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Characterizations of some well-known discrete distributions based on variants of the Rao-Rubin condition. (English) Zbl 0851.62013

Summary: The present note investigates characterizations of some well-known discrete distributions based on variants of the Rao-Rubin condition [C. R. Rao and H. Rubin, ibid. 26, 295-298 (1964; Zbl 0137.36604)]. D. N. Shanbhag and R. M. Clark [Proc. Camb. Philos. Soc. 71, 517-522 (1972; Zbl 0243.62008)], and G. P. Patil and M. V. Ratnaparkhi [Sankyā, Ser. B 39, 65-75 (1977)] have given characterizations for the Poisson, binomial and negative binomial distributions, respectively, replacing the Rao-Rubin condition by conditional expectations of the distributions of the r.v.’s \(X\) and \(Y\).
Our work focusses on unification and generalization as well as on some extensions of these results. A result relating to the R. C. Srivastava and J. Singh [G. Patil et al. (eds.), A modern Course on Statistical Distributions in Scientific Work, Vol. 3, 271-277 (1975; Zbl 0308.60003)] conjecture and modified versions of the results of D. N. Shanbhag [J. Appl. Probab. 14, 640-646 (1977; Zbl 0375.60021)] and A. A. Alzaid [Sankhyā, Ser. A 48, 104-106 (1986; Zbl 0609.60020)] are also given.

MSC:

62E10 Characterization and structure theory of statistical distributions
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