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Nonlinear least-squares estimation. (English) Zbl 1085.62027
Summary: The paper uses empirical process techniques to study the asymptotics of the least-squares estimators (LSE) for fitting of nonlinear regression functions. By combining and extending ideas of C.-F. Wu [Ann. Stat. 9, 501–513 (1981; Zbl 0475.62050)] and S. van de Geer [ibid. 18, No. 2, 907–924 (1990; Zbl 0709.62040)], it establishes new consistency and central limit theorems that hold under only second moment assumptions on the errors. An application to a delicate example of Wu illustrates the use of the new theorems, leading to a normal approximation to the LSE with unusual logarithmic rescalings.

MSC:
62F12 Asymptotic properties of parametric estimators
62J02 General nonlinear regression
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
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References:
[1] Chernoff, H., On the distribution of the likelihood ratio, Ann. math. statist., 25, 573-578, (1954) · Zbl 0056.37102
[2] Jennrich, R.I., Asymptotic properties of non-linear least squares estimators, Ann. math. statist., 40, 633-643, (1969) · Zbl 0193.47201
[3] Lai, T.L., Asymptotic properties of nonlinear least-squares estimates in stochastic regression models, Ann. statist., 22, 1917-1930, (1994) · Zbl 0824.62054
[4] Skouras, K., Strong consistency in nonlinear stochastic regression models, Ann. statist., 28, 871-879, (2000) · Zbl 1105.62355
[5] Van de Geer, S., Estimating a regression function, Ann. statist., 18, 907-924, (1990) · Zbl 0709.62040
[6] Van de Geer, S.; Wegkamp, M., Consistency for the least squares estimator in nonparametric regression, Ann. statist., 24, 2513-2523, (1996) · Zbl 0867.62027
[7] Van der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical process: with applications to statistics, (1996), Springer Berlin · Zbl 0862.60002
[8] M. Wegkamp, Entropy Methods in Statistical Estimation, Center for Mathematics and Computer Science, CWI Tract 125, 1998. · Zbl 0912.62044
[9] Wu, C.-F., Asymptotic theory of nonlinear least squares estimation, Ann. statist., 9, 501-513, (1981) · Zbl 0475.62050
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