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Averaged shifted histograms: Effective nonparametric density estimators in several dimensions. (English) Zbl 0589.62022
Let $$X_ 1,X_ 2,..$$. be i.i.d. from some unknown density f in $${\mathbb{R}}$$. For a given bin-width $$h>0$$ and an integer $$m\geq 1$$ let, for $$0\leq i\leq m-1$$, $$\hat a_ i$$ be the histogram for the grid $$rh+ih/m$$, $$r\in {\mathbb{Z}}.$$
The author proposes $$\hat f_ n$$, the average of the $$\hat a_ i's$$, as an estimator for f, and derives an expansion for the IMSE both as a function of h and m. It turns out that as $$m\to \infty$$, $$\hat f_ n$$ behaves like a kernel estimate with triangular kernel. Also the multivariate case is discussed.
Reviewer: W.Stute

MSC:
 62G05 Nonparametric estimation 62E10 Characterization and structure theory of statistical distributions 62H12 Estimation in multivariate analysis
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