## 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

### Software:

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