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Convergence of linear functionals of the Grenander estimator under misspecification. (English) Zbl 1302.62045

Convergence rates of Grenander estimation of the true misspecified density are treated.
Let \({\mathcal F}\) be the class of decreasing densities on \(\mathbb{R}_+\). Let \(X_1,\dots,X_n\) be an i.i.d. sequence of random variables with density \(f_0\in{\mathcal F}\). Define the KL projection of \(f_0\) as \[ \widehat f_0=\text{argmin}_{g\in{\mathcal F}}\int^\infty_0 f_0(x)\log{f_0(x)\over g(x)}\,dx \] and the Grenander estimation of \(f_0\) as \[ \widehat f_n= \text{argmin}_{g\in{\mathcal F}} \int^\infty_0 \log g(x)\,dF_n(x), \] where \(F_n\) is the empirical distribution function.
Let \(F_0(x)= \int^x_0 f_0(u)\,du\) and \(\widehat F_0\) be the least concave majorant of \(F_0\). Put \({\mathcal M}= \{x\geq 0: \widehat F_0(x)> F(x)\}\) and \({\mathcal W}=\{x\geq 0:\widehat F_0(x)= F(x)\}\). Let \([a, b]\) be the largest interval containing \(x_0\) such that \(\widehat F_0(x)\) is linear on the interval. Let \(\text{gren}_{[a,b]}(g)\) be the (left) derivative of the least concave majorant \(g\) restricted to \([a, b]\). Further, let \(U\) denote a standard Brownian bridge process on \([0, 1]\), \(U_{F_0}(x)= U(F_0(x))\) and \(U^{\text{mod}}_{F_0}(u) = U_{F_0}(u)\) if \(u\in [a,b]\cap{\mathcal W}\), \(U^{\text{mod}}_{F_0}(u) = -\infty\) if \(u\in[a, b]\cap{\mathcal M}\).
The author proves that as \(n\to\infty\) \[ \sqrt{n}(\widehat f_n(x_0)-\widehat f_0(x_0))\to \text{gren}_{[a,b]}(U^{\text{mod}}_{F_0}(x_0)) \] and, in addition, if \([a, b]\cap{\mathcal W}= \{a, b\}\), \[ \sqrt{n}(\widehat f_n(x_0)-\widehat f_0(x_0))\to \sigma Z, \] where \(Z\) is a standard normal random variable and \(\sigma\) is some constant defined by \(\widehat f_0(x_0)\).
The author also shows \(\sqrt{n}\)-convergence of linear functionals of the type
\[ \hat \mu_n(g) = \int_0^\infty g(x) \hat f_n(x) dx \]
and \(\sqrt{n}\)-convergence of the entropy functional in the misspecified case.

MSC:

62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
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