Nonparametric Estimation of the Slope of a Truncated Regression

Abstract

A nonparametric estimate $\beta^\ast$ is presented for the slope of a regression line $Y = \beta_0X + V$ subject to the truncation $Y \leq y_0$. This model is relevant to a cosmological controversy which concerns Hubble's Law in Astronomy. The estimate $\beta^\ast$ corresponds to the zero-crossing of a random function $S_n(\beta)$, which for each $\beta$ is a Mann-Whitney type of statistic designed to measure heterogeneity among the calculated residuals $Y - \beta X$. The asymptotic distribution of $\beta^\ast$ is derived making extensive use of $U$-statistics to show that $S_n(\beta_0)$ is asymptotically normal and then showing that $S_n(\beta)$ behaves like $S_n(\beta_0)$ plus a deterministic term which is locally linear. Results on asymptotic efficiency are compared with finite sample size results by simulation.