No Arabic abstract
In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are performed to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed.
In this paper, we have developed a new class of sampling schemes for estimating parameters of binomial and Poisson distributions. Without any information of the unknown parameters, our sampling schemes rigorously guarantee prescribed levels of precision and confidence.
In this paper, we have developed new multistage tests which guarantee prescribed level of power and are more efficient than previous tests in terms of average sampling number and the number of sampling operations. Without truncation, the maximum sampling numbers of our testing plans are absolutely bounded. Based on geometrical arguments, we have derived extremely tight bounds for the operating characteristic function. To reduce the computational complexity for the relevant integrals, we propose adaptive scanning algorithms which are not only useful for present hypothesis testing problem but also for other problem areas.
In this article, we derive an explicit formula for computing confidence interval for the mean of a bounded random variable. Moreover, we have developed multistage point estimation methods for estimating the mean value with prescribed precision and confidence level based on the proposed confidence interval.
A generalization of the Poisson distribution based on the generalized Mittag-Leffler function $E_{alpha, beta}(lambda)$ 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 $beta$ is suggested.
In this paper we prove the concavity of the $k$-trace functions, $Amapsto (text{Tr}_k[exp(H+ln A)])^{1/k}$, on the convex cone of all positive definite matrices. $text{Tr}_k[A]$ denotes the $k_{mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive tail bounds and expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of a sum of random matrices.