Efficient Inferential Methods in Regression Models with Change Points or High Dimensional Covariates.

Abstract

We present a fast approach for estimating change-point(s) in the broken-stick model efficiently in both cross-sectional and longitudinal settings. Our method, based on local smoothing in a shrinking neighborhood of each change-point, is shown via simulations to be computationally more viable than existing methods that rely on search procedures. The proposed estimates are shown to have root-n-consistency and asymptotic normality. As our motivating application, we study the MBHMS cohort data to describe patterns of change in log estradiol levels around the final menstrual period, for which a two change-points broken-stick model is a good fit.

Though there has been a considerable work done on studying the effects of coarse and fine ambient particles, how the constituent pollutants affect cardiovascular functioning is not clearly understood. We propose using multivariate adaptive elastic-net to capture these effects in an autoregressive model for time series data. Because of the large number of highly correlated pollutants, a reliable method must take into account the high dimensionality as well as multicollinearity issues. This is accomplished by using adaptive elastic-net which deals effectively with the correlated nature of the data. Furthermore, the selection consistency and asymptotic normality properties allow us to provide meaningful statistical inference. As our motivating example, we study the effects of multiple pollutants on several cardiovascular end-points in a rat-study based in Dearborn, Michigan, conducted by the GLACIER.

Finally, we look at problems where there are several covariates and some of the covariates have a simple linear effect while some have a broken-stick effect. We are looking at two separate but similar types of problems: one where we have several covariates each with possibly a broken-stick effect but with only a single change-point and the other where the broken-stick effect is exhibited in a single covariate but the number of change-points is unknown. In both settings we strive for a parsimonious yet accurate model, which necessitates an effective variable selection procedure. In a sparse setting, we illustrate the difficulty in using the popular variable selection methods and propose post local-smoothing thresholded ridge regression as an effective variable selection method.