Interval Estimation in Linear Regression When Both Variables are Subject to Error

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Abstract Define the linear functional relationship with parameters, α, β, by μ = α + βv. This paper considers the problem of confidence interval estimation of α and β separately, based on a sample of independent pairs, (x i , y i ), i = 1, 2, ···, n, with Ex i = vi, Ey i = μ i . The pairs, (x i , y i ), are assumed to be independent (i ≠ i′), to have a common (unknown) covariance matrix Σ, and to follow the bivariate normal distribution. It is known that under these circumstances (providing instrumental variates are available) one can define a variance ratio with 2 and (n − 2) degrees of freedom, F 2,n–2 say, as a function of α, β, sample statistics and instrumental variates to yield a confidence region on (α, β) based on In this paper we show, for interval estimation of α only, that taking limits as (min α, max α) satisfying yields confidence at least (1–γ). A similar result holds for interval estimation of β only. In the latter case, the limits obtained differ from those derived by the well-known proced...