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A characterization of the logarithmic series distribution and its application. (English) Zbl 0553.62015
This paper gives a characterization of the logarithmic series distribution with probability function $$[-\ell n(1-\theta)]^{- 1}\theta^ x/x$$; $$x=1,2,..$$. and $$0<\theta <1$$. Our characterization follows that of Poisson and positive Poisson given by L. N. Bol’shev [Teor. Veroyatn. Primen. 10, 488-499 (1965; Zbl 0202.502)] and the second author [SIAM J. Appl. Math. 34, 545-548 (1978; Zbl 0378.62010)], respectively.
A statistic is suggested as a consequence of the characterization to test whether a random sample $$X_ 1,X_ 2,...,X_ n$$ follows a logarithmic series probability law. The desirability of the test statistic over the usual goodness-of-fit test is discussed. A numerical example is considered for illustration.

##### MSC:
 62E10 Characterization and structure theory of statistical distributions 62G10 Nonparametric hypothesis testing
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##### References:
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