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DECONVOLVING A DENSITY FROM CONTAMINATED

DEPENDENT OBSERVATIONS

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CHRISTIAN H. HESSE

*Mathematisches Institut A, Universität Stuttgart, 70569
Stuttgart, Germany*
(Received August 22, 1994; revised March 20, 1995)

**Abstract.**
The paper studies the performance of
deconvoluting kernel density estimators for estimating the marginal
density of a linear process. The data stem from the linear process and
are partially, respectively fully contaminated by iid errors with a
known distribution. If 1 - *p* denotes the proportion of contaminated
observations (and it is , of course, unknown which observations are
contaminated and which are not) then for 1 - *p* \in (0, 1) and under
mild conditions almost sure deconvolution rates of order
*O*(*n*^{-2/5}(log *n*)^{9/10}) can be achieved for convergence in
\cal L_{\infty}. This rate compares well with the existing rates
for *iid uncontaminated* observations. For *p* = 0 and
exponentially decreasing error characteristic function the
corresponding rates are of merely logarithmic order. As a by-product
the paper also gives a rate of convergence result for the empirical
characteristic function in the linear process context and utilizes
this to demonstrate that deconvoluting kernel density estimators
attain the optimal rate in the dependence case with exponentially
decreasing error characteristic function.

*Key words and phrases*:
Deconvolution, density
estimation, contamination, identifiability, dependence.

**Source**
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