Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators

We compare estimators of the (essential) supremum and the integral of a function $f$ defined on a measurable space when $f$ may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of $f$ by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ295 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Evolutionarily stable strategies of random games, and the vertices of random polygons

An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative (``mutant'') strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for ``almost every large'' game? Letting the entries in the $n\times n$ game matrix be independently randomly chosen according to a distribution $F$, we study the number of ESS with support of size $2.$ In particular, we show that, as $n\to \infty$, the probability of having such an ESS: (i) converges to 1 for distributions $F$ with ``exponential and faster decreasing tails'' (e.g., uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for distributions $F$ with ``slower than exponential decreasing tails'' (e.g., lognormal, Pareto, Cauchy). Our results also imply that the expected number of vertices of the convex hull of $n$ random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii).Comment: Published in at http://dx.doi.org/10.1214/07-AAP455 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

U-statistics and random subgraph counts: Multivariate normal approximation via exchangeable pairs and embedding

In a recent paper by the authors, a new approach--called the "embedding method"--was introduced, which allows to make use of exchangeable pairs for normal and multivariate normal approximation with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs

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

Functional BRK Inequalities, and their Duals, with Applications

Refereed Working Papers / of international relevanc

Monotone Regrouping, Regression, and Simpson’s Paradox

We show in a general setup that if data Y are grouped by a covariate X in a certain way, then under a condition of monotone regression of Y on X, a Simpson’s type paradox is natural rather than surprising. This model was motivated by an observation on recent SAT data which are presented.We show in a general setup that if data Y are grouped by a covariate X in a certain way, then under a condition of monotone regression of Y on X, a Simpson’s type paradox is natural rather than surprising. This model was motivated by an observation on recent SAT data which are presented.Non-Refereed Working Papers / of national relevance onl

On Statistical Inference Under Selection Bias

This note revisits the problem of selection bias, using a simple binomial example. It focuses on selection that is introduced by observing the data and making decisions prior to formal statistical analysis. Decision rules and interpretation of confidence measure and results must then be taken relative to the point of view of the decision maker, i.e., before selection or after it. Such a distinction is important since inference can be considerably altered when the decision maker's point of view changes. This note demonstrates the issue, using both the frequentist and the Bayesian paradigms.This note revisits the problem of selection bias, using a simple binomial example. It focuses on selection that is introduced by observing the data and making decisions prior to formal statistical analysis. Decision rules and interpretation of confidence measure and results must then be taken relative to the point of view of the decision maker, i.e., before selection or after it. Such a distinction is important since inference can be considerably altered when the decision maker's point of view changes. This note demonstrates the issue, using both the frequentist and the Bayesian paradigms.Non-Refereed Working Papers / of national relevance onl