Moments of the sampled autocovariances and autocorrelations for a Gaussian white-noise process

Cumulants, moments about zero, and central moments are obtained for the mean-corrected serial covariances and serial correlations for series realizations of length n from a white-noise Gaussian process. All first and second moments (and some third, fourth, and higher moments) are given explicitly for the serial covariances; and the corresponding moments for the serial correlations are derived either explicitly or implicitly.     

用累积法确定光子相关光谱测量中的超细颗粒粒度

本文扼要介绍了应用光子相关光谱法(PCS)确定超细颗粒粒度的基本理论,详细讨论了PCS测量中用累积法确定超细颗粒粒度的基本方法,分析了这一方法的特点和适应性。并给出了用Pearson I型分布模拟数据的考核计算和对聚苯乙烯标准粒子实测结果。     

General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

Abstract

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.     

Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing

This paper derives the exact confidence intervals for the exponential step-stress accelerated life-testing model as well as the approximate confidence intervals for the k-step exponential step-stress accelerated life-testing model under progressive Type-II censoring. A Monte Carlo simulation study is carried out to examine the performance of these confidence intervals. Finally, an example is given to illustrate the proposed procedures.     

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.     

Cumulant Plots for Assessing the Gamma Distribution

This article introduces graphical procedures for assessing the fit of the gamma distribution. The procedures are based on a standardized version of the cumulant generating function. Plots with bands of 95% simultaneous confidence level are developed by utilizing asymptotic and finite-sample results. The plots have linear scales and do not rely on the use of tables or values of special functions. Further, it is found through simulation, that the goodness-of-fit test implied by these plots compares favorably with respect to power to other known tests for the gamma distribution in samples drawn from lognormal and inverse Gaussian distributions.     

Graphical procedures and goodness-of-fit tests for the compound poisson-exponential distribution

ABSTRACT

The compound Poisson-exponential distribution is a basic model in risk analysis and stochastic hydrology. Graphical procedures for assessing this distribution are proposed which utilize the residuals from a regression involving the moment generating function. Plots furnished with a 95% simultaneous confidence band are constructed. The band and critical points of the equivalent goodness-of-fit test are found by utilizing asymptotic results and fitted regressions involving the supremum of the standardized residuals, the sample size, and the estimated Poisson mean. Simulation results indicate that the tests have good level stability and appreciable power against competing compound Poisson distributions of a mixed type.     

ALL THE GAUSSIAN WHITE NOISE SERIAL COVARIANCE MOMENTS TO ORDER FOUR

Using a recursive method, we obtain all the cumulants, central moments, and moments about zero, up to order 4, for the mean-corrected serial covariances from series realisations of length n, given a Gaussian white noise process. Some implicit higher order results are also derived.