Rabindra N Bhattacharya

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• Bhattacharya, R. N., Kim, H., & Majumdar, M. (2015). Sustainability in the Stochastic Ramsey Model. Journal of Quantitative Economics, 13(2), 169-184.
• Lin, L., Piegorsch, W. W., Bhattacharya, R. N., Lin, L., Piegorsch, W. W., & Bhattacharya, R. N. (2015). Nonparametric benchmark dose estimation with continuous dose-response data. Scandinavian Journal of Statistics, 42(3), 713-731. doi:10.1111/sjos.12132
• Bhattacharya, R. N. (2014). Nonparametric benchmark dose estimation with continuous dose-response data. Scandinavian Journal of Statistics.
• Bhattacharya, R., & Patrangenaru, V. (2014). Rejoinder to the discussion. Journal of Statistical Planning and Inference, 145, 42-48.
• Bhattacharya, R., & Patrangenaru, V. (2014). Statistics on manifolds and landmarks based image analysis: A nonparametric theory with applications. Journal of Statistical Planning and Inference, 145, 1-22.
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  Abstract: This paper provides an exposition of some recent developments in nonparametric inference on manifolds, along with a brief account of an emerging theory on data analysis on stratified spaces. Much of the theory is developed around the notion of Fréceht means with applications, for the most part, to landmark based shape spaces. A number of applications are illustrated with real data in such areas as paleomagnetism, morphometrics and medical diagnostics. Connections to scene recognition and machine vision are also explored. © 2013 Elsevier B.V.
• Piegorsch, W. W., Xiong, H., Bhattacharya, R. N., & Lin, L. (2014). Benchmark dose analysis via nonparametric regression modeling. Risk Analysis, 34(1), 135-151.
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  Abstract: Estimation of benchmark doses (BMDs) in quantitative risk assessment traditionally is based upon parametric dose-response modeling. It is a well-known concern, however, that if the chosen parametric model is uncertain and/or misspecified, inaccurate and possibly unsafe low-dose inferences can result. We describe a nonparametric approach for estimating BMDs with quantal-response data based on an isotonic regression method, and also study use of corresponding, nonparametric, bootstrap-based confidence limits for the BMD. We explore the confidence limits' small-sample properties via a simulation study, and illustrate the calculations with an example from cancer risk assessment. It is seen that this nonparametric approach can provide a useful alternative for BMD estimation when faced with the problem of parametric model uncertainty. © 2013 Society for Risk Analysis.
• Bhattacharya, R. (2013). A nonparametric theory of statistics on manifolds. Springer Proceedings in Mathematics and Statistics, 42, 173-205.
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  Abstract: An expository account of the recent theory of nonparametric inference on manifolds is presented here, with outlines of proofs and examples. Much of the theory centers around Fréchet means; but functional estimation and classification methods using nonparametric Bayes theory are also indicated. Applications in paleomagnetism, morphometrics and medical diagnostics illustrate the theory.
• Bhattacharya, R., & Lin, L. (2013). Recent progress in the nonparametric estimation of monotone curves - With applications to bioassay and environmental risk assessment. Computational Statistics and Data Analysis, 63, 63-80.
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  Abstract: Three recent nonparametric methodologies for estimating a monotone regression function F and its inverse F-1 are (1) the inverse kernel method DNP (Dette et al., 2005; Dette and Scheder, 2010), (2) the monotone spline (Kong and Eubank (2006)) and (3) the data adaptive method NAM (Bhattacharya and Lin, 2010, 2011), with roots in isotonic regression (Ayer et al., 1955; Bhattacharya and Kong, 2007). All three have asymptotically optimal error rates. In this article their finite sample performances are compared using extensive simulation from diverse models of interest, and by analysis of real data. Let there be m distinct values of the independent variable x among N observations y. The results show that if m is relatively small compared to N then generally the NAM performs best, while the DNP outperforms the other methods when m is O(N) unless there is a substantial clustering of the values of the independent variable x. © 2013 Elsevier B.V. All rights reserved.
• Bhattacharya, R., Lin, L., & Majumdar, M. (2013). Rejoinder to: Problems of ruin and survival in economics: Applications of limit theorems in probability. Sankhya: The Indian Journal of Statistics, 75 B(PART 2), 190-194.
• Bhattacharya, R., Majumdar, M., & Lin, L. (2013). Problems of ruin and survival in economics: Applications of limit theorems in probability. Sankhya: The Indian Journal of Statistics, 75 B(PART 2), 145-180.
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  Abstract: In this paper the focus is on characterizing and computing the probabilities of ruin in three mathematical models arising in economics. First, we examine a credit system in which small loans without collaterals are extended to a large number of costumers, and study the probability of collapse due to defaults. Next, we consider a Walrasian model of an exchange economy in which the endowments are random, and analyze the probability that at equilibrium prices an agent does not have the minimum income needed for survival. Finally, the problem of sustaining a constant consumption of a resource the stock of which is augmented by a random input is considered. The steady state of the resulting Markov process, the speed at which it is approached, and the possibility of exhaustion of the stock are examined. © 2013, Indian Statistical Institute.
