Abstract. This chapter provides a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.

this article then is to develop methodology for modeling the nonnormality of the ut, the vt, or both. A second departure from the model specification ( 1 ) is to allow for unknown variances in the state or observational equation, as well as for unknown parameters in the transition matrices Ft and Ht. As a third generalization we allow for nonlinear model structures; that is, X t = ft(Xt-l) q- Ut, and Yt = ht(xt) + vt, t = 1, ..., n, (2) whereft( ) and ht(. ) are given, but perhaps also depend on some unknown parameters. The experimenter may wish to entertain a variety of error distributions. Our goal throughout the article is an analysis for general state-space models that does not resort to convenient assumptions at the expense of model adequacy

A mathematical method called subordination broadens the applicability of the classical advection-dispersion equation for contaminant transport. In this method the time variable is randomized to represent the operational time experienced by different particles. In a highly heterogeneous aquifer the operational time captures the fractal properties of the medium. This leads to a simple, parsimonious model of contaminant transport that exhibits many of the features (heavy tails, skewness, and non-Fickian growth rate) typically seen in real aquifers. We employ a stable subordinator that derives from physical models of anomalous diffusion involving fractional derivatives. Applied to a onedimensional approximation of the MADE-2 data set, the model shows excellent agreement.

We derive useful distributional representations for three discrete laws: the discrete stable distribution of Steutel and Van Harn, the discrete Linnik distribution introduced by Pakes, and a distribution of Sibuya. These representations may be used to obtain simple uniformly fast random variate generators.

Polya has shown that real even continuous functions that are convex on (0,oo), 1 for t -- 0, and decreasing to 0 as t--, o are characteristic functions. Dugu6 and Girault (1955) have shown that the corresponding random variables are distributed as Y/Z where Y is a random variable with density (2)- (sin(x/2)/(x/2)) 2, and Z is independent of Y and bas distribution function 1 - + tk', t > 0. This property allows us to develop fast algorithms for this class of distributions. This is illustrated for the symmetric stable distribution, Linnik's distribution and a few other distributions. We pay special attention to the generation of Y.

Abstract. We develop exact random variate generators for the polynomially and exponentially tilted unilateral stable distributions. The algorithms, which generalize Kanter’s method, are uniformly fast over all choices of the tilting and stable parameters. The key to the solution is a new distribution which we call Zolotarev’s distribution. We also present a novel double rejection method that is useful whenever densities have an integral representation involving an auxiliary variable.

: In this paper, we discuss the issue of estimation of the parameters of stable laws. We present an overview of the known methods and compare them on samples of different sizes and for different values of the parameters. Performance tables are provided. 1 Introduction The Central Limit Theorem, which offers the fundamental justification for approximate normality, points to the importance of ff--stable (sometimes called stable) distributions: they are the only limiting laws of normalized sums of independent, identically distributed random variables. Gaussian distributions, the best known member of the stable family, have long been well understood and widely used in all sorts of problems. However, they do not allow for large fluctuations and are thus inadequate for modeling high variability. Non-Gaussian stable models, on the other hand, do not share such limitations. In general, the upper and lower tails of their distributions decrease like a power function. In literature, this is ofte...

Probability models with heavy tails are used in many diverse fields including finance, physics, hydrology, electrical engineering and computer science. There are many estimators of the index # that governs tail thickness, but few are both shift and scale invariant. In this paper we will look at two new estimators that are both scale and shift invariant. These new estimators give a more useful method of estimating the tail index for applications where a natural center and scale is unknown.

Tail asymptotics of certain mixtures of Poisson distributions show that they are Paretian, their tail index being one of the parameters defining these laws. Estimators similar to those proposed by Press [15] for continuous stable laws are then used for the estimation of the parameters of such laws and asymptotic properties are proved. Inference is based on the empirical probability generating function. As special cases the discrete stable distribution, the discrete Linnik distribution, and the Sibuya distribution are examined. 1 Introduction The theme of the present paper is the asymptotics and the estimation of parameters for mixtures of Poisson distributions as the discrete stable distribution, the discrete Linnik distribution, and the Sibuya distribution. The continuous counterpart for Linnik laws was examined in Jacques, R'emillard, and Theodorescu [8]. A more general approach is to be found in R'emillard and Theodorescu [16]. The plan of the paper is as follows. In Section 2 we s...

Asymptotic properties for de Haan's [13, 14] estimator of the index of a family of laws, containing the Paretian laws, are first studied. Then this estimator as well as others similar to those proposed by Press [32] for stable laws are used for the estimation of the parameters in univariate and multivariate Linnik laws. AMS 1991 subject classifications: Primary 62E10, 62G05; secondary 65C10. Key words: Consistent estimators, distribution theory, Linnik distribution, order statistic, Paretian distribution, stable distribution, tailweight. 1 Introduction Let ff be a positive real number and let P ff denote the family of distribution functions F for which we have the following tailweight property: lim x!1 x ff (1 \Gamma F (x)) = C; (1) where C is a positive constant. Generally C depends on ff and eventually on other parameters occuring in the expression of F . The classical Pareto distribution belongs to this family; according to Mandelbrot [28, 29], we refer to P ff as the Paretia...