A property of the yule distribution and its applications

The relationship Y = RX between two random variables X and Y, where R is distributed independently of X in (0, l), is known to have important consequences in different fields such as income distribution analysis, Inventory decision models, etc.

In this paper it is shown that when X and Y are discrete random variables, relationships of similar nature lead to Yule-type distributions. The implications of the results are studied in connection with problems of income underreporting and inventory decision making.     

Computation of the Generalized F Distribution

Exact expressions for the cumulative distribution function of a random variable of the form ( α ₁ X ₁+ α ₂ X ₂)/ Y are given where X ₁, X ₂ and Y are independent chi-squared random variables. The expressions are applied to the detection of joint outliers and Hotelling's mis-specified T ² distribution.     

Bayesian Estimation of Survival Functions under Stochastic Precedence

When estimating the distributions of two random variables, X and Y, investigators often have prior information that Y tends to be bigger than X. To formalize this prior belief, one could potentially assume stochastic ordering between X and Y, which implies Pr(X < or = z) > or = Pr(Y < or = z) for all z in the domain of X and Y. Stochastic ordering is quite restrictive, though, and this article focuses instead on Bayesian estimation of the distribution functions of X and Y under the weaker stochastic precedence constraint, Pr(X < or = Y) > or = 0.5. We consider the case where both X and Y are categorical variables with common support and develop a Gibbs sampling algorithm for posterior computation. The method is then generalized to the case where X and Y are survival times. The proposed approach is illustrated using data on survival after tumor removal for patients with malignant melanoma.     

Joint analysis of longitudinal data comprising repeated measures and times to events

In biomedical and public health research, both repeated measures of biomarkers Y as well as times T to key clinical events are often collected for a subject. The scientific question is how the distribution of the responses [ T , Y | X ] changes with covariates X . [ T | X ] may be the focus of the estimation where Y can be used as a surrogate for T . Alternatively, T may be the time to drop-out in a study in which [ Y | X ] is the target for estimation. Also, the focus of a study might be on the effects of covariates X on both T and Y or on some underlying latent variable which is thought to be manifested in the observable outcomes. In this paper, we present a general model for the joint analysis of [ T , Y | X ] and apply the model to estimate [ T | X ] and other related functionals by using the relevant information in both T and Y . We adopt a latent variable formulation like that of Fawcett and Thomas and use it to estimate several quantities of clinical relevance to determine the efficacy of a treatment in a clinical trial setting. We use a Markov chain Monte Carlo algorithm to estimate the model's parameters. We illustrate the methodology with an analysis of data from a clinical trial comparing risperidone with a placebo for the treatment of schizophrenia.     

On the effect of overdispersion on exact conditional tests

Let Y ₁, . . ., Y_n denote independent random variables such that Y_j has a one-parameter exponential family distribution with canonical parameter θ _j =λ+ψ X_j ; here X ₁, . . ., X_n are known constants. Consider a test of the null hypothesis ψ=0. Under the null hypothesis, A =Σ Y_j is sufficient for λ and, hence, a test of ψ=0 may be based on the conditional distribution of T =Σ X_j Y_j given A , which is independent of λ. In this paper, the effects of overdispersion due to a mixture model on the conditional distribution of T given A are considered.     

Attaining specified reliability in component design: a stochastic approximation approach

In the design, manufacture and maintenance of components, particular attention is paid to component reliability R, the probability that the strength X of a component will exceed a stress Y to which it will be subjected. The problem addressed here is the design (or redesign) of a compoFent to meet a specified reliability R*. While certain characteristics of the random variables X and Y are assumed (symmetry of X about a unique median for example) it is not assumed that the form of the distribution of (X,Y) is known, nor that X and Y are independent. A design is recomnended based on a variation of the stochastic approximation procedure due to Dupac and Kral (1972) which in general estimates recursively the root of a regression curve assuming both independent and dependent regression variables are subject to experimental error.     

Estimating P[X>Y] from exponential samples containing spurious observations

In this paper we discuss the problem of estimating P[X>Y] when X and Y are independent exponential random variables and the sample from each population contains one spurious observation. The estimates ate derived for exchangeable, identifiable and censored models and their performances are evaluated numerically.     

