Georgios Psarrakos

Some extensions of the residual lifetime and its connection to the cumulative residual entropy

In this article we present a sequence of random variables with weighted tail distribution functions, constructed based on the relevation transform. For this sequence, we prove several recursive formulas and connections to the residual entropy through the unifying framework of the Dickson-Hipp operator. We also give some numerical examples to evaluate our results.

Characterizations based on generalized cumulative residual entropy functions

ABSTRACT The Shannon entropy and the cumulative residual entropy (CRE) of a random variable are useful tools in probability theory. Recently, a new concept called generalized cumulative residual entropy (GCRE) of order n was introduced and studied. It is related with the record values of a sequence of i.i.d. random variables and with the relevation transform. In this paper, we show that, under some assumptions, the GCRE function of a fixed order n uniquely determines the distribution function. Some characterizations of particular probability models are obtained from this general result.

Monotonicity properties and the deficit at ruin in the Sparre Andersen model

Let H u (y) be the (proper) distribution function of the deficit at ruin, given that ruin occurs with initial surplus u, in the Sparre Andersen model of risk theory. Dickson & dos Reis (1996) discussed the monotonicity of H u (y) as a function of u. In this paper, we obtain various monotonicity results for H u (y) and other related quantities for the decreasing/increasing failure rate (DFR/IFR) and the increasing/decreasing mean residual lifetime (IMRL/DMRL) classes of distributions. These results in particular extend and make more concrete the results of Dickson & dos Reis (1996) and Willmot & Lin (1998). A new class of distributions (increasing convolution ratio; ICR) is introduced. This class extends the well-known class of distributions with IFR. Specifically, we show that if the ladder height distribution F in the model is ICR, the ratio is a non-decreasing function of u, where ψ(u) denotes the ruin probability and . Further, we obtain generalizations (expressed in terms of the distribution of the deficit) of the well-known new worse than used (NWU) property of the probability of non-ruin.

Stochastic comparisons of interfailure times under a relevation replacement policy

We provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

On the generalized cumulative residual entropy with applications in actuarial science

Recently, Psarrakos and Navarro (2013) proposed a new measure of uncertainty which extends the cumulative residual entropy (CRE), called the generalized cumulative residual entropy (GCRE). In the present paper, new properties and applications in actuarial risk measures of the GCRE are explored. Bounds, stochastic order properties and characterization results of the new entropy are also discussed. It is shown that the GCRE is invariant under changes in location, and scale directly with scale of a random variable; the same properties also hold for the standard deviation. It is also proved that the GCRE of the first order statistics can uniquely determine the parent distribution. The Weibull distribution, which is commonly used in several fields of applied probability, is characterized by using the mentioned generalized measure. The GCRE is studied as a risk measure and is compared to the standard deviation and the right-tail risk measure, where the latter measure was introduced by Wang (1998). Several examples are also given to illustrate the new results.

On the generalized cumulative residual entropy weighted distributions

ABSTRACT Recently, Feizjavadian and Hashemi (2015) introduced and studied the mean residual weighted (MRW) distribution as an alternative to the length-biased distribution, by using the concepts of the mean residual lifetime and the cumulative residual entropy (CRE). In this article, a new sequence of weighted distributions is introduced based on the generalized CRE. This sequence includes the MRW distribution. Properties of this sequence are obtained generalizing and extending previous results on the MRW distribution. Moreover, expressions for some known distributions are given, and finite mixtures between the new sequence of weighted distributions and the length-biased distribution are studied. Numerical examples are given to illustrate the new results.

A Generalization of the Lundberg Condition in the Sparre Andersen Model and Some Applications

We generalize the well-known Lundberg condition for the existence of the adjustment coefficient, R, in the Sparre Andersen model of risk theory. The new condition is given in terms of the distribution of the deficit at ruin in that model. As an application of this condition, we show that the function e Ru ψ(u) is nonincreasing when the claim size distribution in the model has a decreasing failure rate. Further, we obtain some new bounds for ruin probabilities and stop-loss premiums that are easy to compute and we give some examples to compare these bounds against existing ones in the literature.

RATIO MONOTONICITY FOR TAIL PROBABILITIES IN THE RENEWAL RISK MODEL

A renewal model in risk theory is considered, where H(u;y) is the tail of the distribution of the decit at ruin and F (u) is the tail of the ladder height distribution. Conditions are derived under which the ratio H(u;y)=F (u+y) is nondecreasing in u for any y 0. In particular, it is proven that if the ladder height distribution is stable and DFR or phasetype, then the above ratio is nondecreasing in u. As a byproduct of this monotonicity, an upper bound and an asymptotic result for H(u;y) are derived. Examples are given to illustrate the monotonicity results.

Some results on the joint distribution prior to and at the time of ruin in the classical model

For the classical risk model (i.e. with Poisson arrivals), we study the tail of the joint distribution of the surplus prior to and at ruin. In particular, we obtain some inequalities and monotonicity results for it. Let S be the random variable with distribution function the probability of non-ruin, 1−ψ(u), and the probability the surplus just before ruin exceeds x, given that ruin occurs. We estimate the distance between the residual lifetime of S, Pr(S>u+y∣S>u) and the product , where the tail convolution includes again the random variable S. Finally, based on this distance, we derive a lower bound for the probability of ruin, and we compare this against a bound available in the literature.