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.     

Gamma variates of fractional shape as functioials of a homogeneous multidimensional poisson process

If events are scattered in Rⁿ in accordance with a homogeneous Poisson process and if X is the location of the event with minimal [d]l^P norm, then in the case p = n the n^th absolute powers of the coordinates of X form a sample of size n from a gamma distribution with shape parameter 1/n. In an age of parallel computing, this fact may lead to some attractive simulation methods. One possibility is to generate R = [d]X[d] and U = Y/[d]X[d] independently, perhaps by setting U = Y/[d]Y[d] where Y has any p.d.f. which is a function only of ¦Y¦. We consider for example Y having the uniform distribution in an l^P ball.     

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n₁, …, n_k)th degree, then all the mixed moments up to the order (n₁, …, n_k) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.     

A note on characterizations of the bivariate gamma distribution

Given two independent non-degenerate positive random variables X and Y, Lukacs (1955) proved that X/(X+Y) and X+Y are independent if and only if X and Y are gammally distributed with the same scale parameter.In this work, properties of bivariate gamma distribution are studied. Certain regression version of Lukacs's theorem are given for the bivariate case. Furthermore, characterization of bivariate gamma distribution by the conditions of constancy regression of quadratic statistics is also given.     

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.     

Some inference results on pr(x < y) in the bivariate exponential model

This paper provides three different estimators for Pr(X < Y) when X and Y have a bivariate exponential distribution. The asymptotic variances of the three estimators are also derived. A test for the equality of the means of X and Y and confidence limits for the difference of the two means are presented. Our results are directly applicable in a reliability context with underlying bivariate exponential distribution.     

Some bivariate uniform distributions

Bivariate uniform distributions with dependent components are readily derived by distribution function transformations of the components of non-uniform dependent continuous bivariate random variables (X,Y). Contour plots of joint density functions show the various, and varying, forms of dependence which can arise from different distributional forms for (X,Y) and aids the choice of bivariate uniform distributions as empirical models.     

On univariate and bivariate generalized gamma convolutions

This paper has two parts. In the first part some results for generalized gamma convolutions (GGCs) are reviewed. A GGC is a limit distribution for sums of independent gamma variables. In the second part, bivariate gamma distributions and bivariate GGCs are considered. New bivariate gamma distributions are derived from shot-noise models. The remarkable property hyperbolic complete monotonicity (HCM) for a function is considered both in the univariate case and in the bivariate case.     

Joint distribution of simultaneous exposures to several carcinogens in a case–control study: sample size determination

Consider a population of individuals who are free of a disease under study, and who are exposed simultaneously at random exposure levels, say X,Y,Z,… to several risk factors which are suspected to cause the disease in the populationm. At any specified levels X=x, Y=y, Z=z, …, the incidence rate of the disease in the population ot risk is given by the exposure–response relationship r(x,y,z,…) = P(disease|x,y,z,…). The present paper examines the relationship between the joint distribution of the exposure variables X,Y,Z, … in the population at risk and the joint distribution of the exposure variables U,V,W,… among cases under the linear and the exponential risk models. It is proven that under the exponential risk model, these two joint distributions belong to the same family of multivariate probability distributions, possibly with different parameters values. For example, if the exposure variables in the population at risk have jointly a multivariate normal distribution, so do the exposure variables among cases; if the former variables have jointly a multinomial distribution, so do the latter. More generally, it is demonstrated that if the joint distribution of the exposure variables in the population at risk belongs to the exponential family of multivariate probability distributions, so does the joint distribution of exposure variables among cases. If the epidemiologist can specify the differnce among the mean exposure levels in the case and control groups which are considered to be clinically or etiologically important in the study, the results of the present paper may be used to make sample size determinations for the case–control study, corresponding to specified protection levels, i.e., size α and 1–β of a statistical test. The multivariate normal, the multinomial, the negative multinomial and Fisher's multivariate logarithmic series exposure distributions are used to illustrate our results.     

Proportions,sums and ratios

Data that are proportions arise frequently in all areas of the sciences and engineering. In this paper, the exact distributions of R=X+Y and W=X/(X+Y) and the corresponding moment properties are derived when X and Y are proportions and arise from the most flexible bivariate beta distribution known to date. The associated estimation procedures are developed. Finally, an application is illustrated to compositional data of lavas from Skye.     

