arXiv subjects

Yosef Rinott

Publications and source records attributed to Yosef Rinott.

24 records · Page 2Linked to original sources

arXiv2011

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.

math.PR↗

arXiv2009

Prediction of Ordered Random Effects in a Simple Small Area Model

Prediction of a vector of ordered parameters or part of it arises naturally in the context of Small Area Estimation (SAE). For example, one may want to estimate the parameters associated with the top ten areas, the best or worst area, or a certain percentile. We use a simple SAE model to show that estimation of ordered parameters by the corresponding ordered estimates of each area separately does not yield good results with respect to MSE. Shrinkage-type predictors, with an appropriate amount of shrinkage for the particular problem of ordered parameters, are considerably better, and their performance is close to that of the optimal predictors, which cannot in general be computed explicitly.

stat.ME↗

arXiv2008

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).

math.PR↗

arXiv2007

A smoothing model for sample disclosure risk estimation

When a sample frequency table is published, disclosure risk arises when some individuals can be identified on the basis of their values in certain attributes in the table called key variables, and then their values in other attributes may be inferred, and their privacy is violated. On the basis of the sample to be released, and possibly some partial knowledge of the whole population, an agency which considers releasing the sample, has to estimate the disclosure risk. Risk arises from non-empty sample cells which represent small population cells and from population uniques in particular. Therefore risk estimation requires assessing how many of the relevant population cells are likely to be small. Various methods have been proposed for this task, and we present a method in which estimation of a population cell frequency is based on smoothing using a local neighborhood of this cell, that is, cells having similar or close values in all attributes. We provide some preliminary results and experiments with this method. Comparisons are made to two other methods: 1. a log-linear models approach in which inference on a given cell is based on a ``neighborhood'' of cells determined by the log-linear model. Such neighborhoods have one or some common attributes with the cell in question, but some other attributes may differ significantly. 2 The Argus method in which inference on a given cell is based only on the sample frequency in the specific cell, on the sample design and on some known marginal distributions of the population, without learning from any type of ``neighborhood'' of the given cell, nor from any model which uses the structure of the table.

stat.ME↗

arXiv2005

A permutation test for matching and its asymptotic distribution

We consider a permutation method for testing whether observations given in their natural pairing exhibit an unusual level of similarity in situations where any two observations may be similar at some unknown baseline level. Under a null hypotheses where there is no distinguished pairing of the observations, a normal approximation with explicit bounds and rates is presented for determining approximate critical test levels.

math.ST↗

arXiv2005

Multivariate normal approximations by Stein's method and size bias couplings

Stein's method is used to obtain two theorems on multivariate normal approximation. Our main theorem, Theorem 1.2, provides a bound on the distance to normality for any nonnegative random vector. Theorem 1.2 requires multivariate size bias coupling, which we discuss in studying the approximation of distributions of sums of dependent random vectors. In the univariate case, we briefly illustrate this approach for certain sums of nonlinear functions of multivariate normal variables. As a second illustration, we show that the multivariate distribution counting the number of vertices with given degrees in certain random graphs is asymptotically multivariate normal and obtain a bound on the rate of convergence. Both examples demonstrate that this approach may be suitable for situations involving non-local dependence. We also present Theorem 1.4 for sums of vectors having a local type of dependence. We apply this theorem to obtain a multivariate normal approximation for the distribution of the random $p$-vector which counts the number of edges in a fixed graph both of whose vertices have the same given color when each vertex is colored by one of $p$ colors independently. All normal approximation results presented here do not require an ordering of the summands related to the dependence structure. This is in contrast to hypotheses of classical central limit theorems and examples, which involve e.g., martingale, Markov chain, or various mixing assumptions.

math.PR↗