A NOTE ON EVANESCENT PROCESSES

This note examines the connection between μ-invariant measures for the transition function of a continuous-time Markov chain and those of its q-matrix, Q. The major result establishes a necessary and aufficient condition for a convergent μ-invariant measure for Q to be μ-inhant for the minimal transition function, P, under the assumption that P is honest. This corrects Theorem 6 of Vere-Jones (1969) and the first part of Corollary 1 of Pollett (1986), both of which assert that the above conclusion holds in the absence of this condition. The error was pointed out by E.A. van Doom (1991) and the counterexample which be presented provides the basis for the present arguments. In determining where the error occurred in the original proof, we are able to identify a simple sufficient condition for μ-invariance.     

Computing maximum likelihood estimates from type II doubly censored exponential data

It is well-known that, under Type II double censoring, the maximum likelihood (ML) estimators of the location and scale parameters, θ and δ, of a twoparameter exponential distribution are linear functions of the order statistics. In contrast, when θ is known, theML estimator of δ does not admit a closed form expression. It is shown, however, that theML estimator of the scale parameter exists and is unique. Moreover, it has good large-sample properties. In addition, sharp lower and upper bounds for this estimator are provided, which can serve as starting points for iterative interpolation methods such as regula falsi. Explicit expressions for the expected Fisher information and Cramér-Rao lower bound are also derived. In the Bayesian context, assuming an inverted gamma prior on δ, the uniqueness, boundedness and asymptotics of the highest posterior density estimator of δ can be deduced in a similar way. Finally, an illustrative example is included.     

Multivariate distributions defined in terms of contours

The paper deals with the problem of using contours as the basis for defining probability distributions. First, the most general probability densities with given contours are obtained and the particular cases of circular and elliptical contours are dealt with. It is shown that the so-called elliptically contoured distributions do not include all possible cases. Next, the case of contours defined by polar coordinates is analyzed including its simulation and parameter estimation. Finally, the case of cumulative distribution functions with given contours is discussed. Several examples are used for illustrative purposes.     

Statistical Distributions of Daily Breathing Rates for Narrow Age Groups of Infants and Children

Children may be more susceptible to toxicity from some environmental chemicals than adults. This susceptibility may occur during narrow age periods (windows), which can last from days to years depending on the toxicant. Breathing rates specific to narrow age periods are useful to assess inhalation dose during suspected windows of susceptibility. Because existing breathing rates used in risk assessment are typically for broad age ranges or are based on data not representative of the population, we derived daily breathing rates for narrow age ranges of children designed to be more representative of the current U.S. children's population. These rates were derived using the metabolic conversion method of Layton (1993) and energy intake data adjusted to represent the U.S. population from a relatively recent dietary survey (CSFII 1994–1996, 1998). We calculated conversion factors more specific to children than those previously used. Both nonnormalized (L/day) and normalized (L/kg-day) breathing rates were derived and found comparable to rates derived using energy estimates that are accurate for the individuals sampled but not representative of the population. Estimates of breathing rate variability within a population can be used with stochastic techniques to characterize the range of risk in the population from inhalation exposures. For each age and age-gender group, we present the mean, standard error of the mean, percentiles (50th, 90th, and 95th), geometric mean, standard deviation, 95th percentile, and best-fit parametric models of the breathing rate distributions. The standard errors characterize uncertainty in the parameter estimate, while the percentiles describe the combined interindividual and intra-individual variability of the sampled population. These breathing rates can be used for risk assessment of subchronic and chronic inhalation exposures of narrow age groups of children.     

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution

Summary . A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t -density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.     

BURR CRITICAL VALUE APPROXIMATIONS FOR TESTS OF AUTOCORRELATION AND HETEROSCEDASTICITY

The paper evaluates the accuracy of Burr approximations of critical values and p-values for test a of autocorrelation and heteroscedasticity in the linear regression model.     

TESTING SPHERICITY UNDER A MIXTURE MODEL

The robustness of Mauchly's sphericity test criterion when sampling from a mixture of two multivariate normal distributions is studied. The distribution of the sphericity test criterion when the sample covariance matrix has a non-central Wishart density of rank one is derived in terms of Meijer's G-functions; its distribution under the mixture model is then deduced. The robustness is studied by computing actual significance levels of the test under the mixture model using the critical values under the usual normal model.     

A simple procedure for the selection of significant effects

Summary.  Given a large number of test statistics, a small proportion of which represent departures from the relevant null hypothesis, a simple rule is given for choosing those statistics that are indicative of departure. It is based on fitting by moments a mixture model to the set of test statistics and then deriving an estimated likelihood ratio. Simulation suggests that the procedure has good properties when the departure from an overall null hypothesis is not too small.     

Gini’s nuclear family

The purpose of this paper is to justify the use of the Gini coefficient and two close relatives for summarizing the basic information of inequality in distributions of income. To this end we employ a specific transformation of the Lorenz curve, the scaled conditional mean curve, rather than the Lorenz curve as the basic formal representation of inequality in distributions of income. The scaled conditional mean curve is shown to possess several attractive properties as an alternative interpretation of the information content of the Lorenz curve and furthermore proves to yield essential information on polarization in the population. The paper also provides asymptotic distribution results for the empirical scaled conditional mean curve and the related family of empirical measures of inequality.      

Causal Reasoning from Longitudinal Data*

Abstract. This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so‐called do‐probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.