Some Notes on Refereeing

Analysis of longitudinal data using a general linear mixed model requires the specification of a form for the covariance matrix of within-subject observations. Graphical diagnostics, such as the scatterplot matrix, can be of substantial help in making this specification. I introduce another graphical diagnostic, the Partial-Regression-on-Intervenors Scatterplot Matrix (PRISM), which complements the ordinary scatterplot matrix and which is more useful for identifying certain kinds of correlation structures. PRISMs corresponding to several commonly used correlation structures are displayed. The PRISM's usefulness in model specification is illustrated with an example of longitudinal data from a 100-kilometer road race.     

Confidence Intervals for the Scale Parameter of the Power-Law Process

The power-law process (PLP) is a two-parameter model widely used for modeling repairable system reliability. Results on exact point estimation for both parameters as well as exact interval estimation for the shape parameter are well known. In this paper, we investigate the interval estimation for the scale parameter. Asymptotic confidence intervals are derived using Fisher information matrix and theoretical results by Cocozza-Thivent (1997 Cocozza-Thivent , C. ( 1997 ). Processus Stochastiques et Fiabilité des Systèmes . Berlin : Springer-Verlag . [Google Scholar]). The accuracy of the interval estimation for finite samples is studied by simulation methods.     

Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.     

Inference Under Heteroskedasticity and Leveraged Data

We evaluate the finite-sample behavior of different heteros-ke-das-ticity-consistent covariance matrix estimators, under both constant and unequal error variances. We consider the estimator proposed by Halbert White (HC0), and also its variants known as HC2, HC3, and HC4; the latter was recently proposed by Cribari-Neto (2004 Cribari-Neto , F. ( 2004 ). Asymptotic inference under heteroskedasticity of unknown form . Computat. Statist. Data Anal. 45 : 215 – 233 .[Crossref], [Web of Science ®] , [Google Scholar]). We propose a new covariance matrix estimator: HC5. It is the first consistent estimator to explicitly take into account the effect that the maximal leverage has on the associated inference. Our numerical results show that quasi-t inference based on HC5 is typically more reliable than inference based on other covariance matrix estimators.     

Fast Computation by Block Permanents of Cumulative Distribution Functions of Order Statistics from Several Populations

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution functions of the populations. The computational cost of our formula in the case of two populations is still exponential in the worst case, but it is a dramatic improvement compared to the general formula by Bapat and Beg. In the case when only the joint distribution function of a subset of the order statistics of fixed size is needed, the complexity is polynomial, for the case of two populations.     

Covariance Matrix Formula for Exponential Family Nonlinear Models

This article gives a matrix formula for second-order covariances of maximum likelihood estimators in exponential family nonlinear models, thus generalizing the result of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) valid for generalized linear models with known dispersion parameter. Some simulations show that the second-order covariances for exponential family nonlinear models can be quite pronounced in small to moderate sample sizes.     

A Note on the Prior Distributions of Weibull Parameters for the Reliability Function

In Bayesian Inference it is often desirable to have a posterior density reflecting mainly the information from sample data. To achieve this purpose it is important to employ prior densities which add little information to the sample. We have in the literature many such prior densities, for example, Jeffreys (1967 Jeffreys , H. ( 1967 ). Theory of Probability , 3rd rev. ed. . London : Oxford University Press . [Google Scholar]), Lindley (1956 Lindley , D. V. ( 1956 ). On a measure of the information provided by an experiment . Ann. Mathemat. Statist. 27 : 986 – 1005 .[Crossref] , [Google Scholar]); (1961 Lindley , D. V. ( 1961 ). The use of prior probability distributions in statistical inference and decisions . In: Neyman , J. , ed. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability . Vol. 1. Berkeley : University of California Press , pp. 453 – 468 . [Google Scholar]), Hartigan (1964 Hartigan , J. ( 1964 ). Invariant priors distributions . Ann. Mathemat. Statist. 35 : 836 – 845 .[Crossref] , [Google Scholar]), Bernardo (1979 Bernardo , J. M. ( 1979 ). Reference posterior distributions for Bayesian inference . J. Roy. Statist. Soc. 41 ( 2 ): 113 – 147 . [Google Scholar]), Zellner (1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani (1989 Tibshirani , R. ( 1989 ). Noninformative priors for one parameters of many . Biometrika 76 : 604 – 608 .[Crossref], [Web of Science ®] , [Google Scholar]), etc. In the present article, we compare the posterior densities of the reliability function by using Jeffreys, the maximal data information (Zellner, 1984 Zellner , A. ( 1984 ). Maximal Data Information Prior Distributions, Basic Issues in Econometrics . Chicago : University of Chicago Press . [Google Scholar]), Tibshirani's, and reference priors for the reliability function R(t) in a Weibull distribution.     

A Bivariate Beta-Gamma Autoregressive Process (BVBGAR(1))

In this article, we introduce a bivariate autoregressive process with Gamma marginal distributions using the form of the BGAR(2) process (Risti?, 2005 Risti? , M. M. ( 2005 ). A Beta-Gamma autoregressive process of the second-order (BGAR(2)) . Statist. Probab. Lett. 73 : 403 – 410 . [Google Scholar]) and the Beta-Gamma transformation. Some properties of the process such as the autocovariance matrix, the autocorrelation matrix, and the spectral density matrix are derived. The unknown parameters of the process are estimated using the method of moments and the method of conditional least squares. Some numerical results of the estimators are given. We investigate nonparametric and parametric estimation of the spectral density matrix of this process.     

Bias-Corrected Maximum Likelihood Estimators in Nonlinear Heteroscedastic Models

Nonlinear heteroscedastic models are widely used in econometrics and statistical applications. We derive matrix formulae for the second-order biases of the maximum likelihood estimators of the parameters in the mean and variance response which generalize previous results by Cook et al. (1986 Cook , D. R. , Tsai , C. L. , Wei , B. C. ( 1986 ). Bias in nonlinear regression . Biometrika 73 : 615 – 623 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro (1993 Cordeiro , G. M. ( 1993 ). Bartlett corrections and bias correction for two heteroscedastic regression models . Commun. Statist. Theor. Meth. 22 : 169 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The biases of the estimators are easily obtained as vectors of regression coefficients from suitable weighted linear regressions. The practical use of such biases is illustrated in a simulation study and in an application to a real data set.