Tests of fit for the Rayleigh distribution based on the empirical Laplace transform

In this paper a class of goodness-of-fit tests for the Rayleigh distribution is proposed. The tests are based on a weighted integral involving the empirical Laplace transform. The consistency of the tests as well as their asymptotic distribution under the null hypothesis are investigated. As the decay of the weight function tends to infinity the test statistics approach limit values. In a particular case the resulting limit statistic is related to the first nonzero component of Neyman’s smooth test for this distribution. The new tests are compared with other omnibus tests for the Rayleigh distribution.     

Simulating Multivariate Extreme Value Distributions of Logistic Type

Methods are given for simulating from symmetric and asymmetric versions of the multivariate logistic distribution, and from other multivariate extreme value distributions based on the well known logistic model. We consider two general approaches. The first approach uses transformations to derive random variables with a joint distribution function from which it is easy to simulate. The second approach derives from a specification of conditionally independent marginal components, conditioning on positive stable random variables. This specification extends to models of nested or hierarchical type and leads to an efficient way of incorporating marginal censoring. The algorithms presented in Sections 2 and 3 are available on request from the author. They are also included in the R (Ihaka and Gentleman, 1996) package evd (Stephenson, 2002), which is available from http://www.maths.lancs.ac.uk/~stephena/.     

Maxima of Subexponential Shot-Noise Fields with Finite Radius of Influence

We study the asymptotic behavior of the maxima of shot-noise fields on bounded measurable domains tending to infinity in the sense of van Hove. It is assumed that the radius of influence is finite and the amplitudes are subexponentially distributed. A nondegenerate limiting distribution for the maxima is obtained.     

CONVERGENCE OF CASCADE ALGORITHMS AND SMOOTHNESS OF REFINABLE DISTRIBUTIONS

In this paper, the author at first develops a method to study convergence of the cascade algorithm in a Banach space without stable assumption on the initial (see Theorem 2.1), and then applies the previous result on the convergence to characterizing compactly supported refinable distributions in fractional Sobolev spaces and Holder continuous spaces (see Theorems 3.1, 3.3, and 3.4). Finally the author applies the above characterization to choosing appropriate initial to guarantee the convergence of the cascade algorithm (see Theorem 4.2).     

EXPLICIT EXPRESSIONS FOR SOME DISTRIBUTIONS RELATED TO RUIN PROBLEMS

The classical risk process that is perturbed by diffusion is studied .The explicit expressions for the runi probability and the surplus distribution of the risk process at the time of runi are obtained when the claim amount distribution is a finite mixture of exponential distributions of a Gamma (2,α） distribution.     

GENERALIZED OPERATORS AND OPERATOR-VALUED DISTRIBUTIONS IN QUANTUM FIELD THEORY

The relation between generalized operators and operator-valued distributions is discussed so that these two viewpoints can be used alternatively to explain quantum fields.     

DISTRIBUTIONS OF LIMIT CYCLES FOR POLYNOMIAL VECTOR FIELDS (P_2, Q_3)

The distributions of limit cycles of cubic vector fields (P2, Q3) are considered in this paper, where P2 and Q3 are polynomials of x and y of order two and three, respectively. It is possibly seven different distributions of limit cycles given in [1]. We now prove that in which three kinds of distributions are impossible and other four kinds all can be realized by concrete vector fields of (P2,Q3). Some related results are also given.     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

Computation of the quasi-stationary distributions inM(n)/GI/1/K andGI/M(n)/1/K queues

In this paper, we provide numerical means to compute the quasi-stationary (QS) distributions inM/GI/1/K queues with state-dependent arrivals andGI/M/1/K queues with state-dependent services. These queues are described as finite quasi-birth-death processes by approximating the general distributions in terms of phase-type distributions. Then, we reduce the problem of obtaining the QS distribution to determining the Perron-Frobenius eigenvalue of some Hessenberg matrix. Based on these arguments, we develop a numerical algorithm to compute the QS distributions. The doubly-limiting conditional distribution is also obtained by following this approach. Since the results obtained are free of phase-type representations, they are applicable for general distributions. Finally, numerical examples are given to demonstrate the power of our method.     

One-dimensional non-nearest-neighbor random walks in the presence of traps

A one-dimensional lattice random walk in the presence ofm equally spaced traps is considered. The step length distribution is a symmetric exponential. An explicit analytic expression is obtained for the probability that the random walk will be trapped at thejth trapping site.