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Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. (English) Zbl 0707.62058
Summary: Let W be a normal random variable with mean \(\mu\) and known variance \(\sigma^ 2\). Conditions on the function \(f(\cdot)\) are given under which there exists an unbiased estimator, \(\bar f(W)\), of \(f(\mu)\) for all real \(\mu\). In particular it is shown that \(f(\cdot)\) must be an entire function over the complex plane. Infinite series solutions for \(\bar f(\cdot)\) are obtained which are shown to be valid under growth conditions of the derivatives, \(f^{(k)}(\cdot)\), of \(f(\cdot)\). Approximate solutions are given for the cases in which no exact solution exists. The theory is applied to nonlinear measurement-error models as a means of finding unbiased score functions when measurement error is normally distributed. Relative efficiencies comparing the proposed method to the use of conditional scores are given for the Poisson regression model with canonical link.

MSC:
62F10 Point estimation
62J02 General nonlinear regression
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