Seminars 2009 — Abstracts

Friday, May 15

Speaker: PK Sen, North Carolina Chapel Hill

Title: The Theil-Sen Estimator in a Measurement Error Perspective

Abstract: The classical least squares estimator (LSE), the MLE for normally distributed errors, of the regression slope is known to be nonrobust for outliers, error contamination and departures from the assumed normality. The estimator based on the Kendall tau statistic, known as the Theil-Sen estimator (TSE), is robust, median-unbiased and insensitive to any departure from the assumed normality of the errors (Sen 1968).

This scenario is appraised in a measurement error (ME) model where the unobservable (Y_i^o,X_i^o) satisfy a linear regression model with the slope parameter β. The observable random vectors are (Y_i, X_i) where Y_i = Y_i^o + η_i, X_i = X_i^o + U_i, i = 1, ..., n with η_i, U_i standing for the measurement errors for the dependent and independent variates. Further, let e_i be the error variable associated with the Y_i^o and V_i be the error variable associated with the X_i^o. It is assumed that e_i, η_i, V_i, U_i are all independent. In the classical normal model (Fuller 1987), all these errors are assumed to be normal. Sans this assumed normality, the LSE may not be unbiased nor have simpler distributional properties. It becomes more nonrobust than in the conventional case without ME.

The present study is mainly a characterization of the TSE in a ME setup where the normality of the errors has been dispensed with general regularity assumptions. An important by-product of this characterization of the TSE is the scope for more indepth study of various finite sample to asymptotic properties of the TSE in the ME setup contemplated here. Some applications are also appended.