Consistent estimates of the mode of the probability density function in nonparametric deconvolution problems

We propose two estimates of the mode of the probability density function for nonparametric deconvolution problems. In fact, we observe Y=X+[xi], where [xi] is a measurement error with a known distribution f[xi], and we are interesting by estimating the mode of fX the unknown probability density function of X, where Y1,...,Yn are n i.i.d given observations of Y. We study the asymptotic properties of these mode estimates and the asymptotic normality of the two mode estimates is also given.