An asymptotic expansion for inverse moments of positive binomial and Poisson distributions is derived. The expansion coefficients of the asymptotic series are given by the positive central moments of the distribution. Compared to previous results, a single expansion formula covers all (also non-integer) inverse moments. In addition, the approach can be generalized to other positive distributions.
Certain monotonicity properties of the Poisson approximation to the binomial distribution are established. As a natural application of these results, exact (rather than approximate) tests of hypotheses on an unknown value of the parameter $p$ of the binomial distribution are presented.
We prove the inequality $E[(X/μ)^k] \le (\frac{k/μ}{\log(k/μ+1)})^k \le \exp(k^2/(2μ))$ for sub-Poissonian random variables, such as Binomially or Poisson distributed random variables with mean $μ$. The asymptotics $1+O(k^2/μ)$ can be shown to be tight for small $k$. This improves over previous uniform bounds for the raw moments of those distributions by a factor exponential in $k$.
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.
Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $λ:= p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/π_λ(x)$, where $π_λ$ is the weight function of the Poisson distribution with parameter $λ$.
Compound Poisson distributions and signed compound Poisson measures are used for approximation of the Markov binomial distribution. The upper and lower bound estimates are obtained for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are calculated. For the upper bounds, the smoothing properties of compound Poisson distributions are applied. For the lower bound estimates, the characteristic function method is used.
This paper proposes a generalized binomial distribution with four parameters, which is derived from the finite capacity queueing system with state-dependent service and arrival rates. This distribution is also generated from the conditional Conway-Maxwell-Poisson distribution given a sum of two Conway-Maxwell-Poisson variables. In this paper, we consider the properties about the probability mass function, index of dispersion, skewness and kurtosis and give applications of the proposed distribution from its geneses. The estimation method and simulation study are also considered.
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. Birgé 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.
A new three parameter natural extension of the Conway-Maxwell-Poisson (COM-Poisson) distribution is proposed. This distribution includes the recently proposed COM-Poisson type negative binomial (COM-NB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COM-type Generalization of the Negative Binomial Distribution, Accepted in Communications in Statistics-Theory and Methods] and the generalized COM-Poisson (GCOMP) distribution [Imoto, T. :(2014) A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution, Applied Mathematics and Computation, 247, 824…
We consider the identity testing problem - or goodness-of-fit testing problem - in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ iid samples drawn from an unknown distribution $q$, we investigate how large $ρ>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q) \geq ρ$, where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q) = \|p-q\|_t$ for $t \in [1,2]$ where $\|\cdot\|_t$ is the…
Based on the probability generating function of stuttering Poisson distribution (SPD), this paper considers some equivalent propositions of SPD. From this, we show that some distributions in the application of non-life insurance actuarial science are SPD, such as negative binomial distribution, compound Poisson distribution etc.. By weakening condition of equivalent propositions of SPD, we define the generalized SPD. We consider cumulant estimation of generalized SPD's parameters. As an application, we use SPD with four parameters (4-th SPD) to fit auto insurance claim data. The fitting result…
In this paper, we develop a general theory of truncated inverse binomial sampling. In this theory, the fixed-size sampling and inverse binomial sampling are accommodated as special cases. In particular, the classical Chernoff-Hoeffding bound is an immediate consequence of the theory. Moreover, we propose a rigorous and efficient method for probability estimation, which is an adaptive Monte Carlo estimation method based on truncated inverse binomial sampling. Our proposed method of probability estimation can be orders of magnitude more efficient as compared to existing methods in literature and…
MIT（麻省理工学院）；University of Edinburgh（爱丁堡大学）；Columbia University（哥伦比亚大学）
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 op…
The optimum quality that can be asymptotically achieved in the estimation of a probability p using inverse binomial sampling is addressed. A general definition of quality is used in terms of the risk associated with a loss function that satisfies certain assumptions. It is shown that the limit superior of the risk for p asymptotically small has a minimum over all (possibly randomized) estimators. This minimum is achieved by certain non-randomized estimators. The model includes commonly used quality criteria as particular cases. Applications to the non-asymptotic regime are discussed considerin…
