Wen-Jang Huang

Some skew-symmetric models

The skew-normal distributions have been introduced by many authors, e.g. Azzalini (1985), Arnold et al. (1993), Aigner et al. (1977), Andel et al. (1984). This class of distributions includes the normal distribution and possesses several properties which coincide or are close to the properties of the normal family. However, this class has a skewness parameter which makes it possible to have a reasonable model for a skewed population distribution thus providing a more flexible model which represents the data as adequately as possible. Besides being useful in modeling, they are helpful in studying the robustness, and in Bayesian analysis as priors. The construction of such models is based on the following lemma (see Azzalini, 1985).

Quadratic forms in skew normal variates

In this paper first a characterization of the multivariate skew normal distribution is given. Then the joint moment generating functions of two quadratic forms, and a linear compound and a quadratic form in skew normal variates, have been derived and conditions for their independence are given. Distribution of the ratios of quadratic forms in skew normal variates has also been studied.

Note on a characterization of gamma distributions

Let M>K. If Y1,...,YK are independent; U, YK+1,..., YK+M are independent, U has a [beta](cK/(M - K), 1) distribution, where 0

Characterizations of the Poisson process as a renewal process via two conditional moments

Given two independent positive random variables, under some minor conditions, it is known that fromE(Xr∥X+Y)=a(X+Y)r andE(Xs∥X+Y)=b(X+Y)s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t≥0} be a renewal process, with {Sk, k≥1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, k≤n, ifE(Skr∥A(t)=n)=atr andE(Sks∥A(t)=n)=bts, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t≥0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.

Characterizations based on conditional expectations

For given real functionsg andh, first we give necessary and sufficient conditions such that there exists a random variableX satisfying thatE(g(X)|X≥y)=h(y)rx(y),∀y ∈ Cx, whereCx andTX are the support and the failure rate function ofX, respectively. These extend the results of Ruiz and Navarro (1994) and Ghitany et al. (1995). Next we investigate necessary and sufficient conditions such thath(y)=E(g(X)|X≥y), for a given functionh.

Some characterizations of the Poisson process and geometric renewal process

Let γ t and δ t denote the residual life at t and current life at t , respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G , under mild conditions, as long as holds for a single positive integer n , then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t , we find that for some fixed positive integer n , if is independent of t , then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t .

Characterizations of the Order Statistics Point Process by the Relations between Its Conditional Moments

Let A ≡{A(t), t≥0} be an order statistics point process, with E(A(t)) = m(t) being the mean value function of A(t), t≥0. It is known that m(t) determines the distribution of the process A. In this work, we give some characterizations of m(t) by using certain relations between the conditional moments of the last jump time or current life of A at time t. It is interesting that some results are parallel to those characterizations of Poisson process as a renewal process. Finally, we present some extensions of the results about record values given in Abu-Youssef (2003).

A study of generalized skew-normal distribution

Following the paper by Genton and Loperfido [Generalized skew-elliptical distributions and their quadratic forms, Ann. Inst. Statist. Math. 57 (2005), pp. 389–401], we say that Z has a generalized skew-normal distribution, if its probability density function (p.d.f.) is given by f(z)=2φ p (z; ξ, Ω)π (z−ξ), z∈ℝ p , where φ p (·; ξ, Ω) is the p-dimensional normal p.d.f. with location vector ξ and scale matrix Ω, ξ∈ℝ p , Ω>0, and π is a skewing function from ℝ p to ℝ, that is 0≤π (z)≤1 and π (−z)=1−π (z), ∀ z∈ℝ p . First the distribution of linear transformations of Z are studied, and some moments of Z and its quadratic forms are derived. Next we obtain the joint moment-generating functions (m.g.f.’s) of linear and quadratic forms of Z and then investigate conditions for their independence. Finally explicit forms for the above distributions, m.g.f.’s and moments are derived when π (z)=κ (α′z), where α∈ℝ p and κ is the normal, Laplace, logistic or uniform distribution function.

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.

On some study of skew-t distributions

Abstract In this note, through ratio of independent random variables, new families of univariate and bivariate skew-t distributions are introduced. Probability density function for each skew-t distribution will be given. We also derive explicit forms of moments of the univariate skew-t distribution and recurrence relations for its cumulative distribution function. Finally we illustrate the flexibility of this class of distributions with applications to a simulated data and the volcanos heights data.