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Ad Ridder

Publications and source records attributed to Ad Ridder.

7 recordsLinked to original sources
arXiv2020

New exponential dispersion models for count data -- the ABM and LM classes

In their fundamental paper on cubic variance functions, Letac and Mora (The Annals of Statistics,1990) presented a systematic, rigorous and comprehensive study of natural exponential families on the real line, their characterization through their variance functions and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of variance functions associated with natural exponential families of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As exponential dispersion models are based on natural exponential families, we introduce in this paper two new classes of exponential dispersion models based on their results. For these classes, which are associated with simple variance functions, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero-inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g., Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.

math.ST↗
arXiv2020

Exponential Dispersion Models for Overdispersed Zero-Inflated Count Data

We consider three new classes of exponential dispersion models of discrete probability distributions which are defined by specifying their variance functions in their mean value parameterization. In a previous paper (Bar-Lev and Ridder, 2020a), we have developed the framework of these classes and proved that they have some desirable properties. Each of these classes was shown to be overdispersed and zero inflated in ascending order, making them as competitive statistical models for those in use in statistical modeling. In this paper we elaborate on the computational aspects of their probability mass functions. Furthermore, we apply these classes for fitting real data sets having overdispersed and zero-inflated statistics. Classic models based on Poisson or negative binomial distributions show poor fits, and therefore many alternatives have already proposed in recent years. We execute an extensive comparison with these other proposals, from which we may conclude that our framework is a flexible tool that gives excellent results in all cases. Moreover, in most cases our model gives the best fit.

stat.ME↗
arXiv2018

Monte Carlo Methods for Insurance Risk Computation

In this paper we consider the problem of computing tail probabilities of the distribution of a random sum of positive random variables. We assume that the individual variables follow a reproducible natural exponential family (NEF) distribution, and that the random number has a NEF counting distribution with a cubic variance function. This specific modelling is supported by data of the aggregated claim distribution of an insurance company. Large tail probabilities are important as they reflect the risk of large losses, however, analytic or numerical expressions are not available. We propose several simulation algorithms which are based on an asymptotic analysis of the distribution of the counting variable and on the reproducibility property of the claim distribution. The aggregated sum is simulated efficiently by importancesampling using an exponential cahnge of measure. We conclude by numerical experiments of these algorithms.

math.PR↗
arXiv2011

Counting with Combined Splitting and Capture-Recapture Methods

We apply the splitting method to three well-known counting problems, namely 3-SAT, random graphs with prescribed degrees, and binary contingency tables. We present an enhanced version of the splitting method based on the capture-recapture technique, and show by experiments the superiority of this technique for SAT problems in terms of variance of the associated estimators, and speed of the algorithms.

stat.CO↗
arXiv2010

Approximate results for a generalized secretary problem

A version of the classical secretary problem is studied, in which one is interested in selecting one of the b best out of a group of n differently ranked persons who are presented one by one in a random order. It is assumed that b is a preassigned (natural) number. It is known, already for a long time, that for the optimal policy one needs to compute b position thresholds, for instance via backwards induction. In this paper we study approximate policies, that use just a single or a double position threshold, albeit in conjunction with a level rank. We give exact and asymptotic (as n tends to infinity) results, which show that the double-level policy is an extremely accurate approximation.

math.PR↗
arXiv2010

Finite-state Markov Chains obey Benford's Law

A sequence of real numbers (x_n) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (P^n - P*) and (P^{n+1}-P^n) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonant) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.

math.PR↗