Bodhisattva Sen

 

Change-point and threshold models

In contrast to function estimation, often the goal is to determine a threshold in the domain of the function where some “activity” takes place -- this could either be a rapid change in the function value or a discontinuity. Such models may arise when a stochastic system is subject to sudden external influences and are encountered in almost every field of science. Perhaps, the commonest notion of such a threshold is a change-point; this is an actual discontinuity in the function of interest (or a discontinuity at the derivative level, for example, a wedge-shaped function).


Inference for the change point can be difficult as the usual estimators converge at non-standard rates and have non-normal distributions. In [4], we investigate bootstrap based inference for the least squares estimation of a change point in a stochastic design parametric regression setting. To understand the asymptotic properties of the change point estimator we need results on weak convergence of (smallest) maximizers of certain multi-parameter stochastic processes which are “cadlag” (right continuous with left hand side limits) in one parameter and continuous on the rest, that are developed in [3]. Consistent estimation of the threshold parameter in a more general nonparametric setting is explored in [5] where we use (possibly approximate) p-values.


In [1], we describe an application of such threshold models in astronomy. In [2], we explore least squares estimation and inference for the change point in a regression model where the predictor is functional and behaves like fractional Brownian motion.



References


  1. 1.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. 2.McKeague, I. and Sen, B. (2010). Fractals with point impact in functional linear regression. Ann. Statist., 38, 2559-2586.


  1. 3.Seijo, E. and Sen, B. (2011). A continuous mapping theorem for the smallest argmax functional. Electron. J. Statist., 5, 421-439.


  1. 4.Seijo, E. and Sen, B. (2011). Change point in stochastic design regression and the bootstrap. Ann. Statist., 39, 1580-1607.


  1. 5.Mallik, A., Sen, B., Banerjee, M., and Michailidis, G. (2011). Threshold estimation based on a P-value framework in dose-response and regression settings. Biometrika (to appear).