×

zbMATH — the first resource for mathematics

On a class of discrete distributions arising from the birth-death-with-immigration process. (English) Zbl 0854.60016
Summary: This paper considers a class of distributions which may be regarded as the convolution of a negative binomial and a stopped-sum generalized hypergeometric factorial-moment random variables. Some properties are derived and it is shown that this class of distributions is a subset of distributions for the birth-and-death process with immigration (also reversible counter system). Formulations by mixing, limiting distributions and maximum likelihood equations are also discussed.

MSC:
60E05 Probability distributions: general theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bhattacharya SK (1961) Confluent hypergeometric distributions of discrete and continuous type with applications to accident proneness. Calcutta Statist Ass Bull 15:20–31 · Zbl 0146.41002
[2] Downton F (1970) Bivariate exponential distributions in reliability theory. J Roy Statist Soc Ser B 32:408–417 · Zbl 0226.62101
[3] Erdélyi A (1953) Higher transcendental functions, Vol 1. McGraw-Hill · Zbl 0052.29502
[4] Exton H (1978) Handbook of hypergeometric integrals. Ellis Horwood · Zbl 0377.33001
[5] Getz WM (1975) Optimal control of a birth-and-death process population model. Math Biosciences 23:87–11 · Zbl 0303.60082 · doi:10.1016/0025-5564(75)90122-4
[6] Iosifescu M, Tautu P (1973) Stochastic processes with applications in biology and medicine. Springer-Verlag New York · Zbl 0262.92001
[7] Kemp AW, Kemp CD (1974) A family of discrete distributions defined via their factorial moments. Commun Statist A 3:1187–1196 · Zbl 0329.62012 · doi:10.1080/03610927408827220
[8] Kemp AW (1968) A wide class of discrete distributions and the associated differential equations. Sankhyā A 30:401–410 · Zbl 0186.53004
[9] Laha RG (1954) On some properties of the Bessel function distributions. Bull Calcutta Math Soc 46:59–72 · Zbl 0055.36503
[10] Lai CD (1981) On conditional correlation coefficients of a Wold Markov process of gamma intervals. Aust J Statist 23:232–237 · Zbl 0479.62048 · doi:10.1111/j.1467-842X.1981.tb00781.x
[11] Lampard DG (1968) A stochastic process whose successive intervals between events form a first order Markov chain-I. J Appl Prob 5:648–668 · Zbl 0185.46201 · doi:10.2307/3211928
[12] Lee PA, Ong SH (1986) Higher-order and non-stationary properties of Lampard’s stochastic reversible counter system. Statistics 17:261–278 · Zbl 0595.60100 · doi:10.1080/02331888608801936
[13] Ong SH, Lee PA (1979) The non-central negative binomial distribution. Biom J 21:611–627 · Zbl 0432.62011 · doi:10.1002/bimj.4710210704
[14] Ong SH, Lee PA (1986) On a generalized non-central negative binomial distribution. Commun Statist A 15:1065–1079 · Zbl 0609.60022 · doi:10.1080/03610928608829169
[15] Ong SH (1987) Some notes on the non-central negative binomial distribution. Metrika 34:225–236 · Zbl 0621.60016 · doi:10.1007/BF02613154
[16] Patil GP, Rao CR (1977) The weighted distributions: A survey of their applications. In: Krishnaiah PR (ed.) Applications of Statistics North-Holland Publishing Company 383–405
[17] Phatarfod RM (1971) Some approximate results in renewal and dam theory. J Aust Math Soc 12:426–432 · Zbl 0229.60060
[18] Slater LJ (1966) Generalized hypergeometric functions. Cambridge University Press · Zbl 0135.28101
[19] Sprott DA (1963) A class of contagious distributions and maximum likelihood estimation. Proc Int Symp on Classical and Contagious Discrete Distributions, Statistical Publishing Society Calcutta 337–350
[20] Tripathi RC, Gurland J (1979) Some aspects of the Kemp families of distributions. Commun Statist A 8:855–869 · Zbl 0418.62016 · doi:10.1080/03610927908827804
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.