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Large-sample inference for log-spline models. (English) Zbl 0712.62036
Let Y be a random variable which takes values in a subinterval \({\mathcal Y}\) of \({\mathcal R}\) and has an unknown density f on \({\mathcal Y}\). Let F be the distribution function of f and Q its quantile function. As approximations of f, for each n the author constructs finite-parameter exponential models \(\{\) f(\(\cdot;\theta)\), \(\theta \in \Theta_ n\}\) for f, based on B-splines where \(\Theta_ n\) is a convex open subset of \({\mathcal R}^ J\) and \(J\to \infty\) as \(n\to \infty.\)
Next, using a sample of n independent, identically distributed replications \(Y_ 1,...,Y_ n\) of \({\mathcal Y}\), he considers maximum likelihood estimation of the parameters of the models yielding estimates \(\hat f,\) \(\hat F\) and \(\hat Q\) of f, F and Q, respectively. Then, he shows that under mild conditions these estimates achieve the optimal rate of convergence and clarifies the asymptotic behavior of the corresponding confidence bounds.
Reviewer: K.-i.Yoshihara

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
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