The Generalised Plug-in Algorithm for the Diffeomorphism Kernel Estimate

The optimal value of the smoothing parameter of the Kernel estimator can be obtained by the well known Plug-in algorithm. The optimality is realised in the sense of Mean Integrated Square Error (MISE). In this paper, we propose to generalise this algorithm to the case of the difficult problem of the estimation of a distribution which has a bounded support. The proposed algorithm consists in searching the optimal smoothing parameter by iterations from the expression of MISE of the kernel-diffeomorphism estimator. By some simulations applied to some distribution having a support bounded and semi bounded, we show that the support of the pdf estimator respects the one of the theoretical distribution. We also prove that the proposed method minimizes the Gibbs phenomenon.

Self-Consistent Density Estimation in the Presence of Errors-in-Variables

This paper considers the estimation of the common probability density of independent and identically distributed variables observed with additive measurement errors. The self-consistent estimator of the density function is constructed when the error distribution is known, and a modification of the self-consistent estimation is proposed when the error distribution is unknown. The consistency properties of the proposed estimators and the upper bounds of the mean square error and mean integrated square error are investigated under some suitable conditions. Simulation studies are carried out to assess the performance of our proposed method and compare with the usual deconvolution kernel method. Two real datasets are analyzed for further illustration.

K-nearest neighbour kernel density estimation, the choice of optimal k

ABSTRACT The k-nearest neighbour kernel density estimationmethod is a special type of the kernel density estimation method with the local choice of the bandwidth. An advantage of this estimator is that smoothing varies according to the number of observations in a particular region. The crucial problem is how to estimate the value of the parameter k. In the paper we discuss the problem of choosing the parameter k in a way that minimizes the value of the asymptotic mean integrated square error (AMISE). We define the class of the modified cosine densities that meet the requirements given by the AMISE. The results are compared in a simulation study.

Adaptive Wavelet Estimation of a Biased Density for Strongly Mixing Sequences

The estimation of a biased density for exponentially strongly mixing sequences is investigated. We construct a new adaptive wavelet estimator based on a hard thresholding rule. We determine a sharp upper bound of the associated mean integrated square error for a wide class of functions.