• Bhattacharya, R. N., Ellingson, L., Liu, X., Patrangenaru, V., & Crane, M. (2012). Extrinsic analysis on manifolds is computationally faster than intrinsic analysis with applications to quality control by machinea vision. Applied Stochastic Models in Business and Industry, 28(3), 222-235.
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  Abstract: In our technological era, non-Euclidean data abound, especially because of advances in digital imaging. Patrangenaru ('Asymptotic statistics on manifolds', PhD Dissertation, 1998) introduced extrinsic and intrinsic means on manifolds, as location parameters for non-Euclidean data. A large sample nonparametric theory of inference on manifolds was developed by Bhattacharya and Patrangenaru (J. Stat. Plann. Inferr., 108, 23-35, 2002; Ann. Statist., 31, 1-29, 2003; Ann. Statist., 33, 1211-1245, 2005). A flurry of papers in computer vision, statistical learning, pattern recognition, medical imaging, and other computational intensive applied areas using these concepts followed. While pursuing such location parameters in various instances of data analysis on manifolds, scientists are using intrinsic means, almost without exception. In this paper, we point out that there is no unique intrinsic analysis because the latter depends on the choice of the Riemannian metric on the manifold, and in dimension two or higher, there are infinitely such nonisometric choices. Also, using John Nash's celebrated isometric embedding theorem and an equivariant version, we show that for each intrinsic analysis there is an extrinsic counterpart that is computationally faster and give some concrete examples in shape and image analysis. The computational speed is important, especially in automated industrial processes. In this paper, we mention two potential applications in the industry and give a detailed presentation of one such application, for quality control in a manufacturing process via 3D projective shape analysis from multiple digital camera images. Copyright © 2011 John Wiley & Sons, Ltd. Copyright © 2011 John Wiley & Sons, Ltd.
• Bhattacharya, R., & Wasielak, A. (2012). On the speed of convergence of multidimensional diffusions to equilibrium. Stochastics and Dynamics, 12(1).
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  Abstract: We obtain new criteria for polynomial rates of convergence of ergodic multidimensional diffusions to equilibrium. For this, (1) a method is provided for adapting to continuous-time Markov processes coupling techniques for discrete parameter Harris processes, and (2) estimates of moments of return times to a ball are derived. © 2012 World Scientific Publishing Company.
• Piegorsch, W. W., Xiong, H., Bhattacharya, R. N., & Lin, L. (2012). Nonparametric estimation of benchmark doses in environmental risk assessment. Environmetrics, 23(8), 717-728.
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  Abstract: An important statistical objective in environmental risk analysis is estimation of minimum exposure levels, called benchmark doses (BMDs), which induce a pre-specified benchmark response in a dose-response experiment. In such settings, representations of the risk are traditionally based on a parametric dose-response model. It is a well-known concern, however, that if the chosen parametric form is misspecified, inaccurate and possibly unsafe low-dose inferences can result. We apply a nonparametric approach for calculating BMDs, based on an isotonic dose-response estimator for quantal-response data. We determine the large-sample properties of the estimator, develop bootstrap-based confidence limits on the BMDs, and explore the confidence limits' small-sample properties via a short simulation study. An example from cancer risk assessment illustrates the calculations. © 2012 John Wiley & Sons, Ltd.
• Bhattacharya, R., & Lin, L. (2011). Nonparametric benchmark analysis in risk assessment: A comparative study by simulation and data analysis. Sankhya: The Indian Journal of Statistics, 73(1 B), 144-163.
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  Abstract: We consider the finite sample performance of a new nonparametric method for bioassay and benchmark analysis in risk assessment, which averages isotonic MLEs based on disjoint subgroups of dosages, and whose asymptotic behavior is essentially optimal (Bhattacharya and Lin, Stat Probab Lett 80:1947-1953, 2010). It is compared with three other methods, including the leading kernel-based method, called DNP, due to Dette et al. (J Am Stat Assoc 100:503-510, 2005) and Dette and Scheder (J Stat Comput Simul 80(5):527-544, 2010). In simulation studies, the present method, termed NAM, outperforms the DNP in the majority of cases considered, although both methods generally do well. In small samples, NAM and DNP both outperform the MLE. © Indian Statistical Institute 2011.
• Bhattacharya, R. N. (2010). Comment. Statistica Sinica, 20(1), 58-63.
• Bhattacharya, R., & Lin, L. (2010). An adaptive nonparametric method in benchmark analysis for bioassay and environmental studies. Statistics and Probability Letters, 80(23-24), 1947-1953.
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  Abstract: We present a novel nonparametric method for bioassay and benchmark analysis in risk assessment, which averages isotonic MLEs based on disjoint subgroups of dosages. The asymptotic theory for the methodology is derived, showing that the MISEs (mean integrated squared error) of the estimates of both the dose-response curve F and its inverse F-1 achieve the optimal rate O(N-4/5). Also, we compute the asymptotic distribution of the estimate ζ̃p of the effective dosage ζp=F-1(p) which is shown to have an optimally small asymptotic variance. © 2010 Elsevier B.V.
