Inferences on Correlation Coefficients in Some Classes of Nonnormal Distributions

Correlation coefficients have many applications for studying the relationship among multivariate observations. Classical inferences on correlation coefficients are mainly based on the normality assumption. This assumption is hardly realistic in the real world, which implies that the procedures on correlation coefficients used in many statistical software packages may not be relevant to most data sets in practice. However, we show that the classical procedures, possibly after simple corrections, are also valid in classes of distributions with large skewnesses and heterogeneous marginal kurtoses. A useful class of nonnormal distributions is identified for each of several types of correlation coefficients. The marginals of these distributions may include a variety of univariate distributions with different shapes. The results generalize the classical procedures to much larger classes of distributions than previously known and give a better understanding of the historical controversy regarding the behavior of the sample correlation coefficient. An implication is that one need not be worried so much by the nonnormality of data sets when using these classical procedures, providing simple corrections are evaluated and possibly undertaken.     

Testing for elliptical symmetry in covariance-matrix-based analyses

Many normal-theory test procedures for covariance matrices remain valid outside the family of normal distributions if the matrix of fourth-order moments has structure similar to that of a normal distribution. In particular, for elliptical distributions this matrix of fourth-order moments is a scalar multiple of that for the normal, and for this reason many normal-theory statistics can be adjusted by a scalar multiple so as to retain their asymptotic distributional properties across elliptical distributions. For these analyses, a test for the validity of these scalar-adjusted normal-theory procedures can be viewed as a test on the structure of the matrix of fourth-order moments. In this paper, we develop a Wald statistic for conducting such a test.     

The nonexistence of regularization operators

We show that there are no continuous regularization procedures for the extension of distributions. We also show that there are no continuous projection operators from the spaces of distributions onto subspaces of distributions with support on a given closed set.     

On tests for selection of variables and independence under multivariate regression models

The authors consider various procedures for testing the hypotheses of independence of two sets of variables and certain regression coefficients are zero under multivariate regression model. Various properties of these procedures and the asymptotic distributions associated with these procedures are also considered.     

Truncating the hierarchy of moment equations based on point distribution—application to innovation diffusion

The phenomenon of innovation diffusion, modeled as a nonlinear birth process, leads to a hierarchy of moment equations. For gaining insight into the evolution of moments of a number of adopters in innovation diffusion, truncation procedures based on point distributions are proposed for truncating the hierarchy of moment equations. It is found that the results obtained by employing new truncation procedures are in close agreement with analytical results based on system size expansion for large population size.     

Huber＇s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber‘s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber‘s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal；(ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber‘s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber‘s for heavy-tailed distributions and more efficient than Huber‘s for short-tailed ones both in asymptotics and on finite samples。     

Huber's Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

A brief survey of former and recent results on Huber's minimax approach in robust statistics is given. The least mformative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber's solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavytailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber's procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber's for heavy-tailed distributions and more efficient than Huber's for short-tailed ones both in asymptotics and on finite samples.     

Bayesian nonparametrie inference and monte carlo optimization

As global or combinatorial optimization problems are not effectively tractable by means of deterministic techniques, Monte Carlo methods are used in practice for obtaining ”good“ approximations to the optimum. In order to test the accuracy achieved after a sample of finite size, the Bayesian nonparametric approach is proposed as a suitable context, and the theoretical as well as computational implications of prior distributions in the class of neutral to the right distributions are examined. The feasibility of the approach relatively to particular Monte Carlo procedures is finally illustrated both for the global optimization problem and the {0 - 1} programming problem.     

A load share model for non-identical components of a k-out-of-m system

We consider a k-out-of-m load sharing system when the lifetimes of the components are not necessarily identically distributed random variables. For such systems, a model for the load sharing phenomenon through the exponentiated conditional survival functions of ordered failure times is proposed. This model is more general than the load sharing model with identically distributed component lifetimes and leads to a different family of distributions for ordered random variables. A general expression for the reliability of the system is given. The computations of the reliability for a two component parallel load sharing system corresponding to the exponential and Weibull distributions are discussed. For illustrative purpose, we discuss the inference procedures for a two component parallel load sharing system corresponding to the exponential distributions. A simulation study is carried out to assess the proposed estimation and testing procedures. The applicability of the proposed load sharing model is shown through two data sets.     

