Bivariate Poisson Distribution Research Articles

Estimation in a truncated bivariate poisson distribution

Moment estimators for parameters in a truncated bivariate Poisson distribution are derived in Hamdan (1972) for the special case of λ1 = λ2, Where λ1, λ2 are the marginal means. Here we derive the maximum likelihood estimators for this special case. The information matrix is also obtained which provides asymptotic covariance matrix of the maximum likelihood estimators. The asymptotic covariance matrix of moment estimators is also derived. The asymptotic efficiency of moment estimators is computed and found to be very low.

On bivariate truncated poisson distribution

A bivariate truncated Poisson distribution in terms of Gram Charlier polynomials is defined.A lower truncation in both variables at any points X s 1 X s 2 ,is considered. The probability function of the distribution is expressed in terms of univariate Poisson distribution functions. The probabilities of the bivariate Poisson distribution for given correlation coefficient are tabulated for ρ=0.25,0.5,0.75. Using the cannonical form of the distribution rsth factorial moments of the distribution are obtained. The method of moments is used to find the estimates of the parameters of the marginals λ1,λ2 and the correlation coefficient ρ. The asymtotic variances of these estimates are considered. An example is given to illustrate the application of the estimators and their variances.

The Asymptotic Distribution of Maxima in Bivariate Samples

Abstract This article investigates the joint distribution, as n→ ∞, of the maxima in a sample of n independent observations of a bivariate random variable (X,Y). A method is presented for deriving the asymptotic distribution of the maxima provided that (1) X and Y possess asymptotic extreme-value distributions and (2) the probability element dF(x,y) has a canonical series expansion. Applied to the bivariate normal distribution, this method confirms the known fact that correlated normal maxima are asymptotically uncorrelated. Applied to the bivariate gamma and compound correlated bivariate Poisson distributions, the method establishes that maxima from these distributions are also asymptotically uncorrelated.

The structure of bivariate Poisson distribution

The structure of bivariate Poisson distribution

The diagonal distribution of the bivariate Poisson distribution

The diagonal distribution of the bivariate Poisson distribution

On a characterization of Poisson distributions

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

On a characterization of Poisson distributions

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

Estimation in the Truncated Bivariate Poisson Distribution

We consider the problem of estimating the parameters of the bivariate Poisson distribution from truncated samples. Two kinds of truncation are considered, namely, when zero values may not be observed on one or both marginals. We use the canonical form of the bivariate Poisson distribution to derive the first and second moments of the truncated distribution. Hence, the method of moments is used to obtain estimates for the parameters of the marginals λ1, λ2 and the correlation coefficient ρ. The properties of the estimates are considered in some special cases. An example is given to illustrate the application of the estimation formulae.

Estimation in the Truncated Bivariate Poisson Distribution

We consider the problem of estimating the parameters of the bivariate Poisson distribution from truncated samples. Two kinds of truncation are considered, namely, when zero values may not be observed on one or both marginals. We use the canonical form of the bivariate Poisson distribution to derive the first and second moments of the truncated distribution. Hence, the method of moments is used to obtain estimates for the parameters of the marginals λ1, λ2 and the correlation coefficient ρ. The properties of the estimates are considered in some special cases. An example is given to illustrate the application of the estimation formulae.

Canonical Expansion of the Compound Correlated Bivariate Poisson Distribution

Abstract Edwards and Gurland [4] introduced the compound correlated bivariate Poisson (C.C.B.P.) distribution as a mathematical model applicable to bivariate accident data. It was shown that the marginal distributions of the C.C.B.P. were negative binomials. Moments were derived and regressions were proved to be linear. In the present article, the properties of the orthogonal polynomials with respect to the negative binomial distribution are used to derive the canonical expansion of the C.C.B.P. All properties of the bivariate distribution can be easily derived from its canonical form. As an application, we estimate the correlation coefficient for a contingency table with an underlying C.C.B.P. distribution.

