Characterizations based on \(k\)-th upper and lower record values. (English) Zbl 1057.62005

Bieniek, Mariusz; Szynal, Dominik

Demonstr. Math. 37, No. 2, 463-473 (2004).

Summary: Let \(\{Y_n^{(k)},n\geq 1\}\) and \(\{Z_n^{(k)},n\geq 1\}\) denote, respectively, the sequences of \(k\)-th upper and lower record values of the sequence \(\{X_n,n\geq 1\}\) of independent identically distributed random variables with distribution function \(F\). Let \(n,k\) and \(r\) be given positive integers. We characterize distributions for which one of the conditional expectations

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\(E(Y_{n+r}^{(k)}\mid Y_n^{(k)})\), \(E(Y_n^{(k)}\mid Y_{n+r}^{(k)})\), \(E(Z_{n+r}^{(k)}\mid Z_n^{(k)})\) or \(E(Z_n^{(k)}\mid Z_{n+r}^{(k)})\), is linear. For example, distributions for which \(E(Y_{n+k}^{(k)}\mid Y_n^{(k)})\) has the form \(E(Y^{(k)}_{n+r}\mid Y_n^{(k)})=aY_n^{(k)}+b\) for some \(a,b\in\mathbb{R}\).

© by Mariusz Bieniek

MSC:

62E10	Characterization and structure theory of statistical distributions
62G32	Statistics of extreme values; tail inference

Keywords:

\(k\)-th upper record values; \(k\)-th lower record values; linearity of regression; exponential distribution; Gumbel distribution; power distribution; Pareto distributions; Fréchet distribution; Weibull distribution