The logistic regression model is known to converge to a Poisson point process model if the binary response tends to infinitely imbalanced. In this paper, it is shown that this phenomenon is universal in a wide class of link functions on binomial regression. The proof relies on the extreme value theory. For the logit, probit and complementary log-log link functions, the intensity measure of the point process becomes an exponential family. For some other link functions, deformed exponential families appear. A penalized maximum likelihood estimator for the Poisson point process model is suggested…
We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match up the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations typically are more accurate than Poisson or standard binomial approximations. The …
Sequential estimation of a probability $p$ by means of inverse binomial sampling is considered. For $μ_1,μ_2>1$ given, the accuracy of an estimator $\hat{p}$ is measured by the confidence level $P[p/μ_2\leq\hat{p}\leq pμ_1]$. The confidence levels $c_0$ that can be guaranteed for $p$ unknown, that is, such that $P[p/μ_2\leq \hat{p}\leq pμ_1]\geq c_0$ for all $p\in(0,1)$, are investigated. It is shown that within the general class of randomized or non-randomized estimators based on inverse binomial sampling, there is a maximum $c_0$ that can be guaranteed for arbitrary $p$. A non-randomized est…
Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(η)$ for a certain r.v. $η$. The r.v. $η$ and the class $\mathcal{F}$ depend on the choice of on…
We develop a general method for computing logarithmic and log-gamma expectations of distributions. As a result, we derive series expansions and integral representations of the entropy for several fundamental distributions, including the Poisson, binomial, beta-binomial, negative binomial, and hypergeometric distributions. Our results also establish connections between the entropy functions and to the Riemann zeta function and its generalizations.
Sequential estimation of the success probability $p$ in inverse binomial sampling is considered in this paper. For any estimator $\hat p$, its quality is measured by the risk associated with normalized loss functions of linear-linear or inverse-linear form. These functions are possibly asymmetric, with arbitrary slope parameters $a$ and $b$ for $\hat p<p$ and $\hat p>p$ respectively. Interest in these functions is motivated by their significance and potential uses, which are briefly discussed. Estimators are given for which the risk has an asymptotic value as $p$ tends to $0$, and which guaran…
The binomial and Poisson distributions have interesting relationships with the beta and gamma distributions, respectively, which involve their cumulative distribution functions and the use of conjugate priors in Bayesian statistics. We briefly discuss these relationships and some properties resulting from them which play an important role in the construction of exact nested two-sided confidence intervals and the computation of two-tailed P-values. The purpose of this article is to show that such relationships also exist between the hypergeometric distribution and a special case of the Polya (o…
Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics. The results are then applied to statistics of $m$-dependent $(k_1,k_2)$ events and 2-runs. Heinrich's method and smoothing properties of convolutions are used for the proofs.
Modeling spatial overdispersion requires point processes models with finite dimensional distributions that are overdisperse relative to the Poisson. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. And, though processes based on negative binomial finite dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we chang…
University of Edinburgh（爱丁堡大学）；University of California, San Diego（加州大学圣地亚哥分校）
We give an algorithm for properly learning Poisson binomial distributions. A Poisson binomial distribution (PBD) of order $n$ is the discrete probability distribution of the sum of $n$ mutually independent Bernoulli random variables. Given $\widetilde{O}(1/ε^2)$ samples from an unknown PBD $\mathbf{p}$, our algorithm runs in time $(1/ε)^{O(\log \log (1/ε))}$, and outputs a hypothesis PBD that is $ε$-close to $\mathbf{p}$ in total variation distance. The previously best known running time for properly learning PBDs was $(1/ε)^{O(\log(1/ε))}$. As one of our main contributions, we provide a novel…
Suppose that the distribution of $X_a$ belongs to a natural exponential family concentrated on the nonegative integers and is such that $\E(z^{X_a})=f(az)/f(a)$. Assume that $\Pr(X_a\leq k)$ has the form $c_k\int_a ^{\infty}u^kμ(du)$ for some number $c_k$ and some positive measure $μ,$ both independent of $a.$ We show that this asumption implies that the exponential family is either a binomial, or the Poisson, or a negative binomial family. Next, we study an analogous property for continuous distributions and we find that it is satisfied if and only the families are either Gaussian or Gamma. U…
In this paper we characterize all the $r$-parameter families of count distributions (satisfying mild conditions) that are closed under addition and under binomial subsampling. Surprisingly, few families satisfy both properties and the resulting models consist of the $r$th-order univariate Hermite distributions. Among these, we find the Poisson ($r=1$) and the ordinary Hermite distributions ($r=2$).
