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Asymptotic analysis of penalized likelihood and related estimators. (English) Zbl 0719.62051
The authors present a general approach for analyzing a certain class of curve estimators related to penalized likelihood estimation. Given a fit functional $$l_ n$$ measuring the deviation of our function $$\theta$$ from the data and a penalty functional J, being smaller for more desirable functions, the estimator $${\hat \theta}{}_ n$$ in question is a minimizer of $$l_{n,\lambda}(\theta)=l_ n(\theta | data)+\lambda J(\theta),$$ with some regularization parameter $$\lambda$$.
For certain penalty functionals and fit functionals it is possible to analyze the asymptotic behaviour of the estimates $${\hat \theta}{}_ n$$ using linearization of the quantities involved. The main results give approximations of the systematic - and stochastic error and in particular for log-density estimation, log-hazard estimation and nonparametric logistic regression. Convergence rates are given for $${\hat \theta}{}_ n-\theta_ 0$$ w.r. to spectral norms. The results of D. D. Cox [ibid. 16, No.2, 694-712 (1988; Zbl 0671.62044)] are the basis of the analysis.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62J05 Linear regression; mixed models 41A25 Rate of convergence, degree of approximation 41A35 Approximation by operators (in particular, by integral operators)
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