Details

Autor(en) / Beteiligte

• Belkov, I. V.
• Nevzorov, V. B.

Titel

On the Problem of the Optimal Choice of Record Values

Ist Teil von

• Vestnik, St. Petersburg University. Mathematics, 2018-04, Vol.51 (2), p.107-113

Ort / Verlag

Moscow: Pleiades Publishing

Erscheinungsjahr

2018

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let the independent random variables X 1, X 2, … have the same continuous distribution function. The upper record values X (1) = X 1 < X (2) < … generated by this sequence of variables, as well as the lower record values x (1) = X 1 > x (2) > …, are considered. It is known that in this situation, the mean value c ( n ) of the total number of the both types of records among the first n variables X is given by the equality c ( n )=2(1+1/2+…+1/ n ), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x 1 , x 2 , … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e ( n ) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X r are discussed. The corresponding tables for the values e ( n ) and differences δ( n )= e ( n )– c ( n ) are presented for different values of n . The value of δ= lim n→∞ δ( n )is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.