On one problem of the optimal choice of record values

Abstract

Let independent random variables X1,X2, . . . have the same continuous distribution function. The upper record values X(1) = X1 < X(2) < . . ., generated by this sequence of variables, as well as the lower record values x(1) = X1 > x(2) > . . . are considered. It is known that in this situation the mean value c(n) of the sum of numbers of the both types of records among the first n of X’s is given by equality c(n) = 2(1 + 1 2 + . . . + 1 n ), n = 1, 2, . . .. The problem considered here is how to obtain the maximal mean value e(n) of the considered number of records among the rest of the random variables getting sequentially the observed values x1, x2, . . . of X’s and selecting one of them as the initial point. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element Xr are discussed. The corresponding tables for values e(n) and differences (n) = e(n)−c(n) are presented for different values of n. The value of = limn→∞ (n) is given also. In some sense the considered problem and presented in the paper optimizing procedures are very close to the classical “secretary problem” in which the probability to select the last record value in the set of independent identically distributed X’s is maximized.