The $L_1$ Convergence of Kernel Density Estimates
Devroye, L. P. ; Wagner, T. J.
Ann. Statist., Tome 7 (1979) no. 1, p. 1136-1139 / Harvested from Project Euclid
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Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.
Publié le : 1979-09-14
Classification:  Density estimation,  integral convergence,  kernel estimates,  60F15,  62G05 
@article{1176344796,
     author = {Devroye, L. P. and Wagner, T. J.},
     title = {The $L\_1$ Convergence of Kernel Density Estimates},
     journal = {Ann. Statist.},
     volume = {7},
     number = {1},
     year = {1979},
     pages = { 1136-1139},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176344796}
}

Devroye, L. P.; Wagner, T. J. The $L_1$ Convergence of Kernel Density Estimates. Ann. Statist., Tome 7 (1979) no. 1, pp.  1136-1139. http://gdmltest.u-ga.fr/item/1176344796/