A note on exchangeability of random variables

A sufficient condition for the induced exchangeability or partial exchangeability of linear functions of exchangeable random variables is presented. The use of this result is illustrated through the establishment of conditional exchangeability for two sets of dependent random variables that are important in constructing conditionally distribution–free test procedures for two distinctly different problems.     

Multivariate likelihood ratio orderings between spacings of heterogeneous exponential random variables

Let X ₁, X ₂, ..., X _n be independent exponential random variables such that X _i has failure rate λ for i = 1, ..., p and X _j has failure rate λ^* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D _i:n (p,q) = 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. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D _i:n (p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings. Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.: NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02).     

A law of large numbers for abstract random variables

Summary It was emphasized by Fréchet [8] that in many of the practical applications, we get, as a result of a random experiment, an abstract random variableX belonging to a setX. Givenn determinations of the variableX, an arithmetic meanZ _n is introduced in 2-. It reduces to the classical arithmetic mean, whenX is the set of real numbers. Moreover, if the random variableX is Gaussian, having a meanM(X), and belongs to a certain (4-) familyF, the introduced arithmetic meanZ _n constitutes an exhausting estimation [5] of the meanM(X).     

The association between two random elements: A complete characterization and odds ratio models

For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.Acknowledgements The author thanks both referees for their helpful comments which improved the first draft of the paper.     

On decompositions of some random variables

The aim of this paper is to present several stochastic analogs of classical formulas for the gamma function. The obtained results provide representation of some random variables as finite or infinite products of independent random variables. Examples include generalized gamma, normal, beta and other distributions.     

Non-exchangeability of negatively dependent random variables

We compute the measure of non-exchangeability (with respect to the L ^∞-norm) for a pair of identically distributed continuous random variables that satisfy some negative dependence property, namely quadrant dependence or stochastic decreasingness.     

On the product of gamma random variables

We give expressions for the distribution and density of a product of gamma or equivalently chi-square random variables. In particular, we give the distribution of the product of two independent gamma variables of mean k in terms of the Bessel functions K ₁, … , K _k .     

On the convolution of logistic random variables

An expression is obtained for the distribution of a convolution of independent and identically distributed logistic random variables by directly inverting the characteristic function. This distribution is shown to be closely approximated by a student'st distribution when both distribution are standardized. Moreover, by showing that some of the analytic simplicity and statistical properties that are manifest in the single logistic also obtain in the convolution, an application of the convolution as a dose-response curve in the bio-assay problem is suggested.     

Preference for equivalent random variables: A price for unbounded utilities

Savage's expected utility theory orders acts by the expectation of the utility function for outcomes over states. Therefore, preference between acts depends only on the utilities for outcomes and the probability distribution of states. When acts have more than finitely many possible outcomes, then utility is bounded in Savage's theory. This paper explores consequences of allowing preferences over acts with unbounded utility. Under certain regularity assumptions about indifference, and in order to respect (uniform) strict dominance between acts, there will be a strict preference between some pairs of acts that have the same distribution of outcomes. Consequently in these cases, preference is not a function of utility and probability alone.     

The effects of sampling on measures of association between variables based on sample means

Many studies that involve people's perceptions or behaviors focus on aggregate rather than individual responses. For example, variables describing public perceptions for some set of events may be represented as mean scores for each event. Event mean scores then become the unit of analysis for each variable. The variance of these mean scores for a variable is not only a function of the variation among the events themselves, but is also due to the variation among respondents and their possible responses. This is also the case for the covariances between variables based on event mean scores. In many contexts the variance and covariance components attributable to the sampling of respondents and their responses may be large; these components can be described as measurement error. In this paper we show how to estimate variances and covariances of aggregate variables that are free of these sources of measurement error. We also present a measure of reliability for the event means and examine the effect of the number of respondents on these spurious components. To illustrate how these estimates are computed, forty-two respondents were asked to rate forty events on seven risk perception variables. Computing the variances and covariances for these variables based on event means resulted in relatively large components attributable to measurement error. A demonstration is given of how this error is removed and the resulting effect on our estimates.     

Runs in an ordered sequence of random variables

Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables      

Convergence of information,random variables and noise

Random variables such as state-dependent prices often convey information while entering directly into a decision environment. To understand the convergence properties of behavior based on such random variables, the function mapping any random variable to the information it generates is examined. With respect to convergence in probability of random variables, the information map is continuous with respect to pointwise convergence of information only at completely revealing random variables. Continuity holds when the information map is restricted to any subset of random variables to which the same smooth independent term has been added. Relationships between convergence in distribution of perturbed random variables and convergence in distribution of conditional expectations are also studied.     

Stability and self-decomposability of semi-group valued random variables

A random variable X on IR₊ is said to be self-decomposable, dif for all c∈ (0, 1) there exists a random variable X_c on IR₊ such that X=^dcX+X_c . It is said to be stable if it is self-decomposable and X_c=^d (1 - c)X' , where X and X' are identically and independently distributed. The notions of stability and self-decomposability for infinitely divisible random variables are generalised to abelian semi-groups ( S, + ) with S having an identical involution, by using characteristic functions. The generalised definitions involve semi-groups of scaling operators T . There operators can be interpreted in a slightly different context as generalised continuous-time branching processes (with immigration). The underlying importance of the generator of the semi-groups T in the characterisation of stability and self-decomposability is stressed.     

On the existence of random measure preserving bijections

This paper establishes the existence of an invariant probability measure on the set Π of measure-preserving bijections of the unit interval. In fact, the measure may be further restricted to be a Baire-measure. This result is compared with Aumann's (1967) and Aumann and Shapley's (1974) impossibility theorems.     

Tendency towards normality of linear combinations of random variables

IfX andY are two random variables with the same means and variances, thenX is said to be nearer normal thanY if the absolute values of its cumulants are smaller than the corresponding cumulants ofY. Using this definition, it is shown that a linear combination of a finite number of independent identically distributed random variables is always nearer normal than its constituents, but that this is not necessarily true if not-identically distributed or not-independent variables are used. Some consequences of the results are reached for the testing of normality of time series and for the assumptions frequently made by social scientists about the distribution of their data.