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.

We elucidate the long-term behavior of failure rates for a broad class of frailty models in survival analysis. The class properly includes the proportional hazard frailty model, the additive frailty model, and the accelerated failure time frailty model. A complete asymptotic expansion is derived and compared with the corresponding result for the limiting behavior obtained by Finkelstein and Esaulova (2006a). Several examples are provided to facilitate the comparison and to illustrate both the applicability and the limitations of our approach.

In this paper we provide a sufficient condition for mean residual life ordering of parallel systems with n ≥ 3 heterogeneous exponential components.

Age at first lambing (AFL) plays a key role on the reproductive performance of sheep flocks, although there are no genetic selection programs accounting for this trait in the sheep industry. This could be due to the non-Gaussian distribution pattern of AFL data, which must be properly accounted for by the analytical model. In this manuscript, two different parameterizations were implemented to analyze AFL in the Ripollesa sheep breed, that is, the skew-Gaussian mixed linear model (sGML) and the piecewise Weibull proportional hazards model (PWPH). Data were available from 10 235 ewes born between 1972 and 2013 in 14 purebred Ripollesa flocks located in the north-east region of Spain. On average, ewes gave their first lambing short after their first year and a half of life (590.9 days), and within-flock averages ranged between 523.4 days and 696.6 days. Model fit was compared using the deviance information criterion (DIC; the smaller the DIC statistic, the better the model fit). Model sGML was clearly penalized (DIC=200 059), whereas model PWPH provided smaller estimates and reached the minimum DIC when one cut point was added to the initial Weibull model (DIC=132 545). The pure Weibull baseline and parameterizations with two or more cut points were discarded due to larger DIC estimates (>134 200). The only systematic effect influencing AFL was the season of birth, where summer- and fall-born ewes showed a remarkable shortening of their AFL, whereas neither birth type nor birth weight had a relevant impact on this reproductive trait. On the other hand, heritability on the original scale derived from model PWPH was high, with a model estimate place at 0.114 and its highest posterior density region ranging from 0.079 and 0.143. As conclusion, Gaussian-related mixed linear models should be avoided when analyzing AFL, whereas model PWPH must be viewed as better alternative with superior goodness of fit; moreover, the additive genetic background underlying this reproductive trait supports its inclusion into current genetic selection programs given its economic importance.