Conditional Evaluations of Sums of Sample Maxima and Records

Abstract

We consider sequences of independent and identically absolutely continuously distributed random variables assuming that they have finite expectation and variance. We determine sharp lower and upper bounds on the expectation of the sum of n first sample maxima and n first upper record values under the condition that the value of the jth (1 ≤ j ≤ n) sample maximum and record value, respectively, are known and equal to a given quantile of the parent distribution. The bounds are expressed in terms of the expectation and standard deviation of the parent distribution. Analogous evaluations are presented for the sum of record values in n observations, when the jth sample maximum is known. The theoretical results are numerically compared.