A test for independence in a multivariate exponential distribution with equal correlation coefficients

A rank statistic is considered which may be used for testing for total independence in a p-variate exponential distribution with equal correlation coefficients. Critical values for the statistic are provided for p = 3.4 and sample sizes less than or equal to 20. Finally, the small sample power performance of the rank test relative to that of the locally most powerful similar lest under the exponential alternative is evaluated.     

Nonparametric monitoring of equal predictive ability

This paper introduces a novel sequential approach for online surveillance of the equal predictive ability (EPA) hypothesis presumed to hold for many competing forecasting models. A nonparametric control chart is suggested for providing a decision at every new time point as to whether the EPA hypothesis remains valid. The detection ability of our procedure is evaluated in a Monte Carlo simulation study for various types of deviations from the EPA hypothesis. Our approach enables the quick detection of various shift types, is parsimonious, and robust to misspecifications. Based on these results, we formulate practical recommendations for procedure design.     

面板数据双因素误差回归模型序列相关和随机效应的联合LM检验

对面板数据双因素误差回归模型构造了检验序列相关和随机效应的一种联合LM检验,发现该LM统计量也是检验联合假设H0：σμ^2=λ=0的Baltagi-Li LM统计量和检验假设H0：σv^2=λ=0的Breusch-Pagan-LM统计量之和。当面板数据的个体数N充分大时,该联合LM统计量的渐近分布是χ^2（3）分布;无论双因素误差面板数据回归模型的剩余误差项是AR（1）过程还是MA（1）过程,联合LM检验是相同的,即对随机效应和一阶序列相关的联合LM检验是独立于序列相关的形式。     

A statistical test for the hypothesis of Gaussian random function

A Gaussian random function is a functional version of the normal distribution. This paper proposes a statistical hypothesis test to test whether or not a random function is a Gaussian random function. A parameter that is equal to 0 under Gaussian random function is considered, and its unbiased estimator is given. The asymptotic distribution of the estimator is studied, which is used for constructing a test statistic and discussing its asymptotic power. The performance of the proposed test is investigated through several numerical simulations. An illustrative example is also presented.     

Slope-rotatable designs with equal maximum directional variance for second order response surface models

The concept of sloperotaiability with equal maximum directional vari ance for second order response surface models is introduced as a new design property. This requires that the maximum variance of the estimated slope over all possible directions be only a function of p, which is the distance from the design originif is shown that a rotatable design satisfies this property Also, minimization of tiie maximum variance of the estimated slope over all possible directions is proposed as a new design optirnality criterion, and op¬timal designs are called slope-directional minirnax designs. For the class of cquiradial designs, the slope-directional minirnax designs are compared with D— optimal designs.     

On the correlation coefficient between the first and the r-th smallest order statistics based on n independent exponential random variables

It is proved that the correlation coefficient between the first and the r-th smallest order statistics based on n independent, exponentially distributed random variables is less than or equal to the corresponding correlation coefficient when the n indpendent random variables are identically and exponentially distributed with expectation equal to unity     

Multi - sample test of equal gamma distribution scale parameters in presence of unknown common shape parameter

Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ₁ < … < λ_L and n₁ < … < n_L, where λ₁ …, λ_L are the scale parameters and n₁,…, n_L are the sample sizes.     

A note on the linear discriminant function when group means are equal

The performance of the sample linear discriminant function with known, proportional, covariance matrices and equal but unknown mean vectors is considered. Unconditional misclassification rates are obtained from the Student-t distribution. These results can be used as an aid in verifying simulation programs incorporating the linear discriminant function when Gaussian densities with unequal covariance matrices are used.     

Order statistics from extreme value distribution,i: tables of means,variances and covariances

Let X¹, X², …, Xⁿ be a random sample of size n from an extreme value distribution and X^1:n less than or equal X^2:n less than or equal … less than or equal X^n:n be the order statistics ob-tained from this sample. Tables of the means, variances, and covariances of the order statistics for samples of size n are given for n = 1(1)15(5)30. The computational formulae and procedure used and some checks employed are explained.     

