Bodhisattva Sen


Shape restricted function estimation and inference

Nonparametric function estimation is mostly concerned with understanding the structure (trend/pattern) in the data without making strong parametric assumptions on its form. Most estimation procedures for a nonparametric function (e.g., kernel smoothing), be it a regression function or a probability density, make smoothness assumptions on the underlying function and use local averaging techniques. These estimators depend crucially on tuning parameter(s) (e.g., smoothing bandwidths) and the choice of such parameter(s) can be very problematic.

However, in shape restricted function estimation we can completely by-pass this dependence on tuning parameters and develop estimates that are fully automated in situations where prior knowledge on the shape of the function is available. Such shape constraints arise naturally in numerous applications, e.g., utility/production functions are known to be non-decreasing and concave, distribution functions are known to be non-decreasing, sometimes regression/hazard functions are known to be monotone/unimodal, etc. The goal of my research in this area is to develop and study the properties of the estimated shape restricted functions, especially in multi-dimensions.

Recently in [4], we develop and study the consistency of the nonparametric least squares estimator of a convex regression function when the predictor is multi-dimensional. To the best of my knowledge, this is the first attempt to systematically study the characterization, computation, and consistency of such shape restricted regression estimates in a completely nonparametric setting.

Although the computational and theoretical properties of the shape constrained estimators of monotone/isotonic functions in one-dimension are well understood, the complicated nature of their asymptotic distributions makes inference very difficult. Bootstrap based inference is a natural alternative and is investigated in [3] where we consider the Grenander estimator, the nonparametric maximum likelihood estimator of a decreasing density on the positive real line, that exhibits cube root rate of convergence. See [1] for a likelihood based approach to estimation and inference on a (monotone) distribution function with interval censored data. Other applications of such methods in a stereology, astronomy and econometrics can be found in [6], [5] and [2], respectively.


  1. 1.Sen, B. and Banerjee, M. (2006). A Pseudo-likelihood Method for Analyzing Interval Censored Data. Biometrika, 94, 71-86.

  1. 2.Sen, B., Banerjee, M., Woodroofe, M., Walker, M.G. and Mateo, M. (2009). Streaming Motion in Leo I. Ann. Appl. Statis., 3, 96-116.

  1. 3.Sen, B., Banerjee, M. and Woodroofe, M. (2010). Inconsistency of Bootstrap: the Grenander Estimator. Ann. Statist., 38, 1953-1977.

  1. 4.Seijo, E. and Sen, B. (2011). Nonparametric least squares estimation of a multivariate convex regression. Ann. Statist., 39, 1633-1657.

  1. 5.Sen., B. and Chaudhuri, P. (2011). Mahalanobis’s Fractile Graphs: Some History and New Developments. International Journal of Statistical Sciences, 11, 17-35 (Invited paper for a special issue in honor of Prasanta Chandra Mahalanobis).

  1. 6.Sen, B. and Woodroofe, M. (2011). Bootstrap Confidence Intervals for Isotonic Estimators in a Stereological Problem. Bernoulli (to appear).