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On an Asymptotic Series of Ramanujan  [PDF]
Yaming Yu
Mathematics , 2009, DOI: 10.1007/s11139-009-9169-x
Abstract: An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $\phi(X)$ when the distribution of $X$ belongs to a suitable family indexed by a convolution parameter. Examples include the problem of inverse moments for distribution families such as the binomial or the negative binomial.
Inverse moments of univariate discrete distributions via the Poisson expansion  [PDF]
Koenraad M. R. Audenaert
Mathematics , 2008,
Abstract: In this note we present a series expansion of inverse moments of a non-negative discrete random variate in terms of its factorial cumulants, based on the Poisson-Charlier expansion of a discrete distribution. We apply the general method to the positive binomial distribution and obtain a convergent series for its inverse moments with an error residual that is uniformly bounded on the entire interval 0<=p<=1.
A Winner's Mean Earnings in Lottery and Inverse Moments of the Binomial Distribution  [PDF]
Konstantinos Drakakis
Journal of Probability and Statistics , 2010, DOI: 10.1155/2010/279154
Abstract: We study the mean earnings of a lottery winner as a function of the number of participants in the lottery and of the success probability . We show, in particular, that, for fixed , there exists an optimal value of where the mean earnings are maximized. We also establish a relation with the inverse moments of a binomial distribution and suggest new formulas (exact and approximate) for them. 1. Introduction The game of lottery is both popular and simple. Focusing on the essentials, and leaving aside additional features of secondary importance, which vary across different lottery implementations, the rules of the game are as follows: each player submits to the lottery organizers a ticket consisting of integers (selected by the player, without repetitions, selection order being unimportant) from the range ; within the prespecified time period the game is set to last; upon the expiry of this period no more ticket submissions are accepted, and an -tuple of distinct integers (the “winning set”) is selected uniformly at random by the lottery organizers; each submitted ticket gets compared against the winning set, and, if they match, the corresponding player “wins.” This is known as an lottery system, and the winning probability is clearly . The money the winners earn depends on the number of submitted tickets: each submitted -tuple incurs a certain fee (which we assume, without loss of generality, to be equal to 1), and some fixed ratio of the total sum collected (which we assume, again without loss of generality, to be 100%, namely, the entire sum) is returned as prize money back to the winners and equally split among them. As long as no winner is found, earnings of earlier games accumulate until winners are found, who then split equally the entire sum. This is an important feature in the implementation of lottery systems in practice, known as rollover. For a given success probability , what is the effect of the total number of participants on a winner's mean earnings? This is the object of study of this article. Clearly, more participants lead not only to more prize money, but also to more potential winners. Intuitively, and by the elementary properties of the binomial distribution, we expect that the area of the -plane where is an important borderline: as long as , existence of winners is highly improbable, so most likely the mean earnings will trivially be 0, while, as long as , the law of large numbers applies and suggests that there will be approximately winners; each of when will receive an amount of money equal to . We first analyze the independent
Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions  [PDF]
Michael Short
ISRN Probability and Statistics , 2013, DOI: 10.1155/2013/412958
Abstract: The exact evaluation of the Poisson and Binomial cumulative distribution and inverse (quantile) functions may be too challenging or unnecessary for some applications, and simpler solutions (typically obtained by applying Normal approximations or exponential inequalities) may be desired in some situations. Although Normal distribution approximations are easy to apply and potentially very accurate, error signs are typically unknown; error signs are typically known for exponential inequalities at the expense of some pessimism. In this paper, recent work describing universal inequalities relating the Normal and Binomial distribution functions is extended to cover the Poisson distribution function; new quantile function inequalities are then obtained for both distributions. Exponential bounds—which improve upon the Chernoff-Hoeffding inequalities by a factor of at least two—are also obtained for both distributions. 1. Introduction The Poisson and Binomial distributions are a good approximation for many random phenomena in areas such as telecommunications and reliability engineering, as well as the biological and managerial sciences [1, 2]. Let be a Poisson distributed random variable having mean , and let represent the cumulative distribution function (CDF) of with nonnegative integer support : Similarly, let be a Binomially distributed random variable with parameters and , and let represent the CDF of for integer support : Also, let the th quantiles of and for be obtained from the functions and : Due to numerical and complexity issues, evaluation of the exponential and Binomial summations in (1) and (2) through recursive operations is only practical for small values of the input parameters ( or and ). Instead, a better solution is to evaluate the CDFs directly through either their incomplete Beta/Gamma function representations which can be approximated to high precision by continued fractions or asymptotic expansions [3]. With respect to the quantiles of the distributions given by (3) and (4), no methods to exactly evaluate these functions without iterating the exponential/Binomial sums—or alternately employing a search until the required conditions are satisfied—seem to be known. Typically, a binary search to determine the smallest satisfying (3) or (4) evaluating the respective CDF at each step would be a better general solution, given some initial upper bound for . Such methods (and related variants) are now employed very effectively in modern commercial and research-based statistical packages. In some situations, one may desire simpler solutions to
Asymptotic Comparison of Method of Moments Estimators and Maximum Likelihood Estimators of Parameters in Zero-Inflated Poisson Model  [PDF]
G. Nanjundan, T. Raveendra Naika
Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.36095
Abstract: This paper discusses the estimation of parameters in the zero-inflated Poisson (ZIP) model by the method of moments. The method of moments estimators (MMEs) are analytically compared with the maximum likelihood estimators (MLEs). The results of a modest simulation study are presented.
