A class of bivariate negative binomial distributions with different index parameters in the marginals

Abstract In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.

[1]  Seng-Huat Ong,et al.  On a generalized non-central negative binomial distribution , 1986 .

[2]  J. Gurland,et al.  A Class of Distributions Applicable to Accidents , 1961 .

[3]  Some families of Bessel distributions and their applications , 2006 .

[4]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[5]  P Holgate,et al.  Bivariate generalizations of Neyman's type A distribution. , 1966, Biometrika.

[6]  A. W. Kemp,et al.  A treatise on generating functions , 1984 .

[7]  Skewed Bessel function distributions with application to rainfall data , 2007 .

[8]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[9]  S. Kotz.  Multivariate Distributions at a Cross Road , 1975 .

[10]  John M. Olin.  Markov Chain Monte Carlo Analysis of Correlated Count Data , 2003 .

[11]  J. Juritz,et al.  BIVARIATE COMPOUND POISSON DISTRIBUTIONS , 1987 .

[12]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[13]  S. Ong.  Mixture formulations of a bivariate negative binomial distribution with applications , 1990 .

[14]  H. Srivastava,et al.  Special Functions in Queuing Theory and Related Stochastic Processes , 1982 .

[15]  S. Ong.  Canonical expansions, correlation structure, and conditional distributions of bivariate distributions generated by mixtures , 1993 .

[16]  An extension of the Hille-Hardy formula , 1969 .

[17]  J. F. Barrett,et al.  An expansion for some second-order probability distributions and its application to noise problems , 1955, IRE Trans. Inf. Theory.

[18]  J. Aitchison,et al.  The multivariate Poisson-log normal distribution , 1989 .

[19]  H. O. Lancaster.  Correlations and Canonical Forms of Bivariate Distributions , 1963 .

[20]  H. O. Lancaster.  The Structure of Bivariate Distributions , 1958 .

[21]  Hari M. Srivastava,et al.  Local dependence functions for some families of bivariate distributions and total positivity , 2010, Appl. Math. Comput..

[22]  Hari M. Srivastava,et al.  A unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions , 2005, Appl. Math. Comput..

[23]  John L. Brown,et al.  A criterion for the diagonal expansion of a second-order probability distribution in orthogonal polynomials , 1958 .

[24]  P. W. Karlsson,et al.  Multiple Gaussian hypergeometric series , 1985 .

[25]  Hari M. Srivastava,et al.  Some relationships between the generalized Apostol–Bernoulli polynomials and Hurwitz–Lerch Zeta functions , 2006 .

[26]  S. Ong,et al.  On the bivariate negative binomial distribution of Mitchell and Paulson , 1985 .

[27]  D. R. Jensen.  A Note on Positive Dependence and the Structure of Bivariate Distributions , 1971 .

[28]  C. Lai.  Construction of bivariate distributions by a generalised trivariate reduction technique , 1995 .

[29]  Hari M. Srivastava,et al.  A class of Hurwitz-Lerch Zeta distributions and their applications in reliability , 2008, Appl. Math. Comput..

[30]  Hari M. Srivastava,et al.  Some Integrals of the products of laguerre polynomials , 2001, Int. J. Comput. Math..

[31]  A. G. Arbous,et al.  Accident statistics and the concept of accident-proneness , 1951 .

[32]  Ramesh C. Gupta,et al.  A new generalization of the negative binomial distribution , 2004, Comput. Stat. Data Anal..

[33]  H. M. Srivastava,et al.  Some expansion formulas for a class of generalized Hurwitz–Lerch Zeta functions , 2006 .

[34]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[35]  Hari M. Srivastava,et al.  Some generalizations of the Laplace distribution , 2006, Appl. Math. Comput..

[36]  A. Paulson,et al.  A new bivariate negative binomial distribution , 1981 .

[37]  S. Kocherlakota,et al.  Bivariate discrete distributions , 1992 .

[38]  A. W. Kemp,et al.  Univariate Discrete Distributions , 1993 .

[39]  S. Zeger.  A regression model for time series of counts , 1988 .

[40]  Seng-Huat Ong,et al.  The computer generation of bivariate binomial variables with given marginals and correlation , 1992 .

[41]  Ganapati P. Patil,et al.  Statistical Distributions in Scientific Work , 1981 .

[42]  Kanti V. Mardia,et al.  Families of Bivariate Distributions. , 1971 .

[43]  K. Subrahmaniam,et al.  A Test for “Intrinsic Correlation” in the Theory of Accident Proneness , 1966 .

[44]  Carles M. Cuadras,et al.  Correspondence analysis and diagonal expansions in terms of distribution functions , 2002 .

[45]  Debasis Kundu,et al.  Bivariate generalized exponential distribution , 2009, J. Multivar. Anal..

[46]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[47]  G. Eagleson.  Polynomial Expansions of Bivariate Distributions , 1964 .