Tables of cumulative distribution functions and percentiles of the standardized stable random variables

It is shown that Zolotarev's (1964) integral representation of the cumulative distribution function (c.d.f.) of stable random variables and the IMSL subroutine DCADRE (for numerical integration ) provide a natural and practically simple method for finding the values of c.d.f., the percentiles and the density function of such random variables. For symmetric stable random variables (r.v.'s ) Z_∝, values of P_∝(z) … P(0<Z_∝<z) for z … 0(.02)4.08 and ∝=.1(.2)1.9, as well as percentiles of these r.v.'s for ∝=.5(.1)2 and the percentage points .6, .7(.05).85(.025).9(.01).96(.005).995, are presented. For asymmetric stable r.v.'s we present values of their c.d.f.'s for z … 0(.1)4, ß= ?1(.25)1 and ∝=.1(.2)1.9. These result sare compared with related results of others which were obtained by using different procedure and standardization.     

Limit theorems for random maximum of independent and non-identically distributed random vectors

In this paper, we study the weak convergence of the random maximum of independent and non-identical random vectors. When the random sample size is assumed to be independent of the basic variables and its distribution function is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random maximum are derived. An illustrative example is given.     

Limit distributions of order statistics with random indices in a stationary Gaussian sequence

In this article, we study the limit distributions of the extreme, intermediate, and central order statistics (os) of a stationary Gaussian sequence under equi-correlated setup. When the random sample size is assumed to converge weakly and to be independent of the basic variables, the sufficient (and in some cases the necessary) conditions for the convergence are derived. Finally, we show that the obtained result for the maximum os, with random sample size, is also applicable in the case of the non constant correlation case.     

On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures

We prove a transfer theorem for random sequences with independent random indexes in the double array limit setting under relaxed conditions. We also prove its partial inverse providing the necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed random variables. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance–mean mixtures.     

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.     

Asymptotic behavior of the record value sequence

The class of limit distribution functions of the random record model is fully characterized. Necessary and sufficient conditions as well as the domains of attraction of the limit distribution functions are obtained.     

Limit theory of bivariate dual generalized order statistics with random index

The class of limit distribution functions of bivariate extreme, intermediate and central dual generalized order statistics from independent and identically distributed random variables with random sample size is fully characterized. Two cases are considered. The first case is when the random sample size is assumed to be independent of all basic random variables. The second case is when the interrelation of the random size and the basic random variables is not restricted.     

On Bounds for Probability Generating Functions

Brook (1966) gave an upper bound for the moment generating function (m.g.f.) of a positive random variable (r.v.) in terms of its moments, and used this to obtain an upper bound for the probability generating function (p.g.f.) and hence the extinction probability of a simple branching process. Agresti (1974) rederived this bound of the p.g.f. and used it to obtain a lower bound of the expectation of extinction time of a branching process. In both of these applications the random variable is integer valued, and for this class we improve on Brook's bound by deriving the best upper bound of the p.g.f. Our method, which is a variant of Brook's (1966) is used later to obtain the lower bound of the p.g.f. when the third moment is also known.     

Limit theory of generalized order statistics

Generalized order statistics (gos) were introduced by Kamps [1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart] to unify several models of ordered random variables (rv's), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos that includes oos and sos, the possible limit distribution functions (df's) of the maximum gos are obtained in Nasri-Roudsari [1996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, 281–297]. In this paper, for this subclass, necessary and sufficient conditions of weak convergence, as well as the form of the possible limit df's of extreme, intermediate and central gos are derived. These results are extended to a wider subclass.     

Weak Limit of the Sample Extremal Quotient

Necessary and sufficient conditions are obtained for the weak convergence ( W →) of the sample extremal quotient of independent and identically distributed random variables as the number of observations tends to infinity. The paper fully characterizes the class of possible non-degenerate limit distribution functions of the extremal quotient, under general conditions.     

Uniform asymptotics for ruin probability of a two-dimensional dependent renewal risk model

ABSTRACT

In this paper we consider the tail behavior of a two-dimensional dependent renewal risk model with two dependent classes of insurance business, in which the claim sizes are governed by a common renewal counting process, and their inter-arrival times are dependent, identically distributed. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all time in an infinite interval. Moreover, we point out that the formula still holds uniformly for all time in an infinite interval for widely dependent random variables (r.v.s) under some conditions.     

Some new classes of multivariate survival distribution functions

The concept of the univariate mean remaining life (m.r.l.) function is generalized to the multivariate case. The multivariate mean remaining life (m.m.r.l.) function is utilized to introduce four new classes of multivariate survival distribution functions (s.d.f.'s). Each of these classes is a new generalization of the univariate decreasing mean remaining life (DMRL) class of s.d.f.'s. The duals of these classes are introduced. Some properties, physical interpretation, and relationships among these classes are investigated. Also for each case, the class of s.d.f.'s common in a class and its dual is characterized.     

On some properties of variable size simple random sampling and a limit theorem

This paper introduces a sampling plan for finite populations herein called “variable size simple random sampling” and compares properties of estimators based on it with results from the usual fixed size simple random sampling without replacement. Necessary and sufficient conditions (in the spirit of Hajek (1960)) for the limiting distribution of the sample total (or sample mean) to be normal are given.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

Asymptotic random extremal ratio and product based on generalized order statistics and its dual

The extremal ratio has been used in several fields, most notably in industrial quality control, life testing, small-area variation analysis, and the classical heterogeneity of variance situation. In many biological, agricultural, military activity problems and in some quality control problems, it is almost impossible to have a fixed sample size, because some observations are always lost for various reasons. Therefore, the sample size itself is considered frequently to be an random variable (rv). Generalized order statistics (GOS) have been introduced as a unifying theme for several models of ascendingly ordered rvs. The concept of dual generalized order statistics (DGOS) is introduced to enable a common approach to descendingly ordered rvs. In this article, the limit dfs are obtained for the extremal ratio and product with random indices under non random normalization based on GOS and DGOS. Moreover, this article considers the conditions under which the cases of random and non random indices give the same asymptotic results. Some illustrative examples are obtained, which lend further support to our theoretical results.     

The role of probability generating functions for estimation in incompletely observed random walks

Partial observation of a random walk results in independent convolutions of i.i.d. variables. It is known that under a scheme of sufficiently frequent observation, moments of the random walk can be consistently estimated. In these cases, probability generating functions (p.g.f.'s) can be used to circumvent the difficulties posed by likelihood estimation involving convolutions. Asymptotic properties of the p.g.f. estimates are given, and a comparison is made with the method-of-moments estimator, which is also shown to be asymptotically normal. In a parametric context, the p.g.f. estimator is shown to be asymptotically efficient. Monte Carlo experiments demonstrate that there are advantages to using the p.g.f.-based estimate in small samples as well.     

Lord's MAD Paradox and Jensen's Inequality

Lord (1983) showed that the mean absolute deviation (MAD) measure of dispersion does not always increase when an additional independent source of random variation is added. Jensen's inequality is used to find necessary and sufficient conditions for this paradoxical behavior. The conditions are unlikely to occur in practice.     

A Note On Weighted Distributions

A representation of the "original" random variable (r.v.) in terms of the "weighted" r.v. is given and the Inverse Gaussian distribution is characterized through a distributional property the "weighted" r.v. observed under 1ength biased sampling.