Shape-Restricted Inference for Dependent Data

• Activity Number: 431
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Pramita Bagchi*+ and Stilian A Stoev and Moulinath Banerjee
• Companies: University of Michigan and University of Michigan and University of Michigan
• Keywords: Shape Restricted Inference ; Isotonic Regression ; Weak Dependence ; Strong Dependence ; Time Series Data ; Non-parametric Regression

Abstract

Isotonic regression is an important problem arising in many applications such as climate studies, economics etc. When the data are independent Pooled Adjacent Violators Algorithm provides the isotonic regression estimate (IRE) of the monotone function we want to estimate. Its asymptotic properties are well studied. In many applications, however, the data are dependent. Motivated by the work of Banerjee & Wellner (2001), our goal is to study the RSS statistic and $L_2$ distance between constrained and unconstrained IRE. The limit distribution of these statistics for iid data do not involve the slope of the regression function, which is difficult to estimate and arises in the asymptotic distribution of IRE. Here we have derived the asymptotic distribution of these statistics under both weak and strong dependence of the data. As these two statistics converge jointly the ratio of these two statistics has a limit distribution free of the the slope of the regression function and other scaling factors under both kinds of dependence. We expect to use this ratio statistic to construct confidence interval for the regression function at a particular point for many practical applications.