Rabindra N Bhattacharya
Professor, BIO5 Institute
Professor, Mathematics
Professor, Statistics-GIDP
Primary Department
(520) 626-5647
Research Interest
Rabi N. Bhattacharya is a Professor of Mathematics at the University of Arizona and a Member of BIO5. He is internationally known for his work on asymptotic theories of probability and statistics with diverse applications in many areas of science and engineering. His recent work on statistics on shape spaces has important applications in morphometrics, medical imaging and diagnostics and machine vision, in particular. This original work has inspired a large US and international following. During the 2011-12 academic year he was an invited speaker at international conferences and workshops at the University of Goettingen (Germany), University of Bielefeld (Germany), IMS/APRM 2012 in Tsukuba (Japan), and the Mathematical Biosciences Institute (MBI), Ohio State University. He has published nearly 100 research articles and co-authored four research monographs and two graduate texts. Two of these books have been recently reprinted by SIAM (Society for Industrial and Applied Mathematics) as Classics in Applied Mathematics. He is a Fellow of the IMS and a Member of the AMS, and a recipient of a Humboldt Prize and a Guggenheim Fellowship.Professor Bhattacharya has been intimately associated with the establishment of the Statistics GIDP at the U of A since 2002, and was a member of the Executive Committee of the GIDP for the past six years.

Publications

Bhattacharya, R. N. (2010). Comment. Statistica Sinica, 20(1), 58-63.
Bhattacharya, R., & Majumdar, M. (2004). Stability in distribution of randomly perturbed quadratic maps as Markov processes. Annals of Applied Probability, 14(4), 1802-1809.

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.

Bhattacharya, R. N., & Bhattacharyya, D. K. (1993). Proxy and Instrumental Variable Methods in a Regression Model with One of the Regressors Missing. Journal of Multivariate Analysis, 47(1), 123-138.

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.

Bhattacharya, R., & Patrangenaru, V. (2005). Large sample theory of intrinsic and extrinsic sample means on manifolds-ii. Annals of Statistics, 33(3), 1225-1259.

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.

Bhattacharya, R., Majumdar, M., & Hashimzade, N. (2010). Limit theorems for monotone Markov processes. Sankhya: The Indian Journal of Statistics, 72(1), 170-190.

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.