On the generalization of Gurland distribution

Generalization of the fractional Poisson distribution

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function $E_{α, β}(λ)$ is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter $β$ is suggested.

Star-shaped distributions and their generalizations

Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to ``star-shaped distributions'' with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.

A new generalization of the geometric distribution using Azzalini's mechanism: properties and application

The skewing mechanism of Azzalini for continuous distributions is used for the first time to derive a new generalization of the geometric distribution. Various structural properties of the proposed distribution are investigated. Characterizations, including a new result for the geometric distribution, in terms of the proposed model are established. Extensive simulation experiment is done to evaluate performance of the maximum likelihood estimation method. Likelihood ratio test for the necessity of additional skewing parameter is derived and corresponding simulation based power study is also re…

Mittag - Leffler function distribution - A new generalization of hyper-Poisson distribution

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poison distributions. MLFD is a flexible distribution with varying shapes like non-increasing with unique mode at zero, unimodal with one / two non-zero modes. It can be under, equi or over dispersed. Various d…

单纯形深度的推广：在分布支撑凸包外不消失

Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support

本文首次提出一种在分布支撑凸包外不消失的单纯形深度推广方法，并通过理论、蒙特卡洛模拟和数据集分析验证其性质，解决了传统单纯形深度无法区分凸包外点的问题。

Another Generalization of Unimodality

A general characterization for α-unimodal distributions was provided by Alamatsaz (1985) who later introduced a multivariate extension of them (Alamatsaz 1993). Here, by solving the related equations, another generalization for unimodality is presented. As a result of this generalization, a simpler proof of a conjecture, as well as a characterization for generalized arcsin distributions and some generalizations of the author's earlier works, have been obtained. Last, but not the least, it is shown that some elementary methods can be more powerful than some more advanced techniques.