingly common practice to monitor several related quality characteristics of a product or process using a multivariate control chart procedure. Several types of multivariate control charts, including Hotelling's χ ² and T ² control charts, have been developed in attempts to improve monitoring by using the correlation structure that exists between quality characteristics. The purpose of this paper is to summarize the assumptions made regarding the out-of-control process shift in the economic design of multivariate control charts and to address their consequences. We study the average run length (ARL) properties of the χ ² control chart using a numerical example and show that this chart can perform ineffectively under the assumed out-of-control conditions when designed using the economic approach. Following Healy,[1] Healy, J.D. 1987. A Note on the Multivariate CUSUM Procedures. Technometrics, 29: 409–412. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] we offer an alternative procedure that has improved ARL properties and overall performance. These results can be important to researchers and practitioners who are interested in using the economic design of multivariate control procedures.     

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.     

Identification of out of control quality characteristics in a multivariate manufacturing environment

There are many instances in which the quality of a product or constancy of a process is determined by the joint levels of several attributes or properties. During the conduct of such a process or the production of such a product, one wishes to detect as quickly as possible any departure from a satisfactory state, while at the same time identifying which attributes are responsible for the deviation. In most cases of practical interest, however, there exist correlations among the several properties of interest; this makes it advisable to monitor certain aggregate characteristics of the process, rather than observing its various components separately. When the mean vector of the quality attributes is the major concern, this aggregate monitoring function is most commonly implemented via a T ² chart. The dependencies among attributes, however, complicate the determination of which are responsible when a deviation occurs. This paper presents an approach to help identify aberrant variables when Shewhart type multivariate control charts based on Hotelling's T ² are in use.     

A monte carlo study of variable selection and inferences in a two stage random coefficient linear regression model with unbalanced data

A growth curve analysis is often applied to estimate patterns of changes in a given characteristic of different individuals. It is also used to find out if the variations in the growth rates among individuals are due to effects of certain covariates. In this paper, a random coefficient linear regression model, as a special case of the growth curve analysis, is generalized to accommodate the situation where the set of influential covariates is not known a priori. Two different approaches for seleaing influential covariates (a weighted stepwise selection procedure and a modified version of Rao and Wu’s selection criterion) for the random slope coefficient of a linear regression model with unbalanced data are proposed. Performances of these methods are evaluated by means of Monte-Carlo simulation. In addition, several methods (Maximum Likelihood, Restricted Maximum Likelihood, Pseudo Maximum Likelihood and Method of Moments) for estimating the parameters of the selected model are compared Proposed variable selection schemes and estimators are appliedtotheactualindustrial problem which motivated this investigation.     

Efficiency of tests based on weighted rankings for equality of treatment effects in a complete randomized blocks layout

Quade (1972,1979) proposed a family of nonparametric tests based on a method of weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout.

In this paper we obtain an expression for the Pitman asymp-totic relative efficiency of these tests with respect to the Friedman test.