### On limit distribution theorems for sums of a random number of random variables appearing in the study of rarefaction of a recurrent process

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Power distributions can be characterized by equalities involving three moments of order statistics. Similar equalities involving three moments of k-record values can also be used for such a characterization. The case of samples with random sizes is also considered.

Bayesian estimation for the two parameters of a Gumbel distribution are obtained based on kth lower record values. Prediction, either point or interval, for future kth lower record values is also presented from a Bayesian view point. Some of the results of [4] can be obtained as special cases of our results (k=1).

Relations for the marginal, joint, conditional characteristic functions of k-th upper and lower record values from generalized Pareto distribution and inverse generalized Pareto distribution are given.

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Using characterization conditions of continuous distributions in terms of moments of order statistics given in [12], [23], [6] and [7] we present new goodness-of-fit techniques.

We derive tests of fit from characterizations of continuous distributions via moments of the kth upper record values.

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on m selected kth record values are presented for the parameters of the Gumbel and Burr distributions.

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