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Title: Nonparametric kernel estimation of unrestricted distributions Authors:  Carlos Martins-Filho - University of Colorado at Boulder (United States) [presenting]
Kairat Mynbayev - Kazakh-British Technical University (Kazakhstan)
Abstract: Nonparametric estimation of an unrestricted distribution $F$ which may, or may not, be absolutely continuous, is considered. First, for a point of continuity of $F$, $x$, we consider estimators that can be expressed as $\hat{F}_n(x)=(1/n)\sum_{i=1}^n U((X_i-x)/h)$, for a suitable choice of $U$ and a bandwidth $h>0$. We obtain the rates of convergence of these estimators to $F(x)$. Contrary to the extant literature, we make no restriction on the existence or smoothness of the derivatives of $F$. The key insight for the result is the use of Lebesgue-Stieltjes integrals. A special case of $\hat{F}_n(x)$, that reproduces the traditional kernel estimator, is given when $U(x)=\int_x^\infty K(u)du$ and $K$ is a kernel. Second, for $x$ that is either a point where $F$ has a jump discontinuity, or isolated, we obtain rates of convergence for an estimator $\hat{J}(x)=(1/n)\sum_{i=1}^n W((X_i-x)/h)$ for the jump $F(x)-F(x-)$ and suitable choice of $W$. Once again, no restriction is imposed on $F$ beyond right-continuity. A suitable choice is $W(x)=\int_\mathbb{R}e^{ixu}K(u)du$, the Fourier transform of a kernel $K$. The results are of significant practical use, as there are numerous examples of distributions that have mass points and singularities in Economics, Finance and Biomedicine.