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On Sprent’s generalized least-squares estimator. (English) Zbl 0536.62040
Summary: This paper studies the asymptotic properties of P. Sprent’s [ibid. 28, 278-297 (1966; Zbl 0147.378)] generalized least-squares estimator of the slope parameter in a functional relationship model which allows errors at different data points to be correlated. Consistency is established and the asymptotic variance derived. When the joint distribution of the errors is normal, in which case Sprent’s estimator coincides with the maximum likelihood estimator, the asymptotic variance formula is further simplified and the limiting distribution of the estimator is also shown to be normal.
The variance expression obtained under the normality assumption differs from that of G. R. Dolby [ibid. 34, 393-400 (1972; Zbl 0266.62039)] derived by inverting the information matrix, and the latter procedure is known to be generally invalid in functional relationship problems due to the presence of incidental parameters [cf. W. M. Patefield, Biometrika 65, 535-540 (1978; Zbl 0394.62023)]. An estimator of the asymptotic variance is also suggested.
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
62J99 Linear inference, regression