Recently, an alternative measure of uncertainty called extropy is proposed by Lad et al. [12]. The extropy is a dual of entropy which has been considered by researchers. In this article, we introduce an alternative measure of uncertainty of random variable which we call it cumulative residual extropy. This measure is based on the cumulative distribution function F. Some properties of the proposed measure, such as its estimation and applications, are studied. Finally, some numerical examples for illustrating the theory are included.

The Asmussen–Kroese Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings, where SN is the sum of independent, identically distributed heavy-tailed random variables X1,…,XN and N is a nonnegative, integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the Xi are nonnegative. We also apply our ideas to estimate the quantity E[(SN-u)+].

A three parameter Gaussian exponential approximation to some compound Poisson distributions is considered. It is constructed by specifying the reciprocal of the mean excess function as a linear affine function below some threshold and a positive constant above this threshold. As an analytical approximation to compound Poisson distributions, it is only feasible either for a limited range of the Poisson parameter or for higher coefficients of variation. A semiparametric determination of the unknown threshold parameter is proposed. The analysis of a real-life example from pension fund mathematics displays an improved quality of fit of the new model when compared with other simple good alternative approximations based on the zero gamma, translated gamma and zero translated gamma.