The Finite-time Ruin Probabilities of a Bidimensional risk model with Constant Interest Force and correlated Brownian Motions

We follow some recent works to study bidimensional perturbed compound Poisson risk models with constant interest force and correlated Brownian Motions. Several asymptotic formulae for three different type of ruin probabilities over a finite-time horizon are established. Our approach appeals directly to very recent developments in the ruin theory in the presence of heavy tails of unidimensional risk models and the dependence theory of stochastic processes and random vectors.Comment: 25page

Imputation Estimators Partially Correct for Model Misspecification

Inference problems with incomplete observations often aim at estimating population properties of unobserved quantities. One simple way to accomplish this estimation is to impute the unobserved quantities of interest at the individual level and then take an empirical average of the imputed values. We show that this simple imputation estimator can provide partial protection against model misspecification. We illustrate imputation estimators' robustness to model specification on three examples: mixture model-based clustering, estimation of genotype frequencies in population genetics, and estimation of Markovian evolutionary distances. In the final example, using a representative model misspecification, we demonstrate that in non-degenerate cases, the imputation estimator dominates the plug-in estimate asymptotically. We conclude by outlining a Bayesian implementation of the imputation-based estimation.Comment: major rewrite, beta-binomial example removed, model based clustering is added to the mixture model example, Bayesian approach is now illustrated with the genetics exampl

Nonparametric estimation of a renewal reward process from discrete data

We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Gr\"ubel (2003) and Duval (2012), the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.Comment: arXiv admin note: substantial text overlap with arXiv:1203.313

Stochastic Approximation and Newton's Estimate of a Mixing Distribution

Many statistical problems involve mixture models and the need for computationally efficient methods to estimate the mixing distribution has increased dramatically in recent years. Newton [Sankhya Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating the mixing distribution, which we study as a special case of stochastic approximation (SA). We begin with a review of SA, some recent statistical applications, and the theory necessary for analysis of a SA algorithm, which includes Lyapunov functions and ODE stability theory. Then standard SA results are used to prove consistency of Newton's estimate in the case of a finite mixture. We also propose a modification of Newton's algorithm that allows for estimation of an additional unknown parameter in the model, and prove its consistency.Comment: Published in at http://dx.doi.org/10.1214/08-STS265 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org