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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.

In this work, we introduce a new class of continuous distributions called the generalized poissonfamily which extends the quadratic rank transmutation map. We provide some special models for thenew family. Some of its mathematical properties including Rényi and q-entropies, order statistics andcharacterizations are derived. The estimations of the model parameters is performed by maximumlikelihood method. The Monte Carlo simulations is used for assessing the performance of the maximumlikelihood estimators. The ‡exibility of the proposed family is illustrated by means of two applicationsto real data sets.

In the literature, several continuous univariate distributions have been extensively used for modeling data in many areas such as economics, engineering, biological studies and environmental sciences. However, applied areas such as …nance, lifetime analysis and insurance clearly require extended forms of these distributions. So, several classes of distributions have been constructed by extending common families of continuous distributions. These generalized distributions give more ‡exibility by adding one "or more" parameters to the baseline model. They were pioneered by Gupta et al. (1998) who proposed the exponentiated-G class, which consists of raising the cumulative distribution function (cdf) to a positive power parameter. Many other classes can be cited such as the Marshall-Olkin-G family by Marshall and Olkin (1997), beta generalized-G family by Eugene et al. (2002), exponentiated generalized-G family by Cordeiro et al. (2013), a new method for generating families of continuous distributions by Alzaatreh et al. (2013), transmuted exponentiated generalized-G by Yousof et al. (2015), exponentiated transmuted-G by Merovc et al. (2016), Burr X-G by Yousof et al. (2016), transmuted Weibull G family by Alizadeh et al. (2016), complementary generalized transmuted Poisson-G family by Alizadeh et al. (2016b), transmuted geometric-G by A…fy et al. (2016a), complementary geometric transmuted-G family A…fy et al. (2016b), Kumaraswamy transmuted-G by A…fy et al. (2016c), exponentiated generalized-G Poisson by Aryal and Yousof (2017), Marshall-Olkin generalized family by Yousof et al. (2017a), beta Weibull-G family of distributions by Yousof et al. (2017b), Type I general exponential class of distributions by Hamedani et al. (2017), Topp-Leone odd log-logistic family by de Brito et al. (2017), generalized odd generalized exponential family by Alizadeh et al. (2017), exponentiated Weibull-H family Cordeiro et al. (2017a), generalized transmuted-G by Nofal et al. (2017), Burr XII system of densities by Cordeiro et al. (2017b) and beta transmuted-H family by A…fy et al. (2017), among others.

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A weighted model based on the Rayleigh distribution is proposed and the statistical and reliability properties of this model are presented. Some non-Bayesian and Bayesian methods are used to estimate the β parameter of proposed model. The Bayes estimators are obtained under the symmetric (squared error) and the asymmetric (linear exponential) loss functions using non-informative and reciprocal gamma priors. The performance of the estimators is assessed on the basis of their biases and relative risks under the two above-mentioned loss functions. A simulation study is constructed to evaluate the ability of considered estimation methods. The suitability of the proposed model for a real data is shown by using the Kolmogorov-Smirnov goodness-of-fit test.

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