×

On the bivariate negative binomial distribution of Mitchell and Paulson. (English) Zbl 0594.62016

Summary: The bivariate negative binomial distribution of C. R. Mitchell and A. S. Paulson [ibid. 28, 359-374 (1981; Zbl 0463.60024)] for the case \(b=c=0\) is shown to be equivalent to the accident proneness model of C. B. Edwards and J. Gurland [J. Am. Stat. Assoc. 56, 503-517 (1961; Zbl 0201.528)] and K. Subrahmaniam and K. Subrahmaniam [J. R. Stat. Soc., Ser. B 35, 131-146 (1973; Zbl 0281.62035)]. The diagonal series expansion of its joint probability function is then derived.
Two other formulations of this distribution are also considered: (i) as a mixture model, which showed how it arises as the discrete analogue to the Wicksell-Kibble bivariate gamma distribution, and (ii) as a consequence of the linear birth-and death process with immigration.

MSC:

62E99 Statistical distribution theory
62E10 Characterization and structure theory of statistical distributions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barrett, IRE Transactions on Information Theory IT-1 pp 10– (1955)
[2] and , ”Contributions to the Theory of Accident Proneness, I,” University of California Publications in Statistics, 1952, pp. 215–254.
[3] Brown, IRE Transactions on Information Theory 4 pp 172– (1958) · doi:10.1109/IRETIT.1958.6741953
[4] Edwards, American Statistical Association Journal 56 pp 503– (1961)
[5] Higher Transcendental Function, Vol. 1, McGraw-Hill, New York, 1953.
[6] Higher Transcendental Function, Vol. 2, McGraw-Hill, New York, 1953.
[7] Getz, Mathematical Biosciences 23 pp 87– (1975)
[8] Goodman, Biometrika 61 pp 215– (1974)
[9] Hutchinson, Biometrical Journal 21 pp 553– (1979)
[10] Karlin, Journal of Mathematics and Mechanics 7 pp 643– (1958)
[11] Kotz, Statistical Distributions in Scientific Work 1 pp 247– (1974)
[12] ”A Stochastic Process Whose Successive Intervals between Events Form a First Order Markov Chain–I,” report No. MEE 65-1, Electrical Engineering Department, Monash University, Australia, 1965.
[13] Lampard, Journal of Applied Probability 5 pp 648– (1968)
[14] Lancaster, Annals of Mathematical Statistics 29 pp 719– (1958)
[15] ”A Study of Some Stochastic Processes in Communication Engineering,” Ph.D. Thesis, Monash University, Australia, 1968.
[16] Families of Bivariate Distributions, Hafner Publishing Co., New York, 1970. · Zbl 0223.62062
[17] Mitchell, Naval Research Logistics Quarterly 28 pp 359– (1981)
[18] Special Functions of Mathematical Physics and Chemistry, Oliver and Boyd, Edinburgh, 1961.
[19] Subrahmaniam, Journal of the Royal Statistical Society B28 pp 180– (1966)
[20] Subrahmaniam, Journal of the Royal Statistical Society B35 pp 131– (1973)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.