The Spend-It-All Region and Small Time Results for the Continuous Bomber Problem

A problem of optimally allocating partially effective ammunition $x$ to be used on randomly arriving enemies in order to maximize an aircraft's probability of surviving for time~$t$, known as the Bomber Problem, was first posed by \citet{Klinger68}. They conjectured a set of apparently obvious monotonicity properties of the optimal allocation function $K(x,t)$. Although some of these conjectures, and versions thereof, have been proved or disproved by other authors since then, the remaining central question, that $K(x,t)$ is nondecreasing in~$x$, remains unsettled. After reviewing the problem and summarizing the state of these conjectures, in the setting where $x$ is continuous we prove the existence of a ``spend-it-all'' region in which $K(x,t)=x$ and find its boundary, inside of which the long-standing, unproven conjecture of monotonicity of~$K(\cdot,t)$ holds. A new approach is then taken of directly estimating~$K(x,t)$ for small~$t$, providing a complete small-$t$ asymptotic description of~$K(x,t)$ and the optimal probability of survival

Extreme(ly) mean(ingful): Sequential formation of a quality group

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 is selected, another item is admitted into the group if and only if its score is greater than $\beta$ times the average score of those already selected. Denote by $\bar{Y}_k$ the average of the $k$ first selected items, and by $T_k$ the time it takes to amass them. Some of the key results obtained are: under mild conditions, for the better than average rule, $\bar{Y}_k$ less a suitable chosen function of $\log k$ converges almost surely to a finite random variable. When $1-F(x)=e^{-[x^{\alpha}+h(x)]}$, $\alpha>0$ and $h(x)/x^{\alpha}\stackrel{x\rightarrow \infty}{\longrightarrow}0$, then $T_k$ is of approximate order $k^2$. When $\beta>1$, the asymptotic results for $\bar{Y}_k$ are of a completely different order of magnitude. Interestingly, for a class of distributions, $T_k$, suitably normalized, asymptotically approaches 1, almost surely for relatively small $\beta\ge1$, in probability for moderate sized $\beta$ and in distribution when $\beta$ is large.Comment: Published in at http://dx.doi.org/10.1214/10-AAP684 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Optimal Allocation of a Continuous Resource Using an Iterative Approach and Total Positivity

We study a class of optimal allocation problems, including the well-known Bomber Problem, with the following common probabilistic structure. An aircraft equipped with an amount~$x$ of ammunition is intercepted by enemy airplanes arriving according to a homogenous Poisson process over a fixed time duration~$t$. Upon encountering an enemy, the aircraft has the choice of spending any amount~$0\le y\le x$ of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of $y$. Two different goals have been considered in the literature concerning the optimal amount~$K(x,t)$ of ammunition spent: (i)~Maximizing the probability of surviving for time~$t$, which is the so-called Bomber Problem, and (ii) maximizing the number of enemy airplanes shot down during time~$t$, which we call the Fighter Problem. Several authors have attempted to settle the following conjectures about the monotonicity of $K(x,t)$: [A] $K(x,t)$ is decreasing in $t$, [B] $K(x,t)$ is increasing in $x$, and [C] the amount~$x-K(x,t)$ held back is increasing in $x$. [A] and [C] have been shown for the Bomber Problem with discrete ammunition, while [B] is still an open question. In this paper we consider both time and ammunition continuous, and for the Bomber Problem prove [A] and [C], while for the Fighter we prove [A] and [C] for one special case and [B] and [C] for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (\mbox{TP}_2) in the Bomber and Fighter Problems, respectively. The \mbox{TP}_2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves \mbox{TP}_2 and other properties.Comment: 2 figure

The Noisy Secretary Problem and Some Results on Extreme Concomitant Variables

The classical secretary problem for selecting the best item is studied when the actual values of the items are observed with noise. One of the main appeals of the secretary problem is that the optimal strategy is able to find the best observation with the nontrivial probability of about 0.37, even when the number of observations is arbitrarily large. The results are strikingly di↵erent when the quality of the secretaries are observed with noise. If there is no noise, then the only information that is needed is whether an observation is the best among those already observed. Since observations are assumed to be i.i.d. this is distribution free. In the case of noisy data, the results are no longer distrubtion free. Furthermore, one needs to know the rank of the noisy observation among those already seen. Finally, the probability of finding the best secretary often goes to 0 as the number of obsevations, n, goes to infinity. The results depend heavily on the behavior of pn, the probability that the observation that is best among the noisy observations is also best among the noiseless observations. Results involving optimal strategies if all that is available is noisy data are described and examples are given to elucidate the results

A Sharp Necessary Condition for Admissibility of Sequential Tests-- Necessary and Sufficient Conditions for Admissibility of SPRT\u27S

Consider the problem of sequentially testing the hypothesis that the mean of a normal distribution of known variance is less than or equal to a given value versus the alternative that it is greater than the given value. Impose the linear combination loss function under which the risk becomes a constant c, times the expected sample size, plus the probability of error. It is known that all admissible tests must be monotone--that is, they stop and accept if Sn, the sample sum at stage n, satisfies Sn≤an; stop and reject if Sn≥bn. In this paper we show that any admissible test must in addition satisfy bn−an≤2b¯(c). The bound 2b¯(c) is sharp in the sense that the test with stopping bounds an≡−b¯(c),bn≡b¯(c) is admissible. As a consequence of the above necessary condition for admissibility of a sequential test, it is possible to characterize all sequential probability ratio tests (SPRT\u27s) regarding admissibility. In other words necessary and sufficient conditions for the admissibility of an SPRT are given. Furthermore, an explicit numerical upper bound for b¯(c) is provided