• Bhattacharya, R., & Majumdar, M. (2010). Random iterates of monotone maps. Review of Economic Design, 14(1-2), 185-192.
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  Abstract: 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. © 2008 Springer-Verlag.
• Bhattacharya, R., Majumdar, M., & Hashimzade, N. (2010). Limit theorems for monotone Markov processes. Sankhya: The Indian Journal of Statistics, 72(1), 170-190.
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  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.
• Bandulasiri, A., Bhattacharya, R. N., & Patrangenaru, V. (2009). Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging. Journal of Multivariate Analysis, 100(9), 1867-1882.
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  Abstract: For all p > 2, k > p, a size-and-reflection-shape space S R Σp, 0k of k-ads in general position in Rp, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold R Σp, 0k, a space of orbits of scaled k-ads in general position under the group of isometries of Rp, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p > 2. The Veronese embedding of the planar Kendall shape manifold Σ2k is extended to an equivariant embedding of the size-and-shape manifold S Σ2k, which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory. © 2009 Elsevier Inc. All rights reserved.
• Bhattacharya, A., & Bhattacharya, R. (2008). Statistics on Riemannian manifolds: Asymptotic distribution and curvature. Proceedings of the American Mathematical Society, 136(8), 2959-2967.
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  Abstract: In this article a nonsingular asymptotic distribution is derived for a broad class of underlying distributions on a Riemannian manifold in relation to its curvature. Also, the asymptotic dispersion is explicitly related to curvature. These results are applied and further strengthened for the planar shape space of k-ads. © 2008 American Mathematical Society.
• Bhattacharya, R., & Kong, M. (2007). Consistency and asymptotic normality of the estimated effective doses in bioassay. Journal of Statistical Planning and Inference, 137(3), 643-658.
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  Abstract: In order to estimate the effective dose such as the 0.5 quantile ED50 in a bioassay problem various parametric and semiparametric models have been used in the literature. If the true dose-response curve deviates significantly from the model, the estimates will generally be inconsistent. One strategy is to analyze the data making only a minimal assumption on the model, namely, that the dose-response curve is non-decreasing. In the present paper we first define an empirical dose-response curve based on the estimated response probabilities by using the "pool-adjacent-violators" (PAV) algorithm, then estimate effective doses ED100 p for a large range of p by taking inverse of this empirical dose-response curve. The consistency and asymptotic distribution of these estimated effective doses are obtained. The asymptotic results can be extended to the estimated effective doses proposed by Glasbey [1987. Tolerance-distribution-free analyses of quantal dose-response data. Appl. Statist. 36 (3), 251-259] and Schmoyer [1984. Sigmoidally constrained maximum likelihood estimation in quantal bioassay. J. Amer. Statist. Assoc. 79, 448-453] under the additional assumption that the dose-response curve is symmetric or sigmoidal. We give some simulations on constructing confidence intervals using different methods. © 2006 Elsevier B.V. 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.
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  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.
• Kong, M., Bhattacharya, R. N., James, C., & Basu, A. (2005). A statistical approach to estimate the 3D size distribution of spheres from 2D size distributions. Bulletin of the Geological Society of America, 117(1-2), 244-249.
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  Abstract: Size distribution of rigidly embedded spheres in a groundmass is usually determined from measurements of the radii of the two-dimensional (2D) circular cross sections of the spheres in random flat planes of a sample, such as in thin sections or polished slabs. Several methods have been devised to find a simple factor to convert the mean of such 2D size distributions to the actual 3D mean size of the spheres without a consensus. We derive an entirely theoretical solution based on well-established probability laws and not constrained by limitations of absolute size, which indicates that the ratio of the means of measured 2D and estimated 3D grain size distribution should be r/4 (=.785). Actual 2D size distribution of the radii of submicron sized, pure Fe0 globules in lunar agglutinitic glass, determined from backscattered electron images, is tested to fit the gamma size distribution model better than the log-normal model. Numerical analysis of 2D size distributions of Fe0 globules in 9 lunar soils shows that the average mean of 2D/3D ratio is 0.84, which is very close to the theoretical value. These results converge with the ratio 0.8 that Hughes (1978) determined for millimeter-sized chondrules from empirical measurements. We recommend that a factor of 1.273 (reciprocal of 0.785) be used to convert the determined 2D mean size (radius or diameter) of a population of spheres to estimate their actual 3D size. © 2005 Geological Society of America.
• Bhattacharya, R., & Majumdar, M. (2004). Dynamical systems subject to random shocks: An introduction. Economic Theory, 23(1), 1-12.
• Bhattacharya, R., & Majumdar, M. (2004). Random dynamical systems: A review. Economic Theory, 23(1), 13-38.
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  Abstract: This paper provides a review of some results on the stability of random dynamical systems and indicates a number of applications to stochastic growth models, linear and non-linear time series models, statistical estimation of invariant distributions, and random iterations of quadratic maps.
