Two choice optimal stopping

Let X_1,X_2,... be a sequence of [0,1]-valued i.i.d. random variables, let c\geq 0 be a sampling cost for each observation and let Y_i=X_i-ic, i=1,2,.... For n=1,2,..., let M(Y_1,...,Y_n)=E(max_{1\leq i\leq n}Y_i) and V(Y_1,...,Y_n)=sup_{\tau \in C^n}E(Y_{\tau}), where C^n denotes the set of all stopping rules for Y_1,...,Y_n. Sharp upper bounds for the difference M(Y_1,...,Y_n)-V(Y_1,...,Y_n) are given under various restrictions on c and n.

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables. The objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of "prophet ine...

For $\tau$ a stopping rule adapted to a sequence of $n$ iid observations, we define the loss to be $\ex [ q(R_\tau)]$, where $R_j$ is the rank of the $j$th observation, and $q$ is a nondecreasing function of the rank. This setting covers both the best choice problem with $q(r)={\bf 1}(r>1)$, and Robbins' problem with $q(r)=r$. As $n\to\infty$ the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bo...

In the classical optimal stopping problem, a player is given a sequence of random variables $X_1\ldots X_n$ with known distributions. After observing the realization of $X_i$, the player can either accept the observed reward from $X_i$ and stop, or reject the observed reward from $X_i$ and continue to observe the next variable $X_{i+1}$ in the sequence. Under any fixed ordering of the random variables, an optimal stopping policy, one that maximizes the player's expected rewar...

Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with finite expectation, and for each $k$, $X_k$ is observed at the $k$-th arrival time $S_k$ of a Poisson process with unit rate which is independent of the sequence $\{X_k\}$. For $t>0$, comparisons are made between the expected maximum $M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$ and the optimal stopping value $V(t):=\sup_{\tau\in\TT}\sE[X_\tau \sI(S_\tau\leq t)]$, where $\TT$ is the set of all $\NN$-valued ra...

This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let $M_n:=\max\{X_1,\dots,X_n\}$, and the objective is to select the maximum of the sequence. What is the maximum probability of "stopping at the maximum"? That is, what is the stopping time $\tau$ adapted to $X_1,...,X_n$ that maximizes $P(X_...

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length $n$. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations o...

The present paper studies the limiting behavior of the average score of a sequentially selected group of items or individuals, the underlying distribution of which, $F$, belongs to the Gumbel domain of attraction of extreme value distributions. This class contains the Normal, Lognormal, Gamma, Weibull and many other distributions. The selection rules are the "better than average" ($\beta=1$) and the "$\beta$-better than average" rule, defined as follows. After the first item ...

Let $\boldsymbol W=\{\boldsymbol W_n:n\in\mathbb N\}$ be a sequence of random vectors in $\mathbb R^d$, $d\ge 1$. This paper considers the logarithmic asymptotics of the extremes of $\boldsymbol W$, that is, for any vector $\boldsymbol q>\boldsymbol 0$ in $\mathbb R^d$, we find $$\log\mathbb P\left(\exists{n\in\mathbb N}:\boldsymbol W_n> u \boldsymbol q\right), \quad\text{as} u\to\infty.$$ We follow the approach of the restricted large deviation principle introduced in Duffy ...

Let $X$ be a real-valued random variable with distribution function $F$. Set $X_1,\dots, X_m$ to be independent copies of $X$ and let $F_m$ be the corresponding empirical distribution function. We show that there are absolute constants $c_0$ and $c_1$ such that if $\Delta \geq c_0\frac{\log\log m}{m}$, then with probability at least $1-2\exp(-c_1\Delta m)$, for every $t\in\mathbb{R}$ that satisfies $F(t)\in[\Delta,1-\Delta]$, \[ |F_m(t) - F(t) | \leq \sqrt{\Delta \min\{F(t),1...