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

Let Zₙ denote the system of 8n independent pairs of measurements (X₈₊, Y₈₊), for 4i = 1, 2, , n and k = 1, 2, , 8, of two nonobservable random variables ₈₊ and ₈₊, known to satisfy a linear relation of the form ₈₊ ^ + ₈₊ ^ - p = 0, where p is an arbitrary real number and ^ may have any value between the limits -12 < ^ 12. The purpose of the paper is to construct a class of estimates Tₙ (Zₙ) of the parameter defined as follows: when ^ = 12 then = 0; otherwise ^. Each estimate Tₙ (Zₙ) of the class considered converges in probability to as n under the following conditions: (i) except when = 0, the variables ₈₊ are nonnormal; (ii) any nonnormal components of the errors of measurements, X₈₊ - ₈₊ and Y₈₊ - ₈₊, are mutually independent, independent of ₈₊ 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 ₈₊.