• Bhattacharya, R., & Majumdar, M. (2004). Stability in distribution of randomly perturbed quadratic maps as Markov processes. Annals of Applied Probability, 14(4), 1802-1809.
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  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., Chen, L., Dobson, S., Guenther, R. B., Orum, C., Ossiander, M., Thomann, E., & Waymire, E. C. (2003). Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations. Transactions of the American Mathematical Society, 355(12), 5003-5040.
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  Abstract: A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.
• Bhattacharya, R., & Patrangenaru, V. (2003). Large sample theory of intrinsic and extrinsic sample means on manifolds. I. Annals of Statistics, 31(1), 1-29.
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  Abstract: Sufficient conditions are given for the uniqueness of intrinsic and extrinsic means as measures of location of probability measures Q on Riemannian manifolds. It is shown that, when uniquely defined, these are estimated consistently by the corresponding indices of the empirical Q̂ n. Asymptotic distributions of extrinsic sample means are derived. Explicit computations of these indices of Q̂ n and their asymptotic dispersions are carried out for distributions on the sphere S d (directional spaces), real projective space ℝP N-1 (axial spaces) and ℂP k-2 (planar shape spaces).
• Bhattacharya, R., & Patrangenaru, V. (2002). Nonparametic estimation of location and dispersion on Riemannian manifolds. Journal of Statistical Planning and Inference, 108(1-2), 23-35.
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  Abstract: A central limit theorem for intrinsic means on a complete flat manifold and some asymptotic properties of the intrinsic total sample variance on an arbitrary complete manifold are given. A studentized pivotal statistic and its bootstrap analogue which yield confidence regions for the intrinsic mean on a complete flat manifold are also derived. © 2002 Elsevier Science B.V. All rights reserved.
• Bhattacharya, R. N., & Majumdar, M. (2001). On characterizing the probability of survival in a large competitive economy. Review of Economic Design, 6(2), 133-153.
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  Abstract: We consider a Walrasian exchange economy in which an agent is characterized by a utility function, a random endowment vector, and a function that specifies the minimum expenditure necessary for survival at a given price system. If at any equilibrium price system, the income of the agent is no more than the minimum expenditure for survival, it is ruined. The main results characterize the probability of ruin when the number of agents is large. The implications of stochastic dependence among agents are explored. © Springer-Verlag 2001.
• Bhattacharya, R., & Majumdar, M. (2001). On a Class of Stable Random Dynamical Systems: Theory and Applications. Journal of Economic Theory, 96(1-2), 208-229.
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  Abstract: We consider a random dynamical system in which the state space is an interval, and possible laws of motion are monotone functions. It is shown that if the Markov process generated by this system satisfies a splitting condition, it converges to a unique invariant distribution exponentially fast in the Kolmogorov distance. A central limit theorem on the time-averages of observed values of the states is also proved. As an application we consider a system that captures an interaction of growth and cyclical forces: of two possible laws, one is monotone, but the other is unimodal with two periodic points. Journal of Economic Literature Classification Numbers: C6, D9. © 2001 Academic Press.
• Bhattacharya, R., & Waymire, E. C. (2001). Iterated random maps and some classes of Markov processes. Handbook of Statistics, 19, 145-170.
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  Abstract: After i.i.d. sequences of random variables, one is hard pressed to identify a structure with a more pervasive role in the theory and applications of probability than that of discrete parameter Markov processes. While the historical development is rich in concepts and techniques, efforts to complete the law of large numbers and central limit theory for Markov processes continue down fascinating pathways. In this article we present a contemporary survey of some of the main historical developments on these benchmark problems up through some state of the art theory and methods. © 2001 Elsevier Science B.V. All rights reserved.
• Bhattacharya, R., Thomann, E., & Waymire, E. (2001). A note on the distribution of integrals of geometric Brownian motion. Statistics and Probability Letters, 55(2), 187-192.
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  Abstract: The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At := ∫t0 exp{Zs} ds, t ≥ 0, where {Zs: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ2. In particular, both expected values of the form v(t, x) := Ef(x+At), f homogeneous, as well as the probability density a(t, y) dy := P(At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided. © 2001 Elsevier Science B.V. All rights reserved.
• Almudevar, A., Bhattacharya, R. N., & Sastri, C. C. (2000). Estimating the probability mass of unobserved support in random sampling. Journal of Statistical Planning and Inference, 91(1), 91-105.
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  Abstract: The problem of estimating the probability mass of the support of a distribution not observed in random sampling is considered in the case where the distribution is discrete. An example of a situation in which the problem arises is that of species sampling: suppose that one wishes to determine the species of fish native to a body of water and that, after repeated sampling, one identifies a certain number of species. The problem is to estimate the proportion of the fish population belonging to the unobserved species. Since it is a rare event, ideas from large deviation theory play a role in answering the question. The result depends on the underlying distribution, which is unknown in general. Methods similar to nonparametric bootstrapping are therefore used to prove a limit theorem and obtain a confidence interval for the rate function. © 2000 Elsevier Science B.V.
