Log-concavity of compound distributions with applications in operational and actuarial models

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s, S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters. Copyrigh

Modelling the Frequency of Interarrival Times and Rainfall Depths with the Poisson Hurwitz-Lerch Zeta Distribution

The Poisson-stopped sum of the Hurwitz-Lerch zeta distribution is proposed as a model for interarrival times and rainfall depths. Theoretical properties and characterizations are investigated in comparison with other two models implemented to perform the same task: the Hurwitz-Lerch zeta distribution and the one inflated Hurwitz-Lerch zeta distribution. Within this framework, the capability of these three distributions to fit the main statistical features of rainfall time series was tested on a dataset never previously considered in the literature and chosen in order to represent very different climates from the rainfall characteristics point of view. The results address the Hurwitz-Lerch zeta distribution as a natural framework in rainfall modelling using the additional random convolution induced by the Poisson-stopped model as a further refinement. Indeed the Poisson contribution allows more flexibility and depiction in reproducing statistical features, even in the presence of very different climates

Mean-Variance and Expected Utility: The Borch Paradox

The model of rational decision-making in most of economics and statistics is expected utility theory (EU) axiomatised by von Neumann and Morgenstern, Savage and others. This is less the case, however, in financial economics and mathematical finance, where investment decisions are commonly based on the methods of mean-variance (MV) introduced in the 1950s by Markowitz. Under the MV framework, each available investment opportunity ("asset") or portfolio is represented in just two dimensions by the ex ante mean and standard deviation $(\mu,\sigma)$ of the financial return anticipated from that investment. Utility adherents consider that in general MV methods are logically incoherent. Most famously, Norwegian insurance theorist Borch presented a proof suggesting that two-dimensional MV indifference curves cannot represent the preferences of a rational investor (he claimed that MV indifference curves "do not exist"). This is known as Borch's paradox and gave rise to an important but generally little-known philosophical literature relating MV to EU. We examine the main early contributions to this literature, focussing on Borch's logic and the arguments by which it has been set aside.Comment: Published in at http://dx.doi.org/10.1214/12-STS408 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org