On multiparameter distributions of order k

A multiparameter negative binomial distribution of order k is obtained by compounding the extended (or multiparameter) Poisson distribution of order k by the gamma distribution. A multiparameter logarithmic series distribution of order k is derived next, as the zero truncated limit of the first distribution. Finally a few genesis schemes and interrelationships are established for these three multiparameter distributions of order k. The present work extends several properties of distributions of order k.     

Multidimensional compound Poisson distributions in free probability

Inspired by Speicher's multidimensional free central limit theorem and semicircle families, we prove an in?nite dimensional compound Poisson limit theorem in free probability, and de?ne in?nite dimensional compound free Poisson distributions in a non-commutative probability space. In?nite dimensional free in?nitely divisible distributions are de?ned and characterized in terms of their free cumulants. It is proved that for a sequence of random variables, the following three statements are equivalent:(1) the distribution of the sequence is multidimensional free in?nitely divisible;(2) the sequence is the limit in distribution of a sequence of triangular trays of families of random variables;(3) the sequence has the same distribution as that of {a_1~((i)): i = 1, 2,...}of a multidimensional free L′evy process {{a_1~((i)): i = 1, 2,...} : t≥0}. Under certain technical assumptions, this is the case if and only if the sequence is the limit in distribution of a sequence of sequences of random variables having multidimensional compound free Poisson distributions.     

Approximation and estimation of some compound distributions

The distribution of the total amount claimed up to time t can often be written in the form of a compound distribution G_t(x) = Σp_n(t)F⁽ⁿ⁾(x) where p_n(t) is the probability of exactly n claims while F is the distribution of a single claim. In the actuarial literature one often finds approximations of G_t(x) when the time t is large. It seems more natural to take t fixed and to look for approximations for x large. This paper contains a number of such results for a Poisson process and for a Pascal process. Different hypotheses on the tail behaviour of F(t) yield different expressions to estimate 1 - G_t(x). The results obtained should prove to have wider applicability than suggested by the insurance context. Within it, however, applications to premium calculation principles are immediate.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.     

On some properties of Poisson processes

The Poisson process, i.e., the simple stream, is defined by Khintchine as a stationary, orderly and finite stream without after-effects. A necessary and sufficient condition for a stream to be a simple stream is that the interarrival times are independent random variables with identical exponential distributions. This paper gives a simple and rigorous proof of the necessary and sufficient condition, and discusses the other necessary and sufficient conditions for a renewal process to be a Poisson process.Institute of Applied Mathematics, Academia Sinica     

On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions

Summary The binomial and multinomial distributions are, probably, the best known distributions because of their vast number of applications. The present paper examines some generalizations of these distributions with many practical applications. Properties of these generalizations are studied and models giving rise to them are developed. Finally, their relationship to generalized Poisson distributions is examined and limiting cases are given.     

A simple proof of Feller's characterization of the compound Poisson distributions

A classical result of W. Feller is that every distribution that is infinitely divisible and concentrated on the non-negative integers is compound Poisson. We give a simple proof that uses some of the recursive formulas that have recently become popular among actuaries.     

On some compound distributions with Borel summands

The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.     

Improved recursions for some compound poisson distributions

Calculations by means of the well-known recursion for the compound Poisson distribution are in general time-consuming since each probability depends on all the preceding ones. It is shown that for some claim size distributions a more efficient recursion can be derived.     

Smoothing effect of compound Poisson approximations to the distributions of weighted sums

We investigate the closeness of a compound Poisson approximation to the sum S?=?w ₁ S ₁?+?w ₂ S ₂?+???+?w _N S _N in the Kolmogorov norm. Here S _i are sums of independent or weakly dependent random variables, and w _i are weights. The overall smoothing effect of S on w _i S _i is estimated by Lévy’s concentration function.     

On moments of counting distributions satisfying thekth-order recursion and their compound distributions

We present formulas and recurrence formulas commonly used in insurance mathematics for moments of counting distributions given by the kth-order recursion. Moreover, we develop recurrence formulas for moments of compound distributions with those counting distributions satisfying the kth-order recursion. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part II.     

Generating function analysis of some joint distributions for Poisson loss systems

We present a new approach to derive joint distributions for stationary Poisson loss systems. In particular, for M/M/m/0 and M/M/1/n we find the Laplace transforms (with respect to time t) of the probability that at time t there are i customers in the system and during [0,t], j customers are refused admission; for M/M/m/0 we further determine the LT of the probability that the system was full for less than s time units during [0,t] and serves i customers at time t. Explicit formulas for the corresponding moments are also given.     

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.     

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all \(\psi \) -mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.