• Bhattacharjee, M. C., & Bhattacharya, R. N. (2000). Stochastic equivalence of convex ordered distributions and applications. Probability in the Engineering and Informational Sciences, 14(1), 33-48.
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  Abstract: We consider sufficient conditions for stochastic equivalence of convex ordered random variables. Our main results apply to all convex ordered distributions on the real line and improve on a recent result of Huang and Lin [8] for equality in distribution of convex ordered survival times. Illustrative applications include testing for equality in distribution with convex ordered alternatives and demonstrating several earlier results on stochastic equivalence as special cases.
• Bhattacharya, R. (1999). Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media. Annals of Applied Probability, 9(4), 951-1020.
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  Abstract: Consider diffusions on ℝk, k > 1, governed by the Itô equation d X (t)={b(X(t)) + β(X(t)/a)}dt + σdB(t), where b, β are periodic with the same period and are divergence free, σ is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times 1 ≪ t ≪ a2/3. Under a geometric condition on the velocity field β, the final Gaussian phase occurs for times t ≫ a2(log a)2, and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t ≫ a4(log a)2, or t ≫ a2 log a under additional conditions, with no unbounded growth in dispersion as a function of scale. Examples show the existence of non-Gaussian intermediate phases. These probabilisitic results are applied to analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case b, β are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.
• Bhattacharya, R., & Majumdar, M. (1999). On a Theorem of Dubins and Freedman. Journal of Theoretical Probability, 12(4), 1067-1087.
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  Abstract: Under a notion of "splitting" the existence of a unique invariant probability, and a geometric rate of convergence to it in an appropriate metric, are established for Markov processes on a general state space S generated by iterations of i.i.d. maps on S. As corollaries we derive extensions of earlier results of Dubins and Freedman;(17) Yahav;(30) and Bhattacharya and Lee(6) for monotone maps. The general theorem applies in other contexts as well. It is also shown that the Dubins-Freedman result on the "necessity" of splitting in the case of increasing maps does not hold for decreasing maps, although the sufficiency part holds for both. In addition, the asymptotic stationarity of the process generated by i.i.d. nondecreasing maps is established without the requirement of continuity. Finally, the theory is applied to the random iteration of two (nonmonotone) quadratic maps each with two repelling fixed points and an attractive period-two orbit.
• Bhattacharya, R., Denker, M., & Goswami, A. (1999). Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales. Stochastic Processes and their Applications, 80(1), 55-86.
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  Abstract: The present article analyses the large-time behavior of a class of time-homogeneous diffusion processes whose spatially periodic dynamics, although time independent, involve a large spatial parameter 'a'. This leads to phase changes in the behavior of the process as time increases through different time zones. At least four different temporal regimes can be identified: an initial non-Gaussian phase for times which are not large followed by a first Gaussian phase, which breaks down over a subsequent region of time, and a final Gaussian phase different from the earlier phases. The first Gaussian phase occurs for times 1ta2/3. Depending on the specifics of the dynamics, the final phase may show up reasonably fast, namely, for t≫a2loga; or, it may take an enormous amount of time t≫exp{ca} for some c>0. An estimation of the speed of convergence to equilibrium of diffusions on a circle of circumference 'a' is provided for the above analysis.
• Bhattacharya, R. N. (1997). A hierarchy of Gaussian and non-Gaussian asymptotics of a class of Fokker-Planck equations with multiple scales. Nonlinear Analysis, Theory, Methods and Applications, 30(1), 257-263.
• Bhattacharya, R. N., & Lee, C. (1995). Ergodicity of nonlinear first order autoregressive models. Journal of Theoretical Probability, 8(1), 207-219.
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  Abstract: Criteria are derived for ergodicity and geometric ergodicity of Markov processes satisfying Xn+1 =f(Xn)+σ(Xn)e{open}n+1, where f, σ are measurable, {e{open}n} are i.i.d. with a (common) positive density, E|e{open}n|>∞. In the special case f(x)/x has limits, α, β as x→-∞ and x→+∞, respectively, it is shown that "α
• Bhattacharya, R., & Lee, C. (1995). On geometric ergodicity of nonlinear autoregressive models. Statistics and Probability Letters, 22(4), 311-315.
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  Abstract: A criterion is derived for the geometric Harris ergodicity of general nonlinear autoregressive models, which imposes a condition on the forcing function only at infinity and does not require that the function be continuous. © 1995.
• 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.
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  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. N., & Ghosh, J. K. (1992). A class of U-statistics and asymptotic normality of the number of k-clusters. Journal of Multivariate Analysis, 43(2), 300-330.
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  Abstract: A central limit theorem is proved for a class of U-statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance. As an application we derive the asymptotic normality of the number of Poisson k-clusters in a cube of increasing size in Rd. We also extend earlier results of Jammalamadaka and Janson to general kernels and to general orders k > 2 of the kernel. © 1992.
