Multivariate inverse Gaussian distribution as a limit of multivariate waiting time distributions

Multivariate inverse Gaussian distribution proposed by Minami [2003. A multivariate extension of inverse Gaussian distribution derived from inverse relationship. Commun. Statist. Theory Methods 32(12), 2285–2304] was derived through multivariate inverse relationship with multivariate Gaussian distributions and characterized as the distribution of the location at a certain stopping time of a multivariate Brownian motion. In this paper, we show that the multivariate inverse Gaussian distribution is also a limiting distribution of multivariate Lagrange distributions, which is a family of waiting time distributions, under certain conditions.     

Star-shaped distributions and their generalizations

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to “star-shaped distributions” with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.     

Asymptotic expansions of the null distributions of some test statistics for profile analysis in general distributions

This paper discusses asymptotic expansions for the null distributions of some test statistics for profile analysis under non-normality. It is known that the null distributions of these statistics converge to chi-square distribution under normality [Siotani, M., 1956. On the distributions of the Hotelling's T²$T^2$-statistics. Ann. Inst. Statist. Math. Tokyo 8, 1–14; Siotani, M., 1971. An asymptotic expansion of the non-null distributions of Hotelling's generalized T²$T^2$-statistic. Ann. Math. Statist. 42, 560–571]. We extend this result by obtaining asymptotic expansions under general distributions. Moreover, the effect of non-normality is also considered. In order to obtain all the results, we make use of matrix manipulations such as direct products and symmetric tensor, rather than usual elementwise tensor notation.     

Asymptotics for tests on mean profiles,additional information and dimensionality under non-normality

We consider the comparison of mean vectors for k groups when k is large and sample size per group is fixed. The asymptotic null and non-null distributions of the normal theory likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics are derived under general conditions. We extend the results to tests on the profiles of the mean vectors, tests for additional information (provided by a sub-vector of the responses over and beyond the remaining sub-vector of responses in separating the groups) and tests on the dimension of the hyperplane formed by the mean vectors. Our techniques are based on perturbation expansions and limit theorems applied to independent but non-identically distributed sequences of quadratic forms in random matrices. In all these four MANOVA problems, the asymptotic null and non-null distributions are normal. Both the null and non-null distributions are asymptotically invariant to non-normality when the group sample sizes are equal. In the unbalanced case, a slight modification of the test statistics will lead to asymptotically robust tests. Based on the robustness results, some approaches for finite approximation are introduced. The numerical results provide strong support for the asymptotic results and finiteness approximations.     

Minimum phi-divergence estimators for loglinear models with linear constraints and multinomial sampling

In this paper the family ofφ-divergence estimators for loglinear models with linear constraints and multinomial sampling is studied. This family is an extension of the maximum likelihood estimator studied by Haber and Brown (1986). A simulation study is presented and some alternative estimators to the maximum likelihood are obtained. This work was parcially supported by Grant DGES PB2003-892     

Nonparametric functionals of spectral distributions and their applications to time series analysis

There is a close analogy between empirical distributions of i.i.d. random variables and normalized spectral distributions of wide-sense stationary processes. Herein we make use of this analogy to develop nonparametric comparisons of two spectral distributions and nonparametric tests of stationarity versus change-point alternatives via spectral analysis of a time series.     

Long run variance estimation and robust regression testing using sharp origin kernels with no truncation

A new family of kernels is suggested for use in long run variance (LRV) estimation and robust regression testing. The kernels are constructed by taking powers of the Bartlett kernel and are intended to be used with no truncation (or bandwidth) parameter. As the power parameter (ρ)$(\rho )$ increases, the kernels become very sharp at the origin and increasingly downweight values away from the origin, thereby achieving effects similar to a bandwidth parameter. Sharp origin kernels can be used in regression testing in much the same way as conventional kernels with no truncation, as suggested in the work of Kiefer and Vogelsang [2002a, Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366, 2002b, Heteroskedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation, Econometrica 70, 2093–2095] Analysis and simulations indicate that sharp origin kernels lead to tests with improved size properties relative to conventional tests and better power properties than other tests using Bartlett and other conventional kernels without truncation.     

Nonparametric methods in multivariate factorial designs for large number of factor levels

We propose different multivariate nonparametric tests for factorial designs and derive their asymptotic distribution for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large a, small n case). The tests are based on separate rankings for the different variables, and they are therefore invariant under separate monotone transformations of the individual variables.     

Stationary mixture transition distribution (MTD) models via predictive distributions

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist.29, 657–663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy. Statist. Soc. B.47, 528–539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series.     

