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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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