Existence of Consistent Estimates of the Directional Parameter in a Linear Structural Relation Between Two Varibles

Abstract

Let $Z_n$ denote the system of $8n$ independent pairs of measurements $(X_{ik}, Y_{ik}),$ for $4i = 1, 2, \cdots, n$ and $k = 1, 2, \cdots, 8,$ of two nonobservable random variables $\xi_{ik}$ and $\eta_{ik},$ known to satisfy a linear relation of the form $\xi_{ik} \cos \theta^\ast + \eta_{ik} \sin \theta^\ast - p = 0,$ where $p$ is an arbitrary real number and $\theta^\ast$ may have any value between the limits $-\frac{1}{2}\pi < \theta^\ast \leqq \frac{1}{2}\pi$. The purpose of the paper is to construct a class of estimates $T_n(Z_n)$ of the parameter $\theta$ defined as follows: when $\theta^\ast = \frac{1}{2}\pi$ then $\theta = 0$; otherwise $\theta \equiv \theta^\ast$. Each estimate $T_n(Z_n)$ of the class considered converges in probability to $\theta$ as $n \rightarrow \infty$ under the following conditions: (i) except when $\theta = 0$, the variables $\xi_{ik}$ are nonnormal; (ii) any nonnormal components of the errors of measurements, $X_{ik} - \xi_{ik}$ and $Y_{ik} - \eta_{ik},$ are mutually independent, independent of $\xi_{ik}$ and of the normal components of these errors; (iii) the normal components of the errors may be correlated but as a pair are independent of $\xi_{ik}$.