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Register a new userSome Characterizations and Properties of COM-Poisson Random Variables
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This paper introduces some new characterizations of COM-Poisson random variables. First, it extends Moran-Chatterji characterization and generalizes Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution. Then, it defines the COM-type discrete r.v. ${X_ u }$ of the discrete random variable $X$. The probability mass function of ${X_ u }$ has a link to the Renyi entropy and Tsallis entropy of order $ u $ of $X$. And then we can get the characterization of Stam inequality for COM-type discrete version Fisher information. By using the recurrence formula, the property that COM-Poisson random variables ($ u e 1$) is not closed under addition are obtained. Finally, under the property of not closed under addition of COM-Poisson random variables, a new characterization of Poisson distribution is found.
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In this paper we obtain the limit distribution for partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixing between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between {alpha}-stable distributions and NEF laws is established. We propose estimation of the parameters of the NEF models through the method of moments and also by the maximum likelihood method, which is performed via an Expectation-Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators and an empirical illustration on financial market is presented.
The two parameter Poisson-Dirichlet distribution $PD(alpha,theta)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingmans Poisson-Dirichlet distribution. The two parameter Dirichlet process $Pi_{alpha,theta, u_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(alpha,theta)$ and $Pi_{alpha,theta, u_0}$. The methods used come from the theory of Dirichlet forms.
Given ${X_k}$ is a martingale difference sequence. And given another ${Y_k}$ which has dependency within the sequence. Assume ${X_k}$ is independent with ${Y_k}$, we study the properties of the sums of product of two sequences $sum_{k=1}^{n} X_k Y_k$. We obtain product-CLT, a modification of classical central limit theorem, which can be useful in the study of random projections. We also obtain the rate of convergence which is similar to the Berry-Essen theorem in the classical CLT.
We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, uniform, Kumaraswamy, and triangular distributions.
A new class of two-level non-regular fractional factorial designs is defined. We call this class an {it affinely full-dimensional factorial design}, meaning that design points in the design of this class are not contained in any affine hyperplane in the vector space over $mathbb{F}_2$. The property of the indicator function for this class is also clarified. A fractional factorial design in this class has a desirable property that parameters of the main effect model are simultaneously identifiable. We investigate the property of this class from the viewpoint of $D$-optimality. In particular, for the saturated designs, the $D$-optimal design is chosen from this class for the run sizes $r equiv 5,6,7$ (mod 8).
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