Conditions for quasi-stationarity of the Bayes rule in selection problems with an unknown number of rankable options
Let N be an unknown random number of options which arrive at i.i.d. times \(Z_ 1,...,Z_ N\) with some known continuous distributin F on the interval (0,T). The options are ranked from best (rank 1) to worst, and \(Z_ i\) is the arrival time of the i th best. At any time \(t\in (0,T)\) only the relative ranks of those options which have arrived so far can be observed. The object is to find a stopping rule \(\tau\) based only on the observed relative ranks, which minimizes some risk function E q(R\({}_{\tau})\), where \(R_{\tau}\) is the rank of the option selected by \(\tau\), and q(\(\cdot)\) is a prescribed nondecreasing nonnegative loss function. It is allowed that, with positive probability, \(\tau\) fails to select any option. A loss for not stopping is prescribed by a nondecreasing function Q(\(\cdot)\), with \(Q(0)=0.\) It is shown that only the geometric, among proper distributions on N, has the property that the stopping risk depends just on the elapsed time and not on the number of arrivals so far. This property is called quasi- stationarity. If also the optimal rule has this property we say about stationarity. Quasi-stationarity does not imply stationarity in general. The very special case when quasi-stationarity implies stationarity was considered by \textit{T. J. Stewart} [Oper. Res. 29, 130-145 (1981; Zbl 0454.90042)]. It is shown that it becomes optimal to disregard the number of arrivals so far when that number, say k, is large enough so that (i) the loss for not stopping will not change, \(Q(k)=\sup Q(\cdot)\), and (ii) the loss for stopping remains constant for all ranks greater than \(k+1\), \(q(k+1)=\sup q(\cdot)\). The paper is a continuation of previous authors' investigations [see Ann. Probab. 15, 824-830 (1987; Zbl 0592.60034)].
scientific article; zbMATH DE number 4155702
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