{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,7]],"date-time":"2025-03-07T05:13:37Z","timestamp":1741324417863,"version":"3.38.0"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,10,19]],"date-time":"2024-10-19T00:00:00Z","timestamp":1729296000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,10,19]],"date-time":"2024-10-19T00:00:00Z","timestamp":1729296000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math Meth Oper Res"],"published-print":{"date-parts":[[2025,2]]},"abstract":"Abstract<\/jats:title>\n Suppose N<\/jats:italic> independent Bernoulli trials with success probabilities \n \n $$p_1, p_2,\\ldots $$<\/jats:tex-math>\n \n \n \n p<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n p<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are observed sequentially at times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. The case \n \n $$p_k=1\/k$$<\/jats:tex-math>\n \n \n \n p<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has been studied by many authors as a version of the familiar best choice problem, where both N<\/jats:italic> and the observation times are random. We consider a more general profile \n \n $$p_k=\\theta \/(\\theta +k-1)$$<\/jats:tex-math>\n \n \n \n p<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n \u03b8<\/mml:mi>\n \/<\/mml:mo>\n \n (<\/mml:mo>\n \u03b8<\/mml:mi>\n +<\/mml:mo>\n k<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and assume that the prior distribution of N<\/jats:italic> is negative binomial with shape parameter \n \n $$\\nu $$<\/jats:tex-math>\n \n \u03bd<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, so the arrivals occur at times of a mixed Poisson process. The setting with two parameters offers a high flexibility in understanding the nature of the optimal strategy, which we show is intrinsically related to monotonicity properties of the Gaussian hypergeometric function. Using this connection, we find that the myopic stopping strategy is optimal if and only if \n \n $$\\nu \\ge \\theta $$<\/jats:tex-math>\n \n \n \u03bd<\/mml:mi>\n \u2265<\/mml:mo>\n \u03b8<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. Furthermore, we derive formulas to assess the winning probability and discuss limit forms of the problem for large N<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00186-024-00880-1","type":"journal-article","created":{"date-parts":[[2024,10,19]],"date-time":"2024-10-19T20:30:43Z","timestamp":1729369843000},"page":"1-27","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The last-success stopping problem with random observation times"],"prefix":"10.1007","volume":"101","author":[{"given":"Alexander","family":"Gnedin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3779-6981","authenticated-orcid":false,"given":"Zakaria","family":"Derbazi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,10,19]]},"reference":[{"key":"880_CR1","series-title":"EMS Monographs in Mathematics","doi-asserted-by":"publisher","DOI":"10.4171\/000","volume-title":"Logarithmic combinatorial structures: a probabilistic approach","author":"R Arratia","year":"2003","unstructured":"Arratia R, Barbour AD, Tavar\u00e9 S (2003) Logarithmic combinatorial structures: a probabilistic approach. 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The authors have no Conflict of interest to declare that are relevant to the content of this article. All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors have no financial or proprietary interests in any material discussed in this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Financial Interests"}}]}}