An investigation of the performance of five different estimators in the measurement error regression model

Abstract

In a comprehensive paper by Riggs et al.(1978) the authors analyse the performances of numerous estimators for the regression slope in the measurement error model with positive measurement error variances >0 0 for X and >0 for Y . In particular, using a Monte Carlo simulation, the authors demonstrate that the adjusted geometric mean estimator of Madansky (1959, Equation 4, p. 179), which requires knowledge of both and , performs “much worse than” the maximum likelihood estimator in the normal structural measurement error model which requires only knowledge of the ratio . The second moment estimator, coincides with the maximum likelihood estimator inthe normal structural measurement error model (Madansky (1959)). In practice κ has to be estimated by .In this paper, we show that the bias of is not only dependent on the magnitude of the difference between κ and but also on the magnitude of - . We use a fourth moment estimator to smooth the jump discontinuity in the estimator of Copas (1972) as described in ODriscoll and Ramirez (2011) and use this estimator to find estimates for each error variance and . Our Monte Carlo simulations show that the adjusted geometric mean estimator of Madansky performs much better than the ordinary least squares estimators OLS(y|x) and OLS(x|y) when the error variances are strictly positive and performs equally as well as the geometric estimator, , and the perpendicular estimator of Adcock (1878), , with κ = 1 .