Bayesian tests for matched proportions

A sample of n subjects is observed in each of two states, S₁-and S₂. In each state, a subject is in one of two conditions, X or Y. Thus, a subject may be recorded as showing a change if its condition in the two states is ‘Y,X’ or ‘X,Y’ and, otherwise, the condition is unchanged. We consider a Bayesian test of the null hypothesis that the probability of an ‘X,Y’ change exceeds that of a ‘Y,X’ change by amount k_O. That is, we develop the posterior distribution of k_O, the difference between the two probabilities and reject the null hypothesis if k lies outside the appropriate posterior probability interval. The performance of the method is assessed by Monte Carlo and other numerical studies and brief tables of exact critical values are presented     

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.     

An Efficient Simulation Method for the Computation of a Class of Conditional Expectations

Suppose that the random vector X and the random variable Y are jointly continuous. Also suppose that an observation x of X can be easily simulated and that the probability density function of Y conditional on X = x is known. The paper presents an efficient simulation-based algorithm for estimating E{ g ( X , Y ) | h ( X , Y ) = r } where g and h are real-valued functions. This algorithm is applicable to time series problems in which X = ( X ₁, . . . , X _n−1) and Y = X_n where { x_t } is a discrete time stochastic process for which ( X₁ , . . . , X_n ) is a continuous random vector. A numerical example from time series analysis illustrates the algorithim, for prediction for an ARCH(1) process.     

Computation of the central and noncentral f distributions

Recursion relations suitable for rapid computation are derived for the cumulative distribution of F′ = (X/m)/(Y/n) where X is χ²(λ, m) and Y is independently χ²(n). When n is even no complicated function evaluations are needed. For n odd, a special doubly noncentral t distribution is needed to start the computation. Series representations for this t distribution are given with rigorous bounds on truncation errors. Proper recursion techniques for numerical evaluation of the special functions are given.     

Ordering of Sequentially Sampled Exponential Experiments

Let X ₁, X ₂, ... be a sequence of i.i.d. random variables, X _i∼ F _θ, θ∈Θ. Let N ₁ and N ₂ be two stopping rules. For a class of exponential families { F _θ: θ∈Θ} we show that the experiment Y ₁ = ( X ₁, ..., X _N1) carries more statistical information than Y ₂ = ( X ₁, ..., x _N2) only if N ₁ is stochastically larger then N ₂     

Two Moments of the Logitnormal Distribution

We display the first two moment functions of the Logitnormal(μ, σ²) family of distributions, conveniently described in terms of the Normal mean, μ, and the Normal signal-to-noise ratio, μ/σ, parameters that generate the family. Long neglected on account of the numerical integrations required to compute them, awareness of these moment functions should aid the sensible interpretation of logistic regression statistics and the specification of “diffuse” prior distributions in hierarchical models, which can be deceiving. We also use numerical integration to compare the correlation between bivariate Logitnormal variables with the correlation between the bivariate Normal variables from which they are transformed.     

Probabilistic tolerance design for a subsystem under burr distribution using taguchi's loss functions

Taguchi (1986) has derived tolerances for subcomponents, subsystems, parts and materials in which the relationship between a higher-level (Y) and a lower-level (X) quality characteristic is assumed to be deterministic and linear, namely, Y=_α+βX, without an error term. Tsai (1990) developed a probabilistic tolerance design for a subsystem in which a bivariate normal distribution between the above two quality characteristics as well as Taguchi's quadratic loss function were considered together to develop a closed form solution of the tolerance design for a subsystem. The Burr family is very rich for fitting sample data, and has positive domain. A bivariate Burr distribution can describe a nonlinear relationship between two quality characteristics, hence, it is adopted instead of a bivariate normal distribution and the simple solutions of three probabilistic tolerance desings for a subsystem are obtained for three cases of “nominal-is-best”, “smaller-is-berrer”, and “larger-is-beter” quality characteristics, by using Taguchi’ los functions, respectively.     

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.     

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

A Bivariate Generalization of Gamma Distribution

In this article, a bivariate generalisation of the gamma distribution is proposed by using an unsymmetrical bivariate characteristic function; an extension to the non central case also receives attention. The probability density functions of the product and ratio of the correlated components of this distribution are also derived. The benefits of introducing this generalized bivariate gamma distribution and the distributions of the product and the ratio of its components will be demonstrated by graphical representations of their density functions. An example of this generalized bivariate gamma distribution to rainfall data for two specific districts in the North West province is also given to illustrate the greater versatility of the new distribution.