On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

The Beta Product Distribution with Complex Parameters

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

A new class of skew-symmetric distributions and related families

In this work, we investigate a new class of skew-symmetric distributions, which includes the distributions with the probability density function (pdf) given by g _α(x)=2f(x) G(α x), introduced by Azzalini [A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]. We call this new class as the symmetric-skew-symmetric family and it has the pdf proportional to f(x) G _β(α x), where G _β(x) is the cumulative distribution function of g _β(x). We give some basic properties for the symmetric-skew-symmetric family and study the particular case obtained from the normal distribution.     

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)^α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

A New Type of Discrete Self-Decomposability and Its Application to Continuous-Time Markov Processes for Modeling Count Data Time Series

We propose a family of extended thinning operators, indexed by a parameter γ in [0, 1), with the boundary case of γ=0 corresponding to the well-known binomial thinning operator. The extended thinning operators can be used to construct a class of continuous-time Markov processes for modeling count time series data. The class of stationary distributions of these processes is called generalized discrete self-decomposable, denoted by DSD (γ). We obtain characterization results for the DSD (γ) class and investigate relationships among the classes for different γ's.     

Kshirsagar-Type Lower Bounds for Mean Squared Error of Prediction

Let Y be an observable random vector and Z be an unobserved random variable with joint density f(y, z | θ), where θ is an unknown parameter vector. Considering the problem of predicting Z based on Y, we derive Kshirsagar type lower bounds for the mean squared error of any predictor of Z. These bounds do not require the regularity conditions of Bhattacharyya bounds and hence are more widely applicable. Moreover, the new bounds are shown to be sharper than the corresponding Bhattacharyya bounds. The conditions for attaining the new lower bounds are useful for easy derivation of best unbiased predictors, which we illustrate with some examples.     

Minimum-Distance Estimator for Stable Exponent

Assume that X ₁, X ₂,…, X _n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G _n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.     

Monotonicity Conditions and Inequality Imputation for Sample-Selection and Non-Response Problems

ABSTRACT

Under a sample selection or non-response problem, where a response variable y is observed only when a condition δ = 1 is met, the identified mean E(y|δ = 1) is not equal to the desired mean E(y). But the monotonicity condition E(y|δ = 1) ≤ E(y|δ = 0) yields an informative bound E(y|δ = 1) ≤ E(y), which is enough for certain inferences. For example, in a majority voting with δ being the vote-turnout, it is enough to know if E(y) > 0.5 or not, for which E(y|δ = 1) > 0.5 is sufficient under the monotonicity. The main question is then whether the monotonicity condition is testable, and if not, when it is plausible. Answering to these queries, when there is a ‘proxy’ variable z related to y but fully observed, we provide a test for the monotonicity; when z is not available, we provide primitive conditions and plausible models for the monotonicity. Going further, when both y and z are binary, bivariate monotonicities of the type P(y, z|δ = 1) ≤ P(y, z|δ = 0) are considered, which can lead to sharper bounds for P(y). As an empirical example, a data set on the 1996 U.S. presidential election is analyzed to see if the Republican candidate could have won had everybody voted, i.e., to see if P(y) > 0.5, where y = 1 is voting for the Republican candidate.     

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]^α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.     

A Bivariate Distribution Family with Specified Correlation Coefficient and Given Marginals

For given continuous distribution functions F(x) and G(y) and a Pearson correlation coefficient ρ, an algorithm is provided to construct a sequence of continuous bivariate distributions with marginals equal to F(x) and G(y) and the corresponding correlation coefficient converges to ρ. The algorithm can be easily implemented using S-Plus or R. Applications are given to generate bivariate random variables with marginals including Gamma, Beta, Weibull, and uniform distributions.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.     

Characterizations of Distributions Based on Moments of Residual Life

Let T be a random variable having an absolutely continuous distribution function. It is known that linearity of E(T | T > t) can be used to characterize distributions such as exponential, power and Pareto distribution. In this work, we will extend the above results. More precisely, we characterize the distribution of T by using certain relationships of conditional moments of T. Our results can also be used to obtain new characterization of distributions based on adjacent order statistics or record values.     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Skewed Bessel function distributions with application to rainfall data

Skewed distributions have attracted significant attention in the last few years. In this article, a skewed Bessel function distribution with the probability density function (pdf) f(x)=2 g (x) G (λ x) is introduced, where g (·) and G (·) are taken, respectively, to be the (pdf) and the cumulative distribution function of the Bessel function distribution [McKay, A.T., 1932, A Bessel function distribution, Biometrica, 24, 39–44]. Several particular cases of this distribution are identified and various representations for its moments derived. Estimation procedures by the method of maximum likelihood are also derived. Finally, an application is provided to rainfall data from Orlando, Florida.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

Improved Estimators of the Population Mean for Missing Data

For two dependent random variables X and Y with distributions of convolution equivalence, sufficient conditions are given under which the distribution of the minimum min (X, Y) is still of convolution equivalence. We further extend the result to the multivariate case.     

A general non-central hypergeometric distribution

We construct a general non-central hypergeometric distribution, which models biased sampling without replacement. Our distribution is constructed from the combined order statistics of two samples: one of independent and identically distributed random variables with absolutely continuous distribution F and the other of independent and identically distributed random variables with absolutely continuous distribution G. The distribution depends on F and G only through F○G^{( ? 1)} (F composed with the quantile function of G), and the standard hypergeometric distribution and Wallenius’ non-central hypergeometric distribution arise as special cases. We show in efficient economic markets the quantity traded has a general non-central hypergeometric distribution.