Skew Models II

If g and G are the pdf and the cdf of a distribution symmetric around 0 then the pdf 2g(u)G(λ u) is said to define a skew distribution. In this paper, we provide a mathematical treatment of the skew distributions when g and G are taken to come from one of Pearson type II, Pearson type VII or the generalized t distribution.      

The exponentiated generalized power Lindley distribution: Properties and applications

In this paper, we introduce a new extension of the power Lindley distribution, called the exponentiated generalized power Lindley distribution. Several mathematical properties of the new model such as the shapes of the density and hazard rate functions, the quantile function, moments, mean deviations, Bonferroni and Lorenz curves and order statistics are derived.Moreover, we discuss the parameter estimation of the new distribution using the maximum likelihood and diagonally weighted least squares methods. A simulation study is performed to evaluate the estimators. We use two real data sets to illustrate the applicability of the new model. Empirical findings show that the proposed model provides better fits than some other well-known extensions of Lindley distributions.     

Sums,differences, products,and ratios of hypergeometric beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, differences, products and ratios of independent beta random variables. In this Note we extend Professor Pham-Gia's results when $X_1$ and $X_2$ are independent random variables distributed according to the confluent and Gauss hypergeometric distributions (which are generalizations of the beta distribution). For each of these distributions, we derive exact expressions for the densities of $S=X_1+X_2$, $D=X_1?X_2$, $P=X_1{X}_2$, and $R=X_2/X_1$. The expressions turn out to involve the hypergeometric functions of one and two variables. To cite this article: S. Nadarajah, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Estimation Methods for the Multivariate <Emphasis Type="Italic">t</Emphasis> Distribution

The known estimation and simulation methods for multivariate t distributions are reviewed. A review of selected applications is also provided. We believe that this review will serve as an important reference and encourage further research activities in the area.     

A Pareto model for classical systems

A new Pareto distribution is introduced for pooling knowledge about classical systems. It takes the form of the product of two Pareto probability density functions (pdfs). Various structural properties of this distribution are derived, including its cumulative distribution function (cdf), moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Copyright © 2007 John Wiley & Sons, Ltd.     

The distribution of the maximum of a first order moving average: the continuous case

We give the cumulative distribution function of $M_n$ , the maximum of a sequence of n observations from a first order moving average. Solutions are first given in terms of repeated integrals and then for the case, where the underlying independent random variables have an absolutely continuous probability density function. When the correlation is positive, $P( M_n \leq x ) \ =\ \sum\limits _{j=1}^{\infty } \beta _{j, x} \ \nu _{j, x}^{n},$ where $\{\nu _{j, x}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\beta _{j, x}$ are some coefficients determined later. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $\sum\limits _{i=1}^{\infty } w_i ( \lambda -\theta _i )^{-1}=\lambda -\theta _0$ for certain known weights $\{ w_i\}$ and eigenvalues $\{ \theta _i\}$ of a given matrix. This can be solved by truncating the sum to an increasing number of terms.     

On the Moments of the Absorption Time of Kingman’s Coalescent

Simple integral representations are derived for the moments of the absorption time of Kingman’s coalescent Kingman (J Appl Probab 19:27-43, (1982a)). Their computational efficiency versus known representations is established.