Yosef Rinott

Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions

A function f(x) defined on = 1 - 2 - ... - n where each i is totally ordered satisfying f(x [logical or] y) f(x [logical and] y) >= f(x) f(y), where the lattice operations [logical or] and [logical and] refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP2). A random vector Z = (Z1, Z2,..., Zn) of n-real components is MTP2 if its density is MTP2. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix [Sigma] satisfies -D[Sigma]-1D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP2 properties. The MTP2 property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP2 kernels are presented and amplified by a wide variety of applications.

On two-stage selection procedures and related probability-inequalities

In this paper we discuss a modification of the Dudewicz-Dalal procedure for the problem of selecting the population with the largest mean from k normal populations with unknown variances. We derive some inequalities and use them to lower-bound the probability of correct selection. These bounds are applied to the determination of the second-stage sample size which is required in order to achieve a prescribed probability of correct selection. We discuss the resulting procedure and compare it to that of Dudewicz and Dalai (1975).

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

Steins 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 non-negative 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 for example, martingale, Markov chain or various mixing assumptions.

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

Let X = (X1, X2,..., Xn) be a random vector in Rn (Euclidean n-space) with density f(x). X or f(x) is said to be multivariate reverse rule of order 2 (MRR2) if f(x [curly logical or] y) f(x [curly logical and] y)

Pervasive adaptive protein evolution apparent in diversity patterns around amino acid substitutions in Drosophila simulans.

In Drosophila, multiple lines of evidence converge in suggesting that beneficial substitutions to the genome may be common. All suffer from confounding factors, however, such that the interpretation of the evidence—in particular, conclusions about the rate and strength of beneficial substitutions—remains tentative. Here, we use genome-wide polymorphism data in D. simulans and sequenced genomes of its close relatives to construct a readily interpretable characterization of the effects of positive selection: the shape of average neutral diversity around amino acid substitutions. As expected under recurrent selective sweeps, we find a trough in diversity levels around amino acid but not around synonymous substitutions, a distinctive pattern that is not expected under alternative models. This characterization is richer than previous approaches, which relied on limited summaries of the data (e.g., the slope of a scatter plot), and relates to underlying selection parameters in a straightforward way, allowing us to make more reliable inferences about the prevalence and strength of adaptation. Specifically, we develop a coalescent-based model for the shape of the entire curve and use it to infer adaptive parameters by maximum likelihood. Our inference suggests that ∼13% of amino acid substitutions cause selective sweeps. Interestingly, it reveals two classes of beneficial fixations: a minority (approximately 3%) that appears to have had large selective effects and accounts for most of the reduction in diversity, and the remaining 10%, which seem to have had very weak selective effects. These estimates therefore help to reconcile the apparent conflict among previously published estimates of the strength of selection. More generally, our findings provide unequivocal evidence for strongly beneficial substitutions in Drosophila and illustrate how the rapidly accumulating genome-wide data can be leveraged to address enduring questions about the genetic basis of adaptation.

Shifts in the intensity of purifying selection: An analysis of genome-wide polymorphism data from two closely related yeast species

How much does the intensity of purifying selection vary among populations and species? How uniform are the shifts in selective pressures across the genome? To address these questions, we took advantage of a recent, whole-genome polymorphism data set from two closely related species of yeast, Saccharomyces cerevisiae and S. paradoxus, paying close attention to the population structure within these species. We found that the average intensity of purifying selection on amino acid sites varies markedly among populations and between species. As expected in the presence of extensive weakly deleterious mutations, the effect of purifying selection is substantially weaker on single nucleotide polymorphisms (SNPs) segregating within populations than on SNPs fixed between population samples. Also in accordance with a Nearly Neutral model, the variation in the intensity of purifying selection across populations corresponds almost perfectly to simple measures of their effective size. As a first step toward understanding the processes generating these patterns, we sought to tease apart the relative importance of systematic, genome-wide changes in the efficacy of selection, such as those expected from demographic processes and of gene-specific changes, which may be expected after a shift in selective pressures. For that purpose, we developed a new model for the evolution of purifying selection between populations and inferred its parameters from the genome-wide data using a likelihood approach. We found that most, but not all changes seem to be explained by systematic shifts in the efficacy of selection. One population, the sake-derived strains of S. cerevisiae, however, also shows extensive gene-specific changes, plausibly associated with domestication. These findings have important implications for our understanding of purifying selection as well as for estimates of the rate of molecular adaptation in yeast and in other species.

A Normal Approximation for the Number of Local Maxima of a Random Function on a Graph

Publisher Summary This chapter discusses the normal approximation for the number of local maxima of a random function on a graph. It discusses the conditions for the approximate normality of the distribution of the number of local maxima of a random function on the set of vertices of a graph when the values of the random function are independently identically distributed with a continuous distribution function. For a regular graph, the distribution of the number of local maxima is approximately normal if its variance is large. The basic idea of a normal approximation theorem is to exploit a sum of indicator random variables. The chapter discusses a basic lemma on normal approximation for sums of indicator random variables.

On normal approximation rates for certain sums of dependent random variables

Abstract Let X1, …, Xn be dependent random variables, and set λ = E∑ni=1Xi, and σ2 = Var∑ni=1Xi. In most of the applications of Steins method for normal approximations, the error rate |P((∑ni=1Xi − λ)/σ ⩽ w) − Φ(w)| is of the order of σ− 1 2 . This rate was improved by Stein (1986) and others in some special cases. In this paper it is shown that for certain bounded random variables, a simple refinement of error-term calculations in Steins method leads to improved rates.

Nearest neighbors and Voronoi regions in certain point processes

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering N = Nd for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that lim,_, lim ,Na d Poisson (A = 1). An asymptotic Poisson (A = 1) distribution also holds for many of the other models. On the other hand, we find that limn--lim ,,, N D 0. Related results concern the (random) volume, Vol%, of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that limd.oolimn-.Vol D 1 while lim_,, lim__, Vold 9 0. POISSON PROCESSES; DIRICHLET CELLS

Entropy inequalities for classes of probability distributions I. The univariate case

Entropy functionals of probability densities feature importantly in classifying certain finite-state stationary stochastic processes, in discriminating among competing hypotheses, in characterizing Gaussian, Poisson, and other densities, in describing information processes, and in other contexts. Two general types of problems are considered. For a given parametric family of densities the member of maximal (or sometimes minimal) entropy is ascertained. Secondly, we determine a natural (partial) ordering over for which the entropy functional is monotone. The examples include the multiparameter binomial, multiparameter negative binomial, some classes of log concave densities, and others.