In this paper, we propose a closed form approximation to the mean and variance of a new generalization of negative binomial (NGNB) distribution arising from the Extended COM-Poisson (ECOMP) distribution developed by Chakraborty and Imoto (2016)(see [4]). The NGNB is a special case of the ECOMP distribution and was named so by these authors. This distribution is more flexible in terms of the dispersion index as compared to its ordinary counterparts. It approaches the COM-Poisson distribution (Shmueli et al. 2005) [11] under suitable limiting conditions. The NGNB can also be obtained from the CO…
One of the difficulties in calculating the capacity of certain Poisson channels is that H(lambda), the entropy of the Poisson distribution with mean lambda, is not available in a simple form. In this work we derive upper and lower bounds for H(lambda) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poiss…
National Technical University of Athens（雅典国家技术大学）；Yahoo Research（雅虎研究院）；University of Liverpool（利物浦大学）；CTI
We introduce the problem of simultaneously learning all powers of a Poisson Binomial Distribution (PBD). A PBD of order $n$ is the distribution of a sum of $n$ mutually independent Bernoulli random variables $X_i$, where $\mathbb{E}[X_i] = p_i$. The $k$'th power of this distribution, for $k$ in a range $[m]$, is the distribution of $P_k = \sum_{i=1}^n X_i^{(k)}$, where each Bernoulli random variable $X_i^{(k)}$ has $\mathbb{E}[X_i^{(k)}] = (p_i)^k$. The learning algorithm can query any power $P_k$ several times and succeeds in learning all powers in the range, if with probability at least $1- …
We introduce a class of two-parameter discrete dispersion models, obtained by combining convolution with a factorial tilting operation, similar to exponential dispersion models which combine convolution and exponential tilting. The equidispersed Poisson model has a special place in this approach, whereas several overdispersed discrete distributions, such as the Neyman Type A, Pólya-Aeppli, negative binomial and Poisson-inverse Gaussian, turn out to be Poisson-Tweedie factorial dispersion models with power dispersion functions, analogous to ordinary Tweedie exponential dispersion models with po…
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.
The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.
In this paper, we focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type $(a,b,0)$ class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-…
In this paper, we consider Bayesian point estimation and predictive density estimation in the binomial case. After presenting preliminary results on these problems, we compare the risk functions of the Bayes estimators based on the truncated and untruncated beta priors and obtain dominance conditions when the probability parameter is less than or equal to a known constant. The case where there are both a lower bound restriction and an upper bound restriction is also treated. Then our problems are shown to be related to similar problems in the Poisson case. Finally, numerical studies are presen…
Motivated by the fundamental problem of measuring species diversity, this paper introduces the concept of a cluster structure to define an exchangeable cluster probability function that governs the joint distribution of a random count and its exchangeable random partitions. A cluster structure, naturally arising from a completely random measure mixed Poisson process, allows the probability distribution of the random partitions of a subset of a sample to be dependent on the sample size, a distinct and motivated feature that differs it from a partition structure. A generalized negative binomial …
The $k$th-nearest neighbor rule is arguably the simplest and most intuitively appealing nonparametric classification procedure. However, application of this method is inhibited by lack of knowledge about its properties, in particular, about the manner in which it is influenced by the value of $k$; and by the absence of techniques for empirical choice of $k$. In the present paper we detail the way in which the value of $k$ determines the misclassification error. We consider two models, Poisson and Binomial, for the training samples. Under the first model, data are recorded in a Poisson stream a…
The paper introduces the concept of a cluster structure to define a joint distribution of the sample size and its exchangeable random partitions. The cluster structure allows the probability distribution of the random partitions of a subset of the sample to be dependent on the sample size, a feature not presented in a partition structure. A generalized negative binomial process count-mixture model is proposed to generate a cluster structure, where in the prior the number of clusters is finite and Poisson distributed and the cluster sizes follow a truncated negative binomial distribution. The n…