Rao distance as a measure of influence in the multivariate linear model

Several methods have been suggested to detect influential observations in the linear regression model and a number of them have been extended for the multivariate regression model. In this article we consider the multivariate general linear model, Y = XB + k , which contains the linear regression model and the multivariate regression model as particular cases. Assuming that the random disturbances are normally distributed, the BLUE of v B is also normally distributed. Since the distribution of the BLUE of v B and the distribution of the BLUE of v B in the model with the omission of a set of observations differ, to study the influence that a set of observations has on the BLUE of v B , we propose to measure the distance between both distributions. To do this we use Rao distance.     

A revisit of the distribution of linear combinations of Dirichlet components

Provost and Cheong (2000 Provost, S. B., and Y.-H. Cheong. 2000. On the distribution of linear combinations of the components of a Dirichlet random vector. The Canadian Journal of Statistics 28 (2):417–25.[Crossref], [Web of Science ®] , [Google Scholar]) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. The c-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when its c-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results.     

Confidence intervals on the total variance in an unbalanced two-fold nested classification with equal subsampling

A problem of interest in a variance component analysis is the construction of a confidence interval on the variance of a single observation. This article considers an unbalanced two-fold nested classification with equal subsampling and compares two methods for constructing this interval . Computer simulations indicate that one of these methods in general will provide an interval that has an achieved confidence coefficient at least as great as the stated value.     

Generalized Score Test of Homogeneity Based on Correlated Random Effects Models

We present a family of tests based on correlated random effects models which provides a synthesis and a generalization of recent work on homogeneity testing. In these models each subject has a particular random effect, but the random effects between subjects are correlated. We derive the general form of the score statistic for testing that the random effects have a variance equal to 0. We apply this result to both parametric and semi-parametric models. In both cases we show that under certain conditions the score statistic has an asymptotic normal distribution. We consider several applications of this theory, including overdispersion, heterogeneity between groups, spatial correlations and genetic linkage.     

Law of iterated logarithm and almost sure limit points for random sums

Let {X_n,n≥1} be a sequence of independent identically distributed (i.i.d) random variables with a common distribution function F. When F belongs to the domain of partial attraction of a Semi-Stable law with index ,0<<2, we give complete solution to the results of R. Vasudeva and G. Divanji [Law of iterated logarithm for random subsequences, Statist. Probab. Lett. 12 (1991) 189–194], where they obtained Chover’s form of the law of iterated logarithm for random subsequences. Further, we extended the situation in obtaining almost sure limit points for random subsequences.     

Score tests in the two - way layout of counts

For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µ_ij= E(Y| A=i,B=j) = α_iβ_jγ_ij corresponding respectively to the multiplicative

interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results provide strong evidence that the normal-theory based F-tests typically are very far from nominal size, and that the likelihood ratio test is somewhat more liberal than the score test.     

Alternative expectation formulas for real-valued random vectors

Abstract

When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The random product can be that of strictly increasing functions of random variables. For continuous cases, the derivation based on iterated integrals is employed. It is shown that Hoeffding’s covariance lemma is algebraically equal to a special case of this result. For discrete cases, the elements of a random vector can be non-integers and/or unequally spaced. A discrete version of Hoeffding’s covariance lemma is derived for real-valued random variables.     

Residual-based specification of the random-effects distribution for cluster data

We propose a method for specifying the distribution of random effects included in a model for cluster data. The class of models we consider includes mixed models and frailty models whose random effects and explanatory variables are constant within clusters. The method is based on cluster residuals obtained by assuming that the random effects are equal between clusters. We exhibit an asymptotic relationship between the cluster residuals and variations of the random effects as the number of observations increases and the variance of the random effects decreases. The asymptotic relationship is used to specify the random-effects distribution. The method is applied to a frailty model and a model used to describe the spread of plant diseases.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

All pairwise comparisons of exponential location parameters under type i censoring — equal scale parameter case

In this paper, procedures for all pairwise comparisons of location parameters of negative exponential populations are developed when the common scale parameter is known or unknown using large sample distributional approximations of the relevant random variables. The small sample performance of these procedures are then examined using Monte Carlo simulation.     

A goodness-of-fit test using randomly selected cell boundaries

This paper describes a technique for testing for the goodness-of-fit of a set of observations on a random variable to a hypothesized distribution. The cell boundaries needed for partitioning the range of the random variable are randomly chosen using the computer, thus eliminating an arbitrary choice by the analyst. The power of this test procedure is compared to the test procedure using equal expected cell frequencies