Abstract
AbstractThe full-information best choice problem asks one to find a strategy maximising the probability of stopping at the minimum (or maximum) of a sequence $$X_1,\cdots ,X_n$$
X
1
,
⋯
,
X
n
of i.i.d. random variables with continuous distribution. In this paper we look at more general models, where independent $$X_j$$
X
j
’s may have different distributions, discrete or continuous. A central role in our study is played by the running minimum process, which we first employ to re-visit the classic problem and its limit Poisson counterpart. The approach is further applied to two explicitly solvable models: in the first the distribution of the jth variable is uniform on $$\{j,\cdots ,n\}$$
{
j
,
⋯
,
n
}
, and in the second it is uniform on $$\{1,\cdots , n\}$$
{
1
,
⋯
,
n
}
.
Funder
Wroclaw University of Science and Technology subsidy
Publisher
Springer Science and Business Media LLC
Subject
Economics, Econometrics and Finance (miscellaneous),Engineering (miscellaneous),Statistics and Probability
Reference34 articles.
1. Baryshnikov, Y., Eisenberg, B., Stengle, G.: A necessary and sufficient condition for the existence of the limiting probability of a tie for first place. Stat. Probab. Lett. 23(3), 203–209 (1995)
2. Berezovsky, B.A., Gnedin, A.V.: The best choice problem. Nauka, Moscow (1984)
3. Bojdecki, T.: On optimal stopping of a sequence of independent random variables - probability maximizing approach. Stoch. Proc. Appl. 6, 153–163 (1978)
4. Campbell, G.: The maximum of a sequence with prior information. Comm. Stat. (Part C: Sequential Analysis) 1(3), 177–191 (1982)
5. Chow, Y.S., Robbins, H., Siegmund, D.: The theory of optimal stopping. Dover, 1st edition (1991)