• Bhattacharya, R. N., & Majumdar, M. (1989). Controlled semi-Markov models under long-run average rewards. Journal of Statistical Planning and Inference, 22(2), 223-242.
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  Abstract: Let the state space S be a Borel subset of a complete separable metric space, the action space A compact metric. Existence of stationary optimal policies is proved and a dynamic programming equation derived for general semi-Markov models under the long-run average reward criterion, focusing on it as a limiting case of optimization under discounting as the discount factor goes to one. This extends many earlier results. An example of Reed (1974) on harvesting a natural resource provides an application not covered by earlier results. © 1989.
• Bhattacharya, R. N., & Majumdar, M. (1989). Controlled semi-markov models - the discounted case. Journal of Statistical Planning and Inference, 21(3), 365-381.
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  Abstract: Let the state space S be a Borel subset of a complete separable metric space, the action space A compact metric. Existence of stationary optimal policies is proved for general semi-Markov models with possibly unbounded rewards. The corresponding dynamic programming equations are also derived. The paper presents a synthesis and extensions of earlier results. © 1989.
• Bhattacharya, R. N., & Ghosh, J. K. (1988). On moment conditions for valid formal Edgeworth expansions. Journal of Multivariate Analysis, 27(1), 68-79.
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  Abstract: The validity of formal Edgeworth expansions for statistics which are functions of sample averages was established in R. N. Bhattacharya and J. K. Ghosh (1978, Ann. Statist.6 434-451) under a moment condition which is sometimes too severe. In this article this moment condition is relaxed. Two examples of P. Hall (1983, Ann. Probab.11 1028-1036; 1987, Ann. Probab.15 920-931) are discussed in this context. © 1988.
• Bhattacharya, R. N., & Lee, O. (1988). Ergodicity and central limit theorems for a class of Markov processes. Journal of Multivariate Analysis, 27(1), 80-90.
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  Abstract: We consider a class of discrete parameter Markov processes on a complete separable metric space S arising from successive compositions of i.i.d. random maps on S into itself, the compositions becoming contractions eventually. A sufficient condition for ergodicity is found, extending a result of Dubins and Freedman [8] for compact S. By identifying a broad subset of the range of the generator, a functional central limit theorem is proved for arbitrary Lipschitzian functions on S, without requiring any mixing type condition or irreducibility. © 1988.
• Gupta, V. K., & Bhattacharya, R. N. (1986). Solute dispersion in multidimensional periodic saturated porous media.. Water Resources Research, 22(2), 156-164.
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  Abstract: Suppose in a convection-dispersion equation, governing solute movement in a saturated porous medium of infinite extent, the convection velocity components are periodic functions of spatial coordinates. Then it follows from a general mathematical result that the solute concentration can be asymptotically approximated by a Gaussian density. Two theoretical examples, with and without a constant vertical velocity, are given to illustrate an application of this mathematical result to solute dispersion in a parallel-bedded, 3-dimensional aquifer of infinite extent.-from Authors
• Bhattacharya, R. N., & Gupta, V. K. (1984). ON THE TAYLOR-ARIS THEORY OF SOLUTE TRANSPORT IN A CAPILLARY.. SIAM Journal on Applied Mathematics, 44(1), 33-39.
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  Abstract: A new simple derivation is given of G. I. Taylor's classic theory of solute transport in a straight capillary through which a liquid is flowing in a steady nonturbulent flow. The results derived are stronger, and an explicit representation is provided for the displacement of a solute molecule as the sum of a Brownian motion and the integral of an ergodic Markov process which is asymptotically a Brownian motion. Two curious identities involving zeros of the Bessel function of order one are obtained as a by-product.
• Bhattacharya, R. N., & Gupta, V. K. (1983). A theoretical explanation of solute dispersion in saturated porous media at the Darcy scale.. Water Resources Research, 19(4), 938-944.
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  Abstract: The transport of a nonreactive dilute solute in saturated porous media is explained at three distinct space-time scales. These are the kinetic, microscopic, and Darcy scales. The transition from one scale to the next higher scale, ie, from the kinetic to the microscopic to the Darcy, is a consequence of the central limit theorem of probability theory. -from Authors
• Bhattacharya, R. N., Gupta, V. K., & Waymire, E. (1983). HURST EFFECT UNDER TRENDS.. Journal of Applied Probability, 20(3), 649-662.
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  Abstract: Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) equals c(m plus n)** beta , where m is an arbitrary non-negative parameter and c is not 0. For minus one-half less than beta less than 0 the Hurst exponent is shown to be precisely given by 1 plus beta . For beta less than equivalent to minus one-half and for beta equals 0 the Hurst exponent is 0. 5, while for beta greater than 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.
• Gupta, V. K., & Bhattacharya, R. N. (1983). A new derivation of the Taylor-Aris theory of solute dispersion in a capillary.. Water Resources Research, 19(4), 945-951.
