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