Diagnostics for Autocorrelated Regression Models

This paper discusses the local influence approach to the linear regression model with AR(1) errors. Diagnostics for the autocorrelation models and for the autocorrelation coefficient only are proposed and developed respectively, when simultaneous perturbations of the response vector are allowed. Furthermore, the direction of maximum curvature of local influence analysis is shown to be exactly the same as that in Tsai & Wu (1992) when only the autocorrelation coefficient is of special interest.     

The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables

Abstract

An expression for the exact cumulative distribution function of a ratio of quadratic forms in noncentral normal variable is derived in terms of infinite series of top order invariant polynomials.     

Confidence Bounds for Positive Ratios of Normal Random Variables

Some applications of ratios of normal random variables require both the numerator and denominator of the ratio to be positive if the ratio is to have a meaningful interpretation. In these applications, there may also be substantial likelihood that the variables will assume negative values. An example of such an application is when comparisons are made in which treatments may have either efficacious or deleterious effects on different trials. Classical theory on ratios of normal variables has focused on the distribution of the ratio and has not formally incorporated this practical consideration. When this issue has arisen, approximations have been used to address it. In this article, we provide an exact method for determining (1 ? α) confidence bounds for ratios of normal variables under the constraint that the ratio is composed of positive values and connect this theory to classical work in this area. We then illustrate several practical applications of this method.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

Interval Estimation of the Stress-Strength Reliability with Independent Normal Random Variables

This article develops a procedure to obtain highly accurate confidence interval estimates for the stress-strength reliability R = P(X > Y) where X and Y are data from independent normal distributions of unknown means and variances. Our method is based on third-order likelihood analysis and is compared to the conventional first-order likelihood ratio procedure as well as the approximate methods of Reiser and Guttman (1986 Reiser, B., Guttman, I. (1986). Statistical inference for Pr(Y < X): the normal case. Technometrics 28: 253257.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Guo and Krishnamoorthy (2004 Guo, H., Krishnamoorthy, K. (2004). New approximate inferential methods for the reliability parameter in a stress-strength model: the normal case. Commun. Statist. Theor. Meth. 33: 17151731.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The use of our proposed method is illustrated by an empirical example and its superior accuracy in terms of coverage probability and error rate are examined through Monte Carlo simulation studies.     

A Note on the Distribution of Linear Functions of Two Ordered Correlated Normal Random Variables

We consider the problem of finding the distribution of linear functions of two ordered correlated normal random variables. We derive some distributional properties for these linear statistics and briefly discuss the use of them in location estimation. The connection of the subject with the skew normal distribution is also noted.     

Estimation of gene/haplotype frequencies in genetic marker systems based on phenotype data

Two equivalent methods (gene counting and maximum likelihood) for estimating gene frequencies in a general genetic marker system based on observed phenotype data are derived. Under the maximum likelihood approach, an expression is given for the estimated covariance matrix from which estimated standard errors of the estimators can be found. In addition, consideration is given to the problem of estimating gene frequencies when there are available several independent population data sets.     

Inference on the Common Variance of Correlated Normal Random Variables

ABSTRACT

Scale equivariant estimators of the common variance σ², of correlated normal random variables, have mean squared errors (MSE) which depend on the unknown correlations. For this reason, a scale equivariant estimator of σ² which uniformly minimizes the MSE does not exist. For the equi-correlated case, we have developed three equivariant estimators of σ²: a Bayesian estimator for invariant prior as well as two non-Bayesian estimators. We then generalized these three estimators for the case of several variables with multiple unknown correlations. In addition, we developed a system of confidence intervals which produce the desired coverage probability while being efficient in terms of expected length.     

Density of the Ratio of Two Normal Random Variables and Applications

In reply to a question raised in the literature, and to settle an argument debated in the last decades, we give the exact closed form expression of the density of X/Y, where X and Y are normal random variables, in terms of Hermite and confluent hypergeometric functions. All cases will be considered: standardized and nonstandardized variables, independent or correlated variables. Examples in applied disciplines are presented, and generalizations to ratios of variables from scale mixtures of bivariate normal distributions show the potential of further new applications in applied statistics and operations research.     

