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A generalized Sibuya distribution. (English) Zbl 1398.60028

The authors study the positive integer-valued random variable \(N\) defined by \[ \mathbb{P}(N=n)=\left(1-\frac{\alpha}{\nu+1}\right)\cdots\left(1-\frac{\alpha}{\nu+n-1}\right)\frac{\alpha}{\nu+n}\,, \] for \(n=1,2,\ldots\), and with parameters \(\nu\geq0\) and \(0<\alpha<\nu+1\). This arises as the waiting time for the first success in a sequence of independent Bernoulli trials \(I_1,I_2,\ldots\) whose success probabilities are given by \(\mathbb{P}(I_j=1)=\frac{\alpha}{\nu+j}\). This distribution is termed the generalized Sibuya distribution. The authors show that it may also be constructed as a randomly stopped Poisson process, or as a certain mixed geometric distribution.
The majority of the paper is devoted to fundamental results on the generalized Sibuya distribution, such as moments, tail behaviour, generating functions, divisibility and decomposibility. Some attention is also given to statistical aspects, including method-of-moments and maximum likelihood estimation of the parameters. The authors also present some results on the Sibuya distribution (the special case \(\nu=0\)). This includes some comments on random maxima and minima, a characterization in terms of a pure death process, and a random process whose marginals are Sibuya-distributed.

MSC:

60E05 Probability distributions: general theory
60E07 Infinitely divisible distributions; stable distributions
60G70 Extreme value theory; extremal stochastic processes
62E15 Exact distribution theory in statistics
62E10 Characterization and structure theory of statistical distributions
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