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A characterization of geometric distribution based on weak records

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Abstract

Let \({\{X_n, n\geq 1\}}\) be a sequence of independent and identically distributed non-degenerated random variables with common cumulative distribution function F. Suppose X 1 is concentrated on 0, 1, . . . , N ≤ ∞ and P(X 1 = 1) > 0. Let \({X_{U_w(n)}}\) be the n-th upper weak record value. In this paper we show that for any fixed m ≥ 2, X 1 has Geometric distribution if and only if \({X_{U_{w}(m)}\mathop=\limits^d X_1+\cdots+X_m ,}\) where \({\underline{\underline{d}}}\) denotes equality in distribution. Our result is a generalization of the case m = 2 obtained by Ahsanullah (J Stat Theory Appl 8(1):5–16, 2009).

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Authors and Affiliations

  1. Department of Management Sciences, Rider University, Lawrenceville, NJ, USA

    Mohammad Ahsanullah

  2. Department of Psychiatry, Virginia Institute for Psychiatric and Behavioral Genetics, Virginia Commonwealth University, PO Box 980126, Richmond, VA, 23298-0126, USA

    Fazil Aliev

  3. Ankara University, Ankara, Turkey

    Fazil Aliev

Authors
  1. Mohammad Ahsanullah
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  2. Fazil Aliev
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Correspondence to Fazil Aliev.

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Ahsanullah, M., Aliev, F. A characterization of geometric distribution based on weak records. Stat Papers 52, 651–655 (2011). https://doi.org/10.1007/s00362-009-0274-0

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  • DOI: https://doi.org/10.1007/s00362-009-0274-0

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