SB-Robust Estimators of the Parameters of the Wrapped Normal Distribution

In this article, we study the SB-robustness of various estimators of the mean direction (μ) and the concentration parameter (ρ) of the wrapped normal distribution. The functional corresponding to the sample mean direction is seen to be not SB-robust as an estimator of μ at the family of wrapped normal distributions with varying ρ, whereas the γ-trimmed mean direction is SB-robust at the same family of distributions for the different dispersion measures considered in this article. We also study the SB-robustness of the moment estimator of ρ and also that for a newly introduced trimmed estimator of ρ.     

Confidence Bounds and Hypothesis Tests for Normal Distribution Coefficients of Variation

For normally distributed populations, we obtain confidence bounds on a ratio of two coefficients of variation, provide a test for the equality of k coefficients of variation, and provide confidence bounds on a coefficient of variation shared by k populations.     

On the Ratios of Pathway Random Variables

A study of the densities of ratios of independently distributed random variables following the pathway model, Mathai (2005 Mathai, A.M. (2005). A pathway to matrix - variate gamma and normal densities. Linear Algebra Appl. 396:317–328.[Crossref], [Web of Science ®] , [Google Scholar]), is carried out. The density functions of these random variables are obtained in terms of H-function. The particular cases of the integral forms are shown to be associated with Tsallis statistics and Beck–Cohen superstatistics. Many other special functions coming under the general density are also included. We plotted the density function of the ratio of these random variables for the different values of the pathway parameters. Real-life application of the results in communication theory related to the signal–noise ratio is illustrated.     

Estimators of Certain Functions of the Variance of a Normal Distribution

Estimators of σ^aand log σ which are functions of Σ(x?x)²/d are considered. Besides the usual sampling theory estimators, Bayesian point estimators which are the usual measures of location of the posterior distribution are given, and in each case an exact or asymptotic expression for the divisor d is stated.     

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.     

The Joint Distribution of the Sum and the Maximum of IID Exponential Random Variables

The authors establish the joint distribution of the sum X and the maximum Y of IID exponential random variables. They derive exact formuli describing the random vector (X, Y), including its joint PDF, CDF, and other characteristics; marginal and conditional distributions; moments and related parameters; and stochastic representations leading to further properties of infinite divisibility and self-decomposability. The authors also discuss parameter estimation and include an example from climatology that illustrates the modeling potential of this new bivariate model.     

Some New Results on Likelihood Ratio Orderings for Spacings of Heterogeneous Exponential Random Variables

Let X ₁, X ₂,…, X _n be independent exponential random variables with X _i having failure rate λ _i for i = 1,…, n. Denote by D _i:n  = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…, n, where X _0:n ≡ 0. It is shown that if λ _n+1 ≤ [≥] λ _k for k = 1,…, n then D _n:n  ≤ _lr D _n+1:n+1 and D _1:n  ≤ _lr D _2:n+1 [D _2:n+1 ≤ _lr D _2:n ], and that if λ _i  + λ _j  ≥ λ _k for all distinct i,j, and k then D _n?1:n  ≤ _lr D _n:n and D _n:n+1 ≤ _lr D _n:n , where ≤ _lr denotes the likelihood ratio order. We also prove that D _1:n  ≤ _lr D _2:n for n ≥ 2 and D _2:3 ≤ _lr D _3:3 for all λ _i 's.     

Perturbation diagnostics of autocorrelation coefficients in non linear mixed-effects models with AR(1) errors based on M-estimation

In this work we propose and analyze non linear mixed-effects models for longitudinal data, which are widely used in the fields of economics, biopharmaceuticals, agriculture, and so on. A robust method to obtain maximum likelihood estimates for the parameters is presented, as well as perturbation diagnostics of autocorrelation coefficient in non linear models based on robust estimates and influence curvature. The obtained results are illustrated by plasma concentrations data presented in Davidian and Giltinan, which was analyzed under the non robust situation.     

On the Product and Ratio of Gamma and Beta Random Variables

Summary: The distributions of the product XY and the ratio X/Y are derived when X and Y are gamma and beta random variables distributed independently of each other. Tabulations of the associated percentage points and illustrations of their practical use are also provided. * The authors would like to thank the referee and the editor for carefully reading the paper and for their help in improving the paper.     

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT

The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.     

Misspecifications of the Normal Distribution

Incorrect statements about the normal distribution are discussed and illustrated with counterexamples.