Adrienne W. Kemp

Characterizations of a discrete normal distribution

Abstract The paper obtains a discrete analogue of the normal distribution as the distribution that is characterized by maximum entropy, specified mean and variance, and integer support on (− ∞, ∞). Two alternative characterizations are given, firstly as the distribution of the difference of two related Heine distributions, and secondly as a weighted distribution.

Heine-euler extensions of the poisson distribution

The paper shows that the Heine and Euler distributions (Benkherouf and Bather, 1988) are members of a family of q-series anologues of the Poisson distribution, with similar probability mass functions, but different restrictions on their parameters, and different modes of genesis and properties. The relationships between the Heine, Euler, pseudo-Euler, Poisson and geometric distributions are explored. Illustrative data sets are discussed.

Steady-state Markov chain models for the Heine and Euler distributions

The paper puts forward steady-state Markov chain models for the Heine and Euler distributions. The models for oil exploration strategies that were discussed by Benkherouf and Bather (1988) are reinterpreted as current-age models for discrete renewal processes. Steady-state success-runs processes with non-zero probabilities that a trial is abandoned, Foster processes, and equilibrium random walks corresponding to elective M / M /1 queues are also examined.

Poisson random variate generation

The paper examines the problem of generating Poisson random variates particularly when the parameter x may vary from call to call. A new algorithm based on a unidirectional search from the mode is proposed; the modal probability and modal cumulative probability, when required, are calculated by simple and rapid, yet extremely accurate, asymptotic approximations; a squeeze feature is incorporated. Timings for a Fortran 77 implementation show that the algorithm dominates the current state‐of‐the‐art algorithms for λ

The Discrete Half-Normal Distribution

The discrete half-normal distribution is derived as the maximum entropy distribution on 0,1,. . . with specified mean and variance. It is a limiting q-hyper-Poisson- I distribution that arises from the Morse M/M/1 queue with service-dependent balking. Success runs models are reviewed. A new derivation as a mixture of Heine distributions is given. Finally the moment and other properties are examined.

Classes of Discrete Lifetime Distributions

Abstract Several results that characterize the distribution of a lifetime variable, T, with probability mass function (pmf) p t , where t = 0, 1, 2,…, by its survivor function, S t = ∑ j≥t p j , its hazard function, h t = p t /S t , its cumulative hazard function, Λ t = − lnS t , its accumulated hazard function, , and its mean residual life function, L t = E[(T − t)|T ≥ t] (an initially faulty item is deemed to have a zero lifetime), are presented. These include results that have previously appeared in the literature as well as some new results. Differences in the terminology used by engineers, actuaries, and biostatisticians are pointed out and clarified. Attention is focussed on the relationships between the IFR/DFR, IFRA/DFRA, NBU/NWU, NBUE/NWUE, and IMRL/DMLR classes to which a discrete lifetime distribution and its current age distribution belong.

Weldon's Dice Data Revisited

Abstract This note examines the role of Weldons classical dice data in illustrating the problems of model selection. New, more realistic models are discussed; simple Poisson trials models, with at most two unknown parameters, are fitted. An application to process-oriented inspection sampling for a multiple channel production process is suggested.

Efficient Generation of Logarithmically Distributed Pseudo-random Variables

A one‐line algorithm LB is given for generating samples from the logarithmic series distribution. Two independent uniform random variables are converted into a single logarithmic variable by use of a structural property of the distribution. Faster versions of the method, algorithms LBM and LK, employ simple initial tests which identify those pairs of uniform random variables which yield the most frequently occurring values of the logarithmic variable. Comparison with a search method, algorithm LS, shows that the search method is faster when the parameter a of the logarithmic distribution is less than about 0‐95. However, values of a greater than 095 are often encountered. In ecological contexts a ranges from 0‐9 to 0‐9999; here the search method becomes prohibitively slow, and the algorithms based on the structural method are preferable. Both methods lead to very short, portable, procedures which require a trivial amount of storage.

A family of discrete distributions defined via their factorial moments

This paper examines the family whose probability generating functions have the form of the generalized hypergeometric function, pFq [(a); (b); λ(s-1)] . It includes a number of matching distributions as well as many classic discrete distributions. Properties may be derived from the differential equations satisfied by the various generating functions e.g. useful recurrence formulae for probabilities, cumulants, and moments about an arbitrary point can be obtained.

Patchwork rejection algorithms

Abstract New methods for implementing rejection algorithms are suggested. The objectives are to avoid the generation of unwanted envelope variates, to use fewer pseudorandom uniform variates, and to bypass as far as possible the need to decide whether to accept or to reject. Considerably fewer than two uniforms on average, per attempted output of a target random variable, can be achieved, together with an increased acceptance rate. The generation of tail variates from a normal distribution is used as illustration.