Estimation of bivariate excess probabilities for elliptical models

Let $(X,Y)$ be a random vector whose conditional excess probability $\theta(x,y):=P(Y\leq y | X>x)$ is of interest. Estimating this kind of probability is a delicate problem as soon as $x$ tends to be large, since the conditioning event becomes an extreme set. Assume that $(X,Y)$ is elliptically distributed, with a rapidly varying radial component. In this paper, three statistical procedures are proposed to estimate $\theta(x,y)$ for fixed $x,y$, with $x$ large. They respectively make use of an approximation result of Abdous et al. (cf. Canad. J. Statist. 33 (2005) 317--334, Theorem 1), a new second order refinement of Abdous et al.'s Theorem 1, and a non-approximating method. The estimation of the conditional quantile function $\theta(x,\cdot)^{\leftarrow}$ for large fixed $x$ is also addressed and these methods are compared via simulations. An illustration in the financial context is also given.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ140 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Pairwise alignment incorporating dipeptide covariation

Motivation: Standard algorithms for pairwise protein sequence alignment make the simplifying assumption that amino acid substitutions at neighboring sites are uncorrelated. This assumption allows implementation of fast algorithms for pairwise sequence alignment, but it ignores information that could conceivably increase the power of remote homolog detection. We examine the validity of this assumption by constructing extended substitution matrixes that encapsulate the observed correlations between neighboring sites, by developing an efficient and rigorous algorithm for pairwise protein sequence alignment that incorporates these local substitution correlations, and by assessing the ability of this algorithm to detect remote homologies. Results: Our analysis indicates that local correlations between substitutions are not strong on the average. Furthermore, incorporating local substitution correlations into pairwise alignment did not lead to a statistically significant improvement in remote homology detection. Therefore, the standard assumption that individual residues within protein sequences evolve independently of neighboring positions appears to be an efficient and appropriate approximation

A Pearson-Dirichlet random walk

A constrained diffusive random walk of n steps and a random flight in Rd, which can be expressed in the same terms, were investigated independently in recent papers. The n steps of the walk are identically and independently distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1, 2, 4 . Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D. The previous walk is generalized by considering step lengths which are distributed over the unit (n-1) simplex according to a Dirichlet distribution whose parameters are all equal to q, a given positive value. The walk and the flight above correspond to q=1. For any d >= 3, there exist, for integer and half-integer values of q, two families of Pearson-Dirichlet walks which share a common property. For any n, the d components of the endpoint are jointly distributed as are the d components of a vector uniformly distributed over the surface of a hypersphere of radius l in a space Rk whose dimension k is an affine function of n for a given d. Five additional walks, with a uniform distribution of the endpoint in the inside of a ball, are found from known finite integrals of products of powers and Bessel functions of the first kind. They include four different walks in R3 and two walks in R4. Pearson-Liouville random walks, obtained by distributing the total lengths of the previous Pearson-Dirichlet walks, are finally discussed.Comment: 33 pages 1 figure, the paper includes the content of a recently submitted work together with additional results and an extended section on Pearson-Liouville random walk

Elections as Targeting Contests

This paper develops a model of electoral turnout where parties compensate voters for showing up to the polls. Existence and uniqueness conditions are shown to impose substantial restrictions on the uncertainty about partisan support faced by the parties, and on the distribution of voting costs among citizens. The model predicts that voters in the minority will be more likely to vote, and that turnout increases with the importance of the election. The model can generate the observed correlation between election closeness and electoral turnout, lthough the cause of this correlation may depend on the distribution of voting costs.Elections

Variational Inference in Nonconjugate Models

Mean-field variational methods are widely used for approximate posterior inference in many probabilistic models. In a typical application, mean-field methods approximately compute the posterior with a coordinate-ascent optimization algorithm. When the model is conditionally conjugate, the coordinate updates are easily derived and in closed form. However, many models of interest---like the correlated topic model and Bayesian logistic regression---are nonconjuate. In these models, mean-field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case-by-case basis. In this paper, we develop two generic methods for nonconjugate models, Laplace variational inference and delta method variational inference. Our methods have several advantages: they allow for easily derived variational algorithms with a wide class of nonconjugate models; they extend and unify some of the existing algorithms that have been derived for specific models; and they work well on real-world datasets. We studied our methods on the correlated topic model, Bayesian logistic regression, and hierarchical Bayesian logistic regression

Four moments theorems on Markov chains

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwemake on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.First author draf