Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of X_n.Comment: Published at http://dx.doi.org/10.1214/105051604000000918 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large sample theory of intrinsic and extrinsic sample means on manifolds--II

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere S^d (directional spaces), real projective space RP^{N-1} (axial spaces), complex projective space CP^{k-2} (planar shape spaces) w.r.t. Veronese-Whitney embeddings and a three-dimensional shape space \Sigma_3^4.Comment: Published at http://dx.doi.org/10.1214/009053605000000093 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Iterates of Monotone Maps

In this paper we prove the existence, uniqueness and stability of the invariant distribution of a random dynamical system in which the admissible family of laws of motion consists of monotone maps from a closed subset of a finite dimensional Euclidean space into itself.

Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadtratic maps defines a Markov process: X[subscript n + 1] = epsilon[subscript n + 1] X[subscript n](1 - X[subscript n]), where epsilon[subscript n] are i.i.d. with values in the parameter space [0, 4] of quadratic maps F[subscript theta](x) = theta*x(1 - x). Its study is of significance not only as an important Markov model, but also for dynamical systems defined by the individual quadratic maps themselves. In this article a broad criterion is established for positive Harris recurrence of X[subscript n], whose invariant probability may be viewed as an approximation to the so-called Kolmogorov measure of a dynamical system.

Nonparametric statistics on manifolds with applications to shape spaces

This article presents certain recent methodologies and some new results for the statistical analysis of probability distributions on manifolds. An important example considered in some detail here is the 2-D shape space of k-ads, comprising all configurations of $k$ planar landmarks ($k>2$)-modulo translation, scaling and rotation.Comment: Published in at http://dx.doi.org/10.1214/074921708000000200 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Superiority of Bayes estimators over the MLE in high dimensional multinomial models and its implication for nonparametric Bayes theory

This article focuses on the performance of Bayes estimators, in comparison with the MLE, in multinomial models with a relatively large number of cells. The prior for the Bayes estimator is taken to be the conjugate Dirichlet, i.e., the multivariate Beta, with exchangeable distributions over the coordinates, including the non-informative uniform distribution. The choice of the multinomial is motivated by its many applications in business and industry, but also by its use in providing a simple nonparametric estimator of an unknown distribution. It is striking that the Bayes procedure outperforms the asymptotically efficient MLE over most of the parameter spaces for even moderately large dimensional parameter space and rather large sample sizes.Comment: 49 pages, 11 figures, 12 table