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  Abstract: The transport of a dilute solute under viscous motion in a straight capillary is described at three distinct space-time scales. These are the kinetic, fluid mechanical, and Taylorian scales. The transition from one scale to the next higher scale is shown to be a consequence of the central limit theorem of probability. -from Authors
• Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 60(2), 185-201.
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  Abstract: Let Xt:t≧0 be an ergodic stationary Markov process on a state space S. If  is its infinitesimal generator on L2(S, dm), where m is the invariant probability measure, then it is shown that for all f in the range of {Mathematical expression} converges in distribution to the Wiener measure with zero drift and variance parameter σ2 =-2〈f, g〉=-2〈Âg, g〉 where g is some element in the domain of  such that Âg=f (Theorem 2.1). Positivity of σ2 is proved for nonconstant f under fairly general conditions, and the range of  is shown to be dense in 1⊥. A functional law of the iterated logarithm is proved when the (2+δ)th moment of f in the range of  is finite for some δ>0 (Theorem 2.7(a)). Under the additional condition of convergence in norm of the transition probability p(t, x, d y) to m(dy) as t → ∞, for each x, the above results hold when the process starts away from equilibrium (Theorems 2.6, 2.7 (b)). Applications to diffusions are discussed in some detail. © 1982 Springer-Verlag.
• Bhattacharya, R. N., & Ramasubramanian, S. (1982). Recurrence and ergodicity of diffusions. Journal of Multivariate Analysis, 12(1), 95-122.
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  Abstract: This article attempts to lay a proper foundation for studying asymptotic properties of nonhomogeneous diffusions, extends earlier criteria for transience, recurrence, and positive recurrence, and provides sufficient conditions for the weak convergence of a shifted nonhomogeneous diffusion to a limiting stationary homogenous diffusion. A functional central limit theorem is proved for the class of positive recurrent homogeneous diffusions. Upper and lower functions for positive recurrent nonhomogeneous diffusions are also studied. © 1982.
• Gupta, V. K., Bhattacharya, R. N., & Sposito, G. (1981). A molecular approach to the foundations of the theory of solute transport in porous media. I. Conservative solutes in homogeneous isotropic saturated media. Journal of Hydrology, 50(C), 355-370.
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  Abstract: A molecular model is developed for the transport of a conservative solute at low concentrations in a homogeneous isotropic water-saturated porous medium. In this model, a solute molecule undergoes contact collisions with the solid grains in the medium at successive random times; in between these collisions, the velocity of the molecule is governed by the Langevin equation; the effect of the collisions with the solid grains is to scatter a molecule in a random direction. This molecular model is employed to derive rigorously a parabolic differential equation for the solute concentration at the macroscopic level. In the absence of solute convection, the coefficient of molecular diffusion in a porous medium is proved to be less than the coefficient of molecular diffusion in bulk solution, a finding which is in agreement with experimental observations. For non-zero convection, the assumption of isotropicity of the medium is employed to prove that the solute dispersion tensor is diagonal. For small magnitudes of the liquid velocity, the isotropicity assumption also implies that the coefficients of longitudinal and transverse dispersion are approximately parabolic functions of the liquid velocity. The expressions derived for the dispersion coefficients, when compared with their experimentally observed values, suggest that their dependence on the liquid velocity comes primarily through solute-liquid molecular collisions instead of through collisions of the solute molecules with the grains of the solid phase. The solute convective velocity is shown to be less than or equal to the liquid velocity. Precisely when the difference between the two velocities will be significant remains to be established. © 1981.
• Bhattacharya, R. N., & Gupta, V. K. (1979). ON A STATISTICAL THEORY OF SOLUTE TRANSPORT IN POROUS MEDIA.. SIAM Journal on Applied Mathematics, 37(3), 485-498.
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  Abstract: Consider the motion of a solute molecule in a homogeneous isotropic porous medium saturated with a pure liquid. The following assumptions are made on the velocity of the molecule: the time intervals between successive collisions of the molecule with the solid matter of the medium are i. i. d. random variables; the molecule is scattered by these collisions in random directions which are i. i. d. uniform and independent of the collison times; in between two successive collisions with the solid phase the velocity of the molecule is governed by the Langevin equation. Under these and mild additional assumptions it is proved that the position left brace x(t):t greater than equivalent to 0 right brace of the molecule is approximately a Brownian motion. If the solute molecules are weakly interacting among themselves, then the above result leads to a macroscopic parabolic equation governing solute concentration. If the successive collision times with the solid phase are assumed to be exponential, then the velocity left brace v(t):t greater than equivalent to 0 right brace as well as left brace (v(t),x(t)):t greater than equivalent to 0 right brace are Markovian. This leads to laws of mass, momentum, and energy conservation for solute transport.
• Sposito, G., Gupta, V. K., & Bhattacharya, R. (1979). Foundation theories of solute transport in porous media: a critical review. Advances in Water Resources, 2(C), 59-68.
