Point and confidence interval estimates for a global maximum via extreme value theory

The aim of this paper is to provide some practical aspects of point and interval estimates of the global maximum of a function using extreme value theory. Consider a real-valued function f:D→? defined on a bounded interval D such that f is either not known analytically or is known analytically but has rather a complicated analytic form. We assume that f possesses a global maximum attained, say, at u*∈D with maximal value x*=max _u  f(u)?f(u*). The problem of seeking the optimum of a function which is more or less unknown to the observer has resulted in the development of a large variety of search techniques. In this paper we use the extreme-value approach as appears in Dekkers et al. [A moment estimator for the index of an extreme-value distribution, Ann. Statist. 17 (1989), pp. 1833–1855] and de Haan [Estimation of the minimum of a function using order statistics, J. Amer. Statist. Assoc. 76 (1981), pp. 467–469]. We impose some Lipschitz conditions on the functions being investigated and through repeated simulation-based samplings, we provide various practical interpretations of the parameters involved as well as point and interval estimates for x*.     

Dissemination to research subjects: operationalizing investigator accountability

Recent articles have argued from principles of bioethics for the right of research subjects to receive the results of the studies in which they have participated. We argue that accountability is a powerful tool of meso-level analysis appropriate to reasoning about answerability in research ethics, and that it captures the responsibility of researchers to disseminate study results to research subjects. We offer the following features of the research situation as relevant to the manner of dissemination to study subject, in addition to factors already proposed in the literature (risk and impact on health outcome): (a) features of the research subject in relation to identity, personal investment, disease, and community; (b) characteristics of the research study and field of inquiry in relation to certainty and significance; and (c) relationships among the research subjects and the healthcare workers involved in their care and in the research.     

DYNAMIC OPERATING RULES FOR MOTEL RESERVATIONS

When customers call a motel or a hotel to reserve a particular room (large, small, suites, single-bed, double-bed, etc.) decisions have to be made concerning the allocation of the available rooms. This paper presents a model for determining the maximum number of specific bookings that should be made for each type of room. The objective is to maximize the expected contribution to profit per rental day. The model considers the distribution of requests for reservations over time as well as cancellations. Also included are the effects of late (unbooked) arrivals and overbookings.     

Regular mles for nonregular distributions

Distributions whose extremity values of the support depend on unknown pa¬rameters are usually known as nonregular distributions. In most cases, the MLEs for these parameters cannot be obtained by differentiation. Familiar examples are the uniform distribution on the interval (0,0) and the truncated exponential distribution with truncation parameter 0. However, there exist distributions whose extremity points of the support depend on unknown pa¬rameters, which nevertheless are regular in the sense that the MLEs can be obtained by differentiation. This note provides a method of constructing such nonregular distributions with regular MLEs.     

Stochastic Learning Dynamics and Speed of Convergence in Population Games

We study how long it takes for large populations of interacting agents to come close to Nash equilibrium when they adapt their behavior using a stochastic better reply dynamic. Prior work considers this question mainly for 2 × 2 games and potential games; here we characterize convergence times for general weakly acyclic games, including coordination games, dominance solvable games, games with strategic complementarities, potential games, and many others with applications in economics, biology, and distributed control. If players' better replies are governed by idiosyncratic shocks, the convergence time can grow exponentially in the population size; moreover, this is true even in games with very simple payoff structures. However, if their responses are sufficiently correlated due to aggregate shocks, the convergence time is greatly accelerated; in fact, it is bounded for all sufficiently large populations. We provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies, the length of the better reply paths, the extent to which players can influence the payoffs of others, and the desired degree of approximation to Nash equilibrium.