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There are many practical situations in sampling and testing, when the probability of success varies in a sequence of n independent Bernoulli trials. In many of these cases and for various reasons, we may find it useful to compare the distribution of the number of successes X =Σ Bin (1, p _ i ) in n such trials with a binomial random variable Y=Bin(n, p) for some p . For example, such a comparison might be useful in deciding whether or not stratified sampling is superior (or inferior) to simple random sampling in survey sampling, or whether or not partition (or subdomain) testing is to be preferred to simple random testing in attempting to find faults in software. We will discuss the rationale behind several methods and orders for stochastically comparing the random variables X and Y . These include comparing their means, but also comparing them with respect to the usual stochastic order, the precedence order, the ≥1 order and even the likelihood ratio order. It will be seen that many interesting comparisons between X and Y depend on the relationship between p and various means (harmonic, geometric, arithmetic, complimentary geometric, and complimentary harmonic) of the components in the vector p =( p _1, p _2,... p _ n .

This chapter considers the first-crossing problem of a compound Poisson process with positive integer-valued jumps in a nondecreasing lower boundary . The cases where the boundary is a given linear function, a standard renewal process, or an arbitrary deterministic function are successively examined. Our interest is focused on the exact distribution of the first-crossing level (or time) of the compound Poisson process. It is shown that, in all cases, this law has a simple remarkable form which relies on an underlying polynomial structure. The impact of a raise of a lower deterministic boundary is also discussed.

Various techniques for constructing discrete bivariate distributions are scattered in the literature. We review these methods of construction and group them into some loosely defined clusters.

The normal-Laplace (NL) distribution results from convolving independent normally distributed and Laplace distributed components. It is the distribution of the stopped state of a Brownian motion with a normally distributed starting value if the stopping hazard rate is constant. Properties of the NL distribution discussed in the article include its shape and tail behaviour (fatter than the normal), its moments, and its infinite divisibility. The double Pareto-lognormal distribution is that of an exponentiated normal-Laplace random variable and provides a useful parametric form for modelling size distributions. The generalized normal-Laplace (GNL) distribution is both infinitely divisible and closed under summation. It is possible to construct a Lévy process whose increments follow the GNL distribution. Such a Lévy motion can be used to model the movement of the logarithmic price of a financial asset. An option pricing formula is derived for such an asset.

During the last decade, a substantial part of Barry Arnold’s research effort has been directed towards developing models capable of describing the forms of asymmetry manifested by real data. One general and seemingly elegant means of constructing skew distributions is provided by a lemma presented in ( 1985 ). The now widely known skew-normal distribution is just one special case belonging to the family of distributions generated using the construction implicit in that lemma. In this paper, a simple alternative proof of the lemma is given, and reflections are made upon how the construction arising from it has been employed in the literature. The densities of various special cases are presented, which highlight both the flexibility and limitations of the construction. Likelihood-based inference for the parameters of the location-scale extensions of classes arising from the construction is also considered. General results are given for the solutions to the score equations and for the observed information matrix. For the special case of the skew-normal distribution, it is shown that, for one of the solutions to the score equations, the observed information matrix is always singular.

The generalized three-parameter beta distribution with pdf proportional to x ^ a −1(1− x )^ b −1/{1−(1−λ) x }^a+b is a flexible extension of the classical beta distribution with interesting applications in statistics. In this chapter, several bivariate extensions of this distribution are studied. We propose models with given marginals: a first model consists of a transformation with monotonic components of the Dirichlet distribution and a second model that uses the bivariate Sarmanov-Lee distribution. Next, the class of distributions whose conditionals belong to the generalized three-parameter beta distribution is considered. Two important subfamilies are studied in detail. The first one contains as a particular case the models of ( 1982 ) and ( 2003 ). The second family is more general, and contains among others, the model proposed by ( 1999 ). In addition, using two different conditional schemes, we study conditional survival models. Multivariate extensions are also discussed. Finally, an application to Bayesian analysis is given.

In this chapter, we consider a Kotz-type distribution (of a p -variate random vector X ) which has fatter tail regions than that of multivariate normal distribution, and its probability density function ( pdf ) is given by $$ f(x,\mu ,\Sigma ) = c_p \left| \Sigma \right|^{ - \tfrac{1} {2}} \exp \{ - [(x - \mu )'\Sigma ^{ - 1} (x - \mu )]^{\tfrac{1} {2}} \} , $$ where μ∈ℜ^p, Σ is a positive definite matrix and $$ c_p = \tfrac{{\Gamma (\tfrac{p} {2})}} {{2\pi ^{\tfrac{p} {2}} \Gamma (p)}} $$ . We review various characteristics and provide a simulation algorithm to simulate samples from this distribution. Estimation of the parameters using the maximum likelihood method is discussed. An interesting fact is that the maximum likelihood estimators under this distribution are the generalized spatial median (GSM) estimators as defined by ( 1988 ). Using the asymptotic distribution of the estimates, statistical inferences on the parameters of the distribution are illustrated with an example.

Let X _1,..., X _ d be d ( d ≥3) dependent random variables with finite variances such that X _ j ∼ F _ j . Results on the set S _ d ( F _1,..., F _ d ) of possible correlation matrices with given margins are obtained; this set is relevant for simulating dependent random variables with given marginal distributions and a given correlation matrix. When F _1=...= F _ d = F , we let S _ d ( F ) denote the set of possible correlation matrices. Of interest is the set of F for which S _ d ( F ) is the same as the set of all non-negative definite correlation matrices; using a construction with conditional distributions, we show that this property holds only if F is a (location-scale shift of a) margin of a ( d −1)-dimensional spherical distribution.

In this paper, we apply the theory of pseudodifferential operators and Sobolev spaces to characterize fractional and multifractional probability densities. In the fractional case, local regularity properties of the probability density function are given in terms of fractional moment conditions satisfied by the characteristic function. Conversely, the parameter defining the order of the fractional Sobolev space where the characteristic function lies provides the index of stability in relation to fractional moment conditions of the probability density. The extension to the multifractional case leads to the introduction of new probabilistic models considering the theory of pseudodifferential operators and fractional Sobolev spaces of variables order.

The term “order statistics” was introduced only in 1942, by Wilks. However, the subject is much older, astronomers having long been interested in estimates of location beyond the sample mean. By early in the nineteenth century measures considered included the median, symmetrically trimmed means, the midrange, and related functions of order statistics. In 1818, Laplace obtained (essentially) the distribution of the r th-order statistic in random samples and also derived a condition on the parent density under which the median is asymptotically more efficient than the mean. Other topics considered are of more recent origin: extreme-value theory and the estimation of location and scale parameters by order statistics.