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  Abstract: The theories that have been employed to derive the macroscopic differential equations that describe solute transport through porous media are reviewed critically. These foundational theories may be grouped into three classes: (1) those based in fluid mechanics, (2) those based in kinematic approaches employing the mathematics of the theory of Markov processes, and (3) those based in a formal analogy between statistical thermodynamics and hydrodynamic dispersion. It is shown that the theories of class 1 have had to employ highly artificial models of a porous medium in order to produce a well-defined velocity field in the pore space that can be analysed rigorously or have had to assume that well-defined solutions of the equations of fluid mechanics exist in the pore space of a natural porous medium and then adopt an ad hoc definition of the solute difusivity tensor. The theories of class 2 do not require the validity of fluid mechanics but they suffer from the absence of a firm dynamical basis, at the molecular level, for the stochastic properties they attribute to the velocity of a solute molecule, or they ignore dynamics altogether and make kinematic assumptions directly on the position process of a solute molecule. The theories of class 3 have been purely formal in nature, with an unclear physical content, or have been no different in content from empirically based theories that make use of the analogy between heat and matter flow at the macroscopic level. It is concluded that none of the existing foundational theories has yet achieved the objectives of: (1) deriving, in a physically meaningful and mathematically rigorous fashion, the macroscopic differential equations of solute transport theory, and (2) elucidating the structure of the empirical coefficients appearing in these equations. © 1979.
• Gupta, V. K., Sposito, G., & Bhattacharya, R. N. (1977). TOWARD AN ANALYTICAL THEORY OF WATER FLOW THROUGH INHOMOGENEOUS POROUS MEDIA.. Water Resources Research, 13(1), 208-210.
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  Abstract: Some rigorous mathematical results for the Buckingham-Darcy flux law for water flow through an isotropic, nondeformable, inhomogeneous porous medium are presented. It is shown that the volumetric flux density vector, aside from the component due to gravity, may always be expressed in terms of a scalar and a vector matric flux potential.
• BHATTACHARYA, R. N., GUPTA, V. K., & SPOSITO, G. (1976). ON THE STOCHASTIC FOUNDATIONS OF THE THEORY OF WATER FLOW THROUGH UNSATURATED SOIL.. Water Resources Research, 12(3), JUNE, 1976.
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  Abstract: THE PARABOLIC DIFFERENTIAL EQUATION THAT DESCRIBES THE ISOTHERMAL ISOHALINE TRANSPORT OF WATER THROUGH AN UNSATURATED SOIL IS SHOWN TO BE THE MATHEMATICALLY REGOROUS RESULT OF A FUNDAMENTAL STOCHASTIC HYPOTHESIS: THAT THE TRAJECTORY OF A WATER MOLECULE IS A NONHOMOGENEOUS MARKOV PROCESS CHARACTERIZED BY SPACE- AND TIME-DEPENDENT COEFFICIENTS OF DRIFT AND DIFFUSION.THE DEMONSTRATION IS VALID IN GENERAL FOR HETEROGENEOUS ANISOTROPIC SOILS AND PROVIDES FOR THREE PINCIPAL RESULTS IN THE THEORY OF WATER FLOW THROUGH UNSATURATED MEDIA: (I) A DERIVATION OF THE BUCKINGHAM-DARCY FLUX LAW THAT DOES NOT RELY DIRECTLY ON EXPERIMENT, (II) A NEW THEORETICAL INTERPRETATION OF THE SOIL WATER DIFFUSIVITY AND THE HYDRAULIC CONDUCTIVITY IN MOLECULAR TERMS, AND (III) A PROOF THAT THE SOIL WATER DIFFUSIVITY FOR ANISOTROPIC SOIL IS A SYMMETRIC TENSOR OF THE SECOND RANK.A DYNAMIC ARGUMENT AT THE MOLECULAR LEVEL IS DEVELOPED TO SHOW THAT THE FUNDAMENTAL MARKOVIAN HYPOTHESIS IS PHYSICALLY REASONABLE IN THE CASE OF WATER MOVEMENT THROUGH AN UNSATURATED SOIL.(A).
• Bhattacharya, R. N., Gupta, V. K., & Sposito, G. (1976). ON THE STOCHASTIC FOUNDATIONS OF THE THEORY OF WATER FLOW THROUGH UNSATURATED SOIL.. Water Resources Research, 12(3), 503-512.
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  Abstract: The parabolic differential equation that describes the isothermal isohaline transport of water through an unsaturated soil is shown to be the mathematically rigorous result of a fundamental stochastic hypothesis: that the trajectory of a water molecule is a nonhomogeneous Markov process characterized by space- and time-dependent coefficients of drift and diffusion. The demonstration is valid in general for heterogeneous anisotropic soils and provides for three principal results in the theory of water flow through unsaturated media. A dynamic argument at the molecular level is developed to show that the fundamental Markovian hypothesis is physically reasonable in the case of water movement through an unsaturated soil.
• Bhattacharya, R. N., & Majumdar, M. (1973). Random exchange economies. Journal of Economic Theory, 6(1), 37-67.
• Bhattacharya, R. N. (1972). Speed of convergence of the n-fold convolution of a probability measure on a compact group. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 25(1), 1-10.