Derivation of mixture distributions and weighted likelihood function as minimizers of KL-divergence subject to constraints

In this article, mixture distributions and weighted likelihoods are derived within an information-theoretic framework and shown to be closely related. This surprising relationship obtains in spite of the arithmetic form of the former and the geometric form of the latter. Mixture distributions are shown to be optima that minimize the entropy loss under certain constraints. The same framework implies the weighted likelihood when the distributions in the mixture are unknown and information from independent samples generated by them have to be used instead. Thus the likelihood weights trade bias for precision and yield inferential procedures such as estimates that can be more reliable than their classical counterparts.     

Sequential procedures for a class of distributions related to the uniform

Summary The main purpose of this paper is to examine some sequential inference procedures, mainly in the one-sided hypothesis testing situation, for the parameter of a class of distributions related to the Uniform distribution on (0, θ). The procedures are all based on the sequence of maxima of independent and identically distributed random variables and, for the case of a single (upper) boundary, procedures which are optimal in the sense of minimizing the average sample size are discussed. The impossibility of using two boundaries is demonstrated, thus leaving the best procedure with one boundary as the optimal procedure. The novelty of this procedure is that it uniformly minimizes the average sample size.     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.     

Comparison of statistical procedures in analytic hierarchy process using a ranking test

Several statistical procedures for estimation of the priority parameters in the setup of the Analytic Hierarchy Process (AHP) exist in the literature. The purpose of this article is to make appropriate comparisons of such statistical methods. Pairwise comparison matrices are simulated using different statistical distributions of the error part used in the procedures. Priority parameters are estimated for each simulated pairwise comparison matrix using the method suggested. Standard nonparametric statistical procedures are applied to check whether the order of the priority estimates is consistent with that of their parameter values irrespective of the choice of particular statistical procedure. Statistical procedures based on the reciprocal matrices are also compared with the eigenvalue method.     

The PH/PH/1 queue at epochs of queue size change

The PH/PH/1 queue is considered at embedded epochs which form the union of arrival and departure instants. This provides us with a new, compact representation as a quasi-birth-and-death process, where the order of the blocks is the sum of the number of phases in the arrival and service time distributions. It is quite easy to recover, from this new embedded process, the usual distributions at epochs of arrival, or epochs of departure, or at arbitrary instants. The quasi-birth-and-death structure allows for efficient algorithmic procedures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Performance analysis for assemble-to-order systems with general renewal arrivals and random batch demands

We study multi-product and multi-item assemble-to-order systems under general assumptions on demand patterns and replenish leadtime distributions. We only assume that the demand process of each product being a renewal process, and the replenish leadtimes follow general distributions. Based upon techniques from renewal theory, we developed procedures for approximating key performance measures of these inventory systems, such as average inventory and immediate order fill rate. We also obtain qualitative results that reveal the impacts of changes in demand patterns and leadtime variability upon the performance of the systems.     

Point processes and multivariate extreme values

A new model for point processes is developed which assumes that the interarrival times are exponentially distributed and follow joint multivariate extreme value distributions. It is shown that such processes may arise via natural generating procedures, and that, under very weak assumptions, that they can be approximated as closely as desired by appropriate finite models.     

Modified weighted squared error estimation procedures with special emphasis on the stable laws

Two families of parameter estimation procedures for the stablelaws based on a variant of the characteristic function are provided. The methodology which produces viable computational procedures for the stable laws is generally applicable to other families of distributions across avariety of settings. Both families of procedures may be described as a modified weighted chi-squared minimization procedure, and both explicitlytake account of constraints on the parameter space. Influence functions for and efficiencies of the estimators a r e given. If _l, x₂, ..., x_n is a random sample from an unknown distribution F, a method for determining the stable law to which F is attracted is developed. Procedures for regression and autoregression with stable error structurear e provided. A number of examples are given.     

Heuristic solution methods for the stochastic flow shop problem

We investigate the stochastic flow shop problem with m machines and general distributions for processing times. No analytic method exists for solving this problem, so we looked instead at heuristic methods. We devised three constructive procedures with modest computational requirements, each based on approaches that have been successful at solving the deterministic counterpart. We compared the performance of these procedures experimentally on a set of test problems and found that all of them achieve near-optimal performance.     

Double shrinkage estimation of ratio of scale parameters

The problems of estimating ratio of scale parameters of two distributions with unknown location parameters are treated from a decision-theoretic point of view. The paper provides the procedures improving on the usual ratio estimator under strictly convex loss functions and the general distributions having monotone likelihood ratio properties. In particular,double shrinkage improved estimators which utilize both of estimators of two location parameters are presented. Under order restrictions on the scale parameters, various improvements for estimation of the ratio and the scale parameters are also considered. These results are applied to normal, lognormal, exponential and pareto distributions. Finally, a multivariate extension is given for ratio of covariance matrices.     

Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.