Pedro Jodrá

Original article: Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function

We provide procedures to generate random variables with Lindley distribution, and also with Poisson-Lindley or zero-truncated Poisson-Lindley distribution, as simple alternatives to the existing algorithms. Our procedures are based on the fact that the quantile functions of these probability distributions can be expressed in closed form in terms of the Lambert W function. As a consequence, the extreme order statistics from the above distributions can also be computer generated in a straightforward manner.

A note on the traveling repairman problem

Given a finite set of N nodes and the time required for traveling among nodes, in the traveling repairman problem, we seek a route that minimizes the sums of the delays for reaching each node. In this note, we present a linear algorithm for solving the traveling repairman problem when the underlying graph is a path, improving the Θ(N2) time and space complexity of the previously best algorithm for this problem. We also provide a linear algorithm for solving the walk problem with deadlines (WPD) on paths.

Exact Kolmogorov and total variation distances between some familiar discrete distributions

We give exact closed-form expressions for the Kolmogorov and the total variation distances between Poisson, binomial, and negative binomial distributions with different parameters. In the Poisson case, such expressions are related with the Lambert function.

A closed-form expression for the quantile function of the Gompertz-Makeham distribution

In this note, we give a closed-form expression in terms of the Lambert W function for the quantile function of the Gompertz-Makeham distribution. This probability distribution has frequently been used to describe human mortality and to establish actuarial tables. The analytical expression provided for the quantile function is helpful to generate random samples drawn from the Gompertz-Makeham distribution by means of the inverse transform method.

On a connection between the polylogarithm function and the Bass diffusion model

The purpose of this paper is to establish a connection between the polylogarithm function and a continuous probability distribution. We provide closed-form expressions in terms of the polylogarithm function for all the moments of a continuous random variable related to the Bass diffusion model, which was introduced by Bass and is widely used in marketing science. In addition, a new integral representation of the polylogarithm of order n is achieved from a probabilistic formulation.

A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

The shifted Gompertz distribution was introduced by Bemmaor (1994) as a random model of adoption timing of innovations in a market. The purpose of this article is threefold. We provide explicit expressions for the expectation and variance of this model, which are functions of the Euler constant. In addition, for simulation purposes, we derive a closed-form expression for the quantile function of the shifted Gompertz distribution in terms of the Lambert W function. Finally, the limit distributions and computer generation of extreme order statistics are considered. In particular, we show that the domains of attraction for maxima and minima are the Gumbel and Weibull distributions, respectively.

A class of tests for the two-sample problem for count data

A class of tests for the two-sample problem for count data whose test statistic is an L 2 -norm of the difference between the empirical probability generating functions associated with each sample is considered. The tests can be applied to count data of any arbitrary fixed dimension. Since the null distribution of the test statistic is unknown, some approximations are investigated. Specifically, the bootstrap, permutation and weighted bootstrap estimators are examined. All of them provide consistent estimators. A simulation study analyzes the performance of these approximations for small and moderate sample sizes. This study also includes a comparison with other two-sample tests whose test statistic is a weighted integral of the difference between the empirical characteristic functions of the samples.

On the computer generation of the Erlang and negative binomial distributions with shape parameter equal to two

We provide closed-form expressions for the quantile functions of the Erlang and negative binomial distributions with shape parameter equal to two. These expressions are related to the Lambert W function.

An efficient algoritm for on-line searching of minima in Monge path-decomposable tridimensional arrays

Abstract We consider the problem of computing the recurrence E[i] = minj = 1,…,m min1 ⩽ k ⩽ i {b(i, j) + c(j, k) + E[k − 1]}, i = 1,…,n, where E[0] is known and B = {b(i, j)} and C = {c(j, k)} are known weight Monge matrices of size n × m and m × n, respectively. We provide an Θ(m + n)-algorithm for calculating the E[i] values. This algorithm allows us to linearly solve the two following problems: Finding the minimum Hamiltonian curve from point p1 to point pm for N points on a convex polygon, and solving the traveling salesman problem for N points on a convex polygon and a segment line inside it, improving the previous Θ(N log N)-algorithms for both these problems.

On the Muth Distribution

The Muth distribution is a continuous random variable introduced in the context of reliability theory. In this paper, some mathematical properties of the model are derived, including analytical expressions for the moment generating function, moments, mode, quantile function and moments of the order statistics. In this regard, the generalized integro-exponential function, the Lambert W function and the golden ratio arise in a natural way. The parameter estimation of the model is performed by the methods of maximum likelihood, least squares, weighted least squares and moments, which are compared via a Monte Carlo simulation study. A natural extension of the model is considered as well as an application to a real data set.