DECONVOLVING A DENSITY FROM CONTAMINATED DEPENDENT OBSERVATIONS

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 ∈ (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 L_{∞}. 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.