On some properties of bivariate weighted distributions

Let X^w and Y^w be weighted random variables arising from the distribution of (X,Y). We explore implications of independence of X and Y on the dependence structure of (X^w, Y^w). We also show that when X and Y are independent and the weight function is symmetric, identical distribution of X^w and Y^w implies that of X and Y. We discuss application of these results to the study of a renewal process.     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

Moments for a ratio of correlated gamma variates

This paper considers the finite integral moments for the ratio, R = X/Y, where X and Y re correlated gamma distributed variables. An analytical and numerical comparison is given for two classes of underlying bivariate gamma distributions. It is shown that the two bivariate gamma structures provide indentical experessions for the mth unadjussted moment, E(R^m), if and only if either of the following conditions hold : 1) X and Y are uncorrelated of 2) m=1. A numerical evaluation is performed to determine the extent that the two methods differ whenever the variables are correlated     

Symmetrically distributed and unbiased estimators in linear models

Let Y be distributed symmetrically about Xβ. Natural generalizations of odd location statistics, say T‘Y’, and even location-free statistics, say W‘Y’, that were used by Hogg ‘1960, 1967)’ are introduced. We show that T‘Y’ is distributed symmetrically about β and thus E[T‘Y’] = β and that each element of T‘Y’ is uncorrelated with each element of W‘Y’. Applications of this result are made to R-estiraators and the result is extended to a multivariate linear model situation.     

Tests for Bivariate Symmetry Against Ordered Alternatives in Square Contingency Tables

Let X and Y denote two ordinal response variables, each having I levels. When subjects are classified on both variables, there are I 2 possible combinations of classifications. Let pij = Pr (X = i, Y = j) . This paper introduces a family of tests based on φ –divergence measures for testing H₀: pij = pji against H1: pij ≥ pji (I≥ j) ; and for testing H1 against H2: pij unrestricted. A simulation study assesses some of the family of tests introduced in this paper in comparison to the likelihood ratio test.     

SOME CHARACTERIZATIONS OF THE CAUCHY DISTRIBUTION

If X and Y are independent standard Cauchy random variables then (i) Y and (X+Y)/(1-Xu) are independent, (ii) X and (X + Y)/(1 -XU) are identically distributed, and (iii) X and 2X/(1-X²) are identically distributed. Each of these three properties is shown to characterize the Cauchy distribution among absolutely continuous distributions. Some related uniform characterizations are discussed.     

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {( X _n , Y _n )} is a sequence of pairs of cector-valued stochastic variables which converges weakly to ( X , Y ), and that { y _n } converges to y . Sufficient conditions for the conditional distribution of X _n given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.     

Estimation of p(y<x) in the burr case: a comparative study

This paper provides a simulation study which compares three estimators for R = P(Y<X) when Y and X are two independent but not identically distributed Burr random variables. These estimators are the minimum variance unbiased, the maximum likelihood and Bayes estimators. Moreover, the sensitivity of Bayes estimator to the prior parameters is considered.     

Approximate Identifiability, Moments and Censored Data

It is well known that the joint distribution of a pair of random variables ( X,Y ) is not identifiable on the basis of the joint distribution of the function (min ( X,Y ), 1[ X < Y ]). This paper introduces the concept of approximate identifiability and studies its relevance to the function (min ( X,Y ), Y ). It shows that the distribution of ( X,Y ) is approximately identifiable on the basis of the distribution of (min ( X,Y ), Y ). The identification is explicitly executed by a method of moments. The method is applied to the analysis of censored distributions arising in the theory of clinical trials and is compared to the standard method of Kaplan and Meier.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

A characterization of and new consistent tests for symmetry

Let X and Y be independent and identically distributed random variables having a continuous distribution function. We study new consistent tests for symmetry around a known median based on the fact that the distribution of X is symmetric around 0 if, and only if, |X| and |max(X,Y)| have the same distribution.     

Non-linear canonical correlation analysis with a simulated annealing solution

An exploratory tool is introduced to examine potential non-linear relation-ships between two sets of variables, X andY, in a sample of multivariate data. Simulated annealing is applied to find canonical coefficient vectors a and b such that a squared non-linear correlation between a'Xand b'Y is maximiSed. A measure of non-linear correlation is developed for this optimization which utilies a nearest-neighbor regression estimate for the unknown functional relationship. In addition to examining potential relations between the canonical variables, this method can identify the important variables in each set.     

Estimation of pr{y<x) for truncation parameter distributions

Blackwell-Rao-Lehmann-Scheffe theory is used to derive the minimum variance ur biased estimator of P=Pr{Y<X} when the independent random variables X and Y follow thf truncation parameter distributions The two-parameter exponential, Pareto, power function and uniform distributions are considered in examples.