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Second-order approximation of the entropy in nonlinear least-squares estimation. (English) Zbl 0812.62071
Kybernetika 30, No. 2, 187-198 (1994); erratum ibid. 32, No. 1, 104 (1996).
Summary: Measures of variability of the least-squares estimator \(\widehat {\theta}\) are essential to assess the quality of the estimation. In nonlinear regression, an accurate approximation of the covariance matrix of \(\widehat {\theta}\) is difficult to obtain. A second-order approximation of the entropy of the distribution of \(\widehat {\theta}\) is proposed, which is only slightly more complicated than the widely used bias approximation of M. J. Box [J. R. Stat. Soc, Ser. B 33, 171- 201 (1971; Zbl 0232.62029)]. It is based on the “flat” or “saddle- point approximation” of the density of \(\widehat {\theta}\). The neglected terms are of order \({\mathcal O} (\sigma^ 4)\), while the classical first order approximation neglects terms of order \({\mathcal O} (\sigma^ 2)\). Various illustrative examples are presented, including the use of the approximate entropy as a criterion for experimental design.

MSC:
62J02 General nonlinear regression
62E17 Approximations to statistical distributions (nonasymptotic)
62E20 Asymptotic distribution theory in statistics
62K05 Optimal statistical designs
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