Abstract:

Iteration of randomly chosen quadratic maps defines a Markov process: X n+1 = ε n+1 X n(1 - X n), where ε n are i.i.d. with values in the parameter space [0,4] of quadratic maps F θ(x) = θx(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. © Institute of Mathematical Statistics, 2004.

Abstract:

Suppose Y has a linear regression on X1, X2, but observations are only available on (Y, X1). If large scale data on (X1, X2) are available, which do not include Y, and if the regression of X2, given X1, is nonlinear, then one may estimate the regression coefficients of Y by using the proxy g(X1) {colon equals} E(X2|X1) for X2, or an instrument φ(X1) which is uncorrelated with X2. Both methods provide estimators which are asymptotically normal around the true parameter values under appropriate assumptions. A computation of the optimal instrument is provided, and the asymptotic relative efficienties of the two types of estimators compared. © 1993 Academic Press. All rights reserved.

Abstract:

This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Fréchet 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 ∑ 3⁴. © Institute of Mathematical Statistics, 2005.

Abstract:

This article considers the convergence to steady states of Markov processes generated by the action of successive i.i.d. monotone maps on a subset S of an Eucledian space. Without requiring irreducibility or Harris recurrence, a "splitting" condition guarantees the existence of a unique invariant probability as well as an exponential rate of convergence to it in an appropriate metric. For a special class of Harris recurrent processes on [0,∞) of interest in economics, environmental studies and queuing theory, criteria are derived for polynomial and exponential rates of convergence to equilibrium in total variation distance. Central limit theorems follow as consequences. © 2010, Indian Statistical Institute.