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Recurrence relations for single and product moments of \(k\)-th record values from Pareto, generalized Pareto and Burr distributions. (English) Zbl 0932.62015

Summary: We give recurrence relations satisfied by single and product moments of \(k\)-th upper record values from the Pareto, generalized Pareto and Burr distributions. From these relations one can obtain all the single and product moments of all \(k\)-th record values and at the same time all record values \((k=1)\). Moreover, we see that the single and product moments of all \(k\)-th record values from these distributions can be expressed in terms of the moments of the minimal statistic of a \(k\)-sample from these distributions.
We also show that similar results established recently by us [J. Appl. Stat. Sci. 7, No. 1, 53-61 (1998; Zbl 0901.62023)] for the \(k\)-th upper record values from the exponential distribution may be deduced by letting the shape parameter \(\beta\) tend to 0. At the end we give characterizations of the three distributions considered. These results generalize, among other things, those given by N. Balakrishnan and M. Ahsanullah [ibid. 23, No. 10, 2841-2852 (1994)].

MSC:

62E10 Characterization and structure theory of statistical distributions

Citations:

Zbl 0901.62023
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References:

[1] Ahsanullah M., Record Statistics (1995) · Zbl 0907.62017
[2] Arnold B.C., Pareto distributions (1983)
[3] Balakrishnan, R. and Ahsanullah, M. 1994.Recurrence relations for sin{\(\neg\)}gle and product moments of record values from generalized Pareto distribution, 2841–2852. Commun. Statist. Theory Meth. · Zbl 0850.62118
[4] Balakrishnan R., Relations for single and prod{\(\neg\)}uct moments of record values from exponential distribution (1995)
[5] Beirlant J., Practical Analysis of Extreme Values (1996) · Zbl 0888.62003
[6] Dziubdziela, W. and Kopociiiski, B. 1976.Limiting properties of the k-th record value, 187–190. Appl. Math. · Zbl 0337.60023
[7] Grudzien Z., UMCS Lublin
[8] Lin, G.D. 1986.On a moment problem, 595–598. Tohoku Math. Journal. · Zbl 0602.42016
[9] Pawlas, P. and Szynal, D. 1998.Relations for single and product moments of k-th record values from exponential and Gumbel distributions, 53–62. J. Appl. Statist. Sc. · Zbl 0901.62023
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