Analysis of middle-censored data with exponential lifetime distributions

Recently Jammalamadaka and Mangalam [2003. Non-parametric estimation for middle censored data. J. Nonparametric Statist. 15, 253–265] introduced a general censoring scheme called the “middle-censoring” scheme in non-parametric set up. In this paper we consider this middle-censoring scheme when the lifetime distribution of the items is exponentially distributed and the censoring mechanism is independent and non-informative. In this set up, we derive the maximum likelihood estimator and study its consistency and asymptotic normality properties. We also derive the Bayes estimate of the exponential parameter under a gamma prior. Since a theoretical construction of the credible interval becomes quite difficult, we propose and implement Gibbs sampling technique to construct the credible intervals. Monte Carlo simulations are performed to evaluate the small sample behavior of the techniques proposed. A real data set is analyzed to illustrate the practical application of the proposed methods.     

Progressively Type-II right censored order statistics from discrete distributions

Progressively Type-II right censored order statistics from continuous distributions have been studied rather extensively in the literature; see Balakrishnan and Aggarwala [2000. Progressive Censoring: Theory, Methods and Applications. Birkhäuser, Boston]. In this paper, we derive the joint and marginal distributions of progressively Type-II right censored order statistics from discrete distributions. We then use these distributions to show the non-Markovian property as well as to discuss some properties in the special case of the geometric distribution.     

Finite mixtures of multivariate Poisson distributions with application

In the present paper we examine finite mixtures of multivariate Poisson distributions as an alternative class of models for multivariate count data. The proposed models allow for both overdispersion in the marginal distributions and negative correlation, while they are computationally tractable using standard ideas from finite mixture modelling. An EM type algorithm for maximum likelihood (ML) estimation of the parameters is developed. The identifiability of this class of mixtures is proved. Properties of ML estimators are derived. A real data application concerning model based clustering for multivariate count data related to different types of crime is presented to illustrate the practical potential of the proposed class of models.     

Connections of the Poisson weight function to overdispersion and underdispersion

In this paper, we establish several connections of the Poisson weight function to overdispersion and underdispersion. Specifically, we establish that the logconvexity (logconcavity) of the mean weight function is a necessary and sufficient condition for overdispersion (underdispersion) when the Poisson weight function does not depend on the original Poisson parameter. We also discuss some properties of the weighted Poisson distributions (WPD). We then introduce a notion of pointwise duality between two WPDs and discuss some associated properties. Next, we present some illustrative examples and provide a discussion on various Poisson weight functions used in practice. Finally, some concluding remarks are made.     

Generalized exponential distribution: Existing results and some recent developments

Mudholkar and Srivastava [1993. Exponentiated Weibull family for analyzing bathtub failure data. IEEE Trans. Reliability 42, 299–302] introduced three-parameter exponentiated Weibull distribution. Two-parameter exponentiated exponential or generalized exponential distribution is a particular member of the exponentiated Weibull distribution. Generalized exponential distribution has a right skewed unimodal density function and monotone hazard function similar to the density functions and hazard functions of the gamma and Weibull distributions. It is observed that it can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. The genesis of this model, several properties, different estimation procedures and their properties, estimation of the stress-strength parameter, closeness of this distribution to some of the well-known distribution functions are discussed in this article.     

Stochastic volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

On a multiple endpoints testing problem

In a clinical trial comparing drug with placebo, where there are multiple primary endpoints, we consider testing problems where an efficacious drug effect can be claimed only if statistical significance is demonstrated at the nominal level for all endpoints. Under the assumption that the data are multivariate normal, the multiple endpoint-testing problem is formulated. The usual testing procedure involves testing each endpoint separately at the same significance level using two-sample t-tests, and claiming drug efficacy only if each t-statistic is significant. In this paper we investigate properties of this procedure. We show that it is identical to both an intersection union test and the likelihood ratio test. A simple expression for the p-value is given. The level and power function are studied; it is shown that the test may be conservative and that it is biased. Computable bounds for the power function are established.     

Independence in multi-way contingency tables: S.N. Roy's breakthroughs and later developments

In the mid-1950s S.N. Roy and his students contributed two landmark articles to the contingency table literature [Roy, S.N., Kastenbaum, M.A., 1956. On the hypothesis of no “interaction” in a multiway contingency table. Ann. Math. Statist. 27, 749–757; Roy, S.N., Mitra, S.K., 1956. An introduction to some nonparametric generalizations of analysis of variance and multivariate analysis. Biometrika 43, 361–376]. The first article generalized concepts of interaction from 2×2×2$2\times 2\times 2$ contingency tables to three-way tables of arbitrary size and to larger tables. In the second article, which is the source of our primary focus, various notions of independence were clarified for three-way contingency tables, Roy's union–intersection test was applied to construct chi-squared tests of hypotheses about the structure of such tables, and the chi-squared statistics were shown not to depend on the distinction between response and explanatory variables. This work pre-dates by many years later developments that expressed such results in the context of loglinear models. It pre-dates by a quarter century the development of graphical models. We summarize the main results in these key articles and discuss the connection between them and the later developments of loglinear modeling and of graphical modeling. We also mention ways in which these later developments have themselves been further generalized.