Testing Poisson Binomial Distributions  [PDF]
Jayadev Acharya,Constantinos Daskalakis
Computer Science , 2014,
Abstract: A Poisson Binomial distribution over $n$ variables is the distribution of the sum of $n$ independent Bernoullis. We provide a sample near-optimal algorithm for testing whether a distribution $P$ supported on $\{0,...,n\}$ to which we have sample access is a Poisson Binomial distribution, or far from all Poisson Binomial distributions. The sample complexity of our algorithm is $O(n^{1/4})$ to which we provide a matching lower bound. We note that our sample complexity improves quadratically upon that of the naive "learn followed by tolerant-test" approach, while instance optimal identity testing [VV14] is not applicable since we are looking to simultaneously test against a whole family of distributions.
Learning Poisson Binomial Distributions  [PDF]
Constantinos Daskalakis,Ilias Diakonikolas,Rocco A. Servedio
Computer Science , 2011,
Abstract: We consider a basic problem in unsupervised learning: learning an unknown \emph{Poisson Binomial Distribution}. A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$ is the distribution of a sum of $n$ independent Bernoulli random variables which may have arbitrary, potentially non-equal, expectations. These distributions were first studied by S. Poisson in 1837 \cite{Poisson:37} and are a natural $n$-parameter generalization of the familiar Binomial Distribution. Surprisingly, prior to our work this basic learning problem was poorly understood, and known results for it were far from optimal. We essentially settle the complexity of the learning problem for this basic class of distributions. As our first main result we give a highly efficient algorithm which learns to $\eps$-accuracy (with respect to the total variation distance) using $\tilde{O}(1/\eps^3)$ samples \emph{independent of $n$}. The running time of the algorithm is \emph{quasilinear} in the size of its input data, i.e., $\tilde{O}(\log(n)/\eps^3)$ bit-operations. (Observe that each draw from the distribution is a $\log(n)$-bit string.) Our second main result is a {\em proper} learning algorithm that learns to $\eps$-accuracy using $\tilde{O}(1/\eps^2)$ samples, and runs in time $(1/\eps)^{\poly (\log (1/\eps))} \cdot \log n$. This is nearly optimal, since any algorithm {for this problem} must use $\Omega(1/\eps^2)$ samples. We also give positive and negative results for some extensions of this learning problem to weighted sums of independent Bernoulli random variables.
Poisson inverse problems  [PDF]
Anestis Antoniadis,Jéremie Bigot
Mathematics , 2006, DOI: 10.1214/009053606000000687
Abstract: In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet--vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347--1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback--Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.
Compendium of Distributions, I: Beta, Binomial, Chi-Square, F, Gamma, Geometric, Poisson, Student's t, and Uniform.  [cached]
The Python Papers Source Codes , 2009,
Abstract: This manuscript illustrates the implementation and testing of nine statistical distributions, namely Beta, Binomial, Chi-Square, F, Gamma, Geometric, Poisson, Student s t and Uniform distribution, where each distribution consists of three common functions ì Probability Density Function (PDF), Cumulative Density Function (CDF) and the inverse of CDF (inverseCDF).
Segmentation of the Poisson and negative binomial rate models: a penalized estimator  [PDF]
Alice Cleynen,Emilie Lebarbier
Statistics , 2013,
Abstract: We consider the segmentation problem of Poisson and negative binomial (i.e. overdispersed Poisson) rate distributions. In segmentation, an important issue remains the choice of the number of segments. To this end, we propose a penalized log-likelihood estimator where the penalty function is constructed in a non-asymptotic context following the works of L. Birg\'e and P. Massart. The resulting estimator is proved to satisfy an oracle inequality. The performances of our criterion is assessed using simulated and real datasets in the RNA-seq data analysis context.
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