A Note on Positive Dependence and the Structure of Bivariate Distributions

Let X and Y be random variables whose marginal distributions $\mu ( \cdot )$ are identical. If their joint distribution function admits a series expansion in appropriate sets of functions with nonnegative coefficients, it is shown that \[ P\{ X \in A,Y \in A\} \geqq P\{ X \in A\} P\{ Y \in A\} \] for every $\mu $-measurable set A. This property is demonstrated for a variety of known two-dimensional distributions whose marginal measures are either discrete or absolutely continuous with respect to Lebesgue measure. Examples include bivariate Gaussian, exponential, gamma, chi-square, binomial, Poisson, negative binomial, and hypergeometric distributions.

Canonical Expansion of the Compound Correlated Bivariate Poisson Distribution

Canonical Expansion of the Compound Correlated Bivariate Poisson Distribution

A problem of the Allocation of Particles in Cells and Cycles of Random Permutations

Previous article Next article A problem of the Allocation of Particles in Cells and Cycles of Random PermutationsV. F. KolchinV. F. Kolchinhttps://doi.org/10.1137/1116005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] John Riordan, An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons Inc., New York, 1958xi+244 MR0096594 0078.00805 Google Scholar[2] V. L. Goncharov, From the field of combinatorial analysis, Izv. Akad. Nauk SSSR, Ser. Mat., 8 (1944), 3–48, (In Russian.) 0129.31503 Google Scholar[3] V. L. Goncharov, On the distribution of cycles in permutations, Dokl. Akad. Nauk SSSR, 35 (1942), 299–301, (In Russian.) Google Scholar[4] L. A. Shepp and , S. P. Lloyd, Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc., 121 (1966), 340–357 MR0195117 0156.18705 CrossrefGoogle Scholar[5] V. F. Kolchin, A class of limit theorems for conditional distributions, Litovsk. Mat. Sb., VIII (1968), 53–63, (In Russian.) 0235.60023 Google Scholar[6] V. F. Kolchin, On the limiting behavior of extreme order statistics in a polynomial scheme, Theory Prob. Applications, 14 (1969), 458–469 10.1137/1114058 LinkGoogle Scholar[7] B. A. Sevast'yanov and , V. P. Chistyakov, Asymptotic normality in the classical ball problem, Theory Prob. Applications, 9 (1964), 198–211 10.1137/1109034 0142.14704 LinkGoogle Scholar[8] V. P. Chistyakov, Local limit theorems in the theory of branching random processes, Theory Prob. Applications, 2 (1957), 345–363 10.1137/1102024 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A prior for record linkage based on allelic partitionsComputational Statistics & Data Analysis, Vol. 172 | 1 Aug 2022 Cross Ref Central limit theorem for the prefix exchange distance under Ewens sampling formulaDiscrete Mathematics, Vol. 344, No. 2 | 1 Feb 2021 Cross Ref Random Partition Models for Microclustering TasksJournal of the American Statistical Association, Vol. 3 | 8 December 2020 Cross Ref Probabilistic divide-and-conquer: Deterministic second halfAdvances in Applied Mathematics, Vol. 92 | 1 Jan 2018 Cross Ref Counts of Bernoulli success strings in a multivariate frameworkStatistics & Probability Letters, Vol. 119 | 1 Dec 2016 Cross Ref On runs, bivariate Poisson mixtures and distributions that arise in Bernoulli arraysElectronic Communications in Probability, Vol. 19, No. none | 1 Jan 2014 Cross Ref Asymptotic behavior of some statistics in Ewens random permutationsElectronic Journal of Probability, Vol. 18, No. none | 1 Jan 2013 Cross Ref Joint Distributions of Counts of Strings in Finite Bernoulli SequencesJournal of Applied Probability, Vol. 49, No. 03 | 4 February 2016 Cross Ref Joint Distributions of Counts of Strings in Finite Bernoulli SequencesJournal of Applied Probability, Vol. 49, No. 3 | 4 February 2016 Cross Ref Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random A-PermutationsA. L. YakymivTheory of Probability & Its Applications, Vol. 54, No. 1 | 17 February 2012AbstractPDF (202 KB)Connections Between Bernoulli Strings and Random PermutationsThe Legacy of Alladi Ramakrishnan in the Mathematical Sciences | 19 July 2010 Cross Ref A study of counts of Bernoulli strings via conditional Poisson processesProceedings of the American Mathematical Society, Vol. 137, No. 6 | 30 December 2008 Cross Ref The number of components in a logarithmic combinatorial structureThe Annals of Applied Probability, Vol. 10, No. 2 | 1 May 2000 Cross Ref Limit Theorems for Combinatorial Structures via Discrete Process ApproximationsRandom Structures and Algorithms, Vol. 3, No. 3 | 1 Jan 1992 Cross Ref Goodness-of-Fit Tests for Generalized Urn Schemes Based on Separable StatisticsA. F. RonzhinTheory of Probability & Its Applications, Vol. 33, No. 1 | 17 July 2006AbstractPDF (980 KB)Local Limit Theorems for the Number of Components of Random Permutations and MappingsA. I. PavlovTheory of Probability & Its Applications, Vol. 33, No. 1 | 17 July 2006AbstractPDF (384 KB)Generation of a random partition of a finite set by an urn modelJournal of Combinatorial Theory, Series A, Vol. 35, No. 2 | 1 Sep 1983 Cross Ref The Asymptotic Distribution of Maximum Tree Size in a Random ForestYu. L. PavlovTheory of Probability & Its Applications, Vol. 22, No. 3 | 28 July 2006AbstractPDF (726 KB)A Problem of the Allocation of Particles in Cells and Random MappingsV. F. KolchinTheory of Probability & Its Applications, Vol. 21, No. 1 | 17 July 2006AbstractPDF (1050 KB) Volume 16, Issue 1| 1971Theory of Probability & Its Applications1-198 History Submitted:13 May 1969Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1116005Article page range:pp. 74-90ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Estimation for the bivariate Poisson distribution

This paper is concerned with the estimation of the covariance parameter of the bivariate Poisson distribution. It is shown that the method of moments has low efficiency for distributions with appreciable correlation, and an iterative method of solving the likelihood equation is described. A further method of estimation is described which is more efficient than the method of moments for certain regions of the parameter space. Finally, the numerical results of applying the maximum-likelihood method to two sets of data are given as illustrations.

A Class of Distributions Applicable to Accidents

Abstract This paper presents an extension of the mathematical model used to justify accident proneness. It assumes that the distribution of accidents incurred by an individual in non-overlapping intervals is a correlated bivariate Poisson (C.B.P.). On compounding this correlated bivariate Poisson through a Gamma distribution an extended bivariate negative binomial or, more precisely, a compound correlated bivariate Poisson (C.C.B.P.) distribution is obtained. Recurrence relations and expressions for the required probabilities are illustrated for two sets of data. The C.C.B.P. proved to fit as well as the bivariate negative binomial when the estimate of B12 was close to zero, and much better than the latter distribution when the estimate of B12 was not close to zero.

A Class of Distributions Applicable to Accidents

Abstract This paper presents an extension of the mathematical model used to justify accident proneness. It assumes that the distribution of accidents incurred by an individual in non-overlapping intervals is a correlated bivariate Poisson (C.B.P.). On compounding this correlated bivariate Poisson through a Gamma distribution an extended bivariate negative binomial or, more precisely, a compound correlated bivariate Poisson (C.C.B.P.) distribution is obtained. Recurrence relations and expressions for the required probabilities are illustrated for two sets of data. The C.C.B.P. proved to fit as well as the bivariate negative binomial when the estimate of B12 was close to zero, and much better than the latter distribution when the estimate of B12 was not close to zero.