Adaptive wavelet estimation of a density from mixtures under multiplicative censoring

Abstract

In this paper, a mixture model under multiplicative censoring is considered. We investigate the estimation of a component of the mixture (a density) from the observations. A new adaptive estimator based on wavelets and a hard thresholding rule is constructed for this problem. Under mild assumptions on the model, we study its asymptotic properties by determining an upper bound of the mean integrated squared error over a wide range of Besov balls. We prove that the obtained upper bound is sharp.

A Note on the Adaptive Estimation of a Multiplicative Separable Regression Function

“…Then we adapt our methodology to propose an efficient and adaptive procedure. It is based on two wavelet thresholding estimators following the construction studied in Chaubey et al [6]. It has the features to be adaptive for a wide class of unknown functions and enjoy nice MISE properties.…”

“…To be more specific, we use the "double thresholding" wavelet technique, introduced by Delyon and Juditsky [15] then recently improved by Chaubey et al [6]. The role of the second thresholding (appearing in the definition of the wavelet estimator for , ) is to relax assumption on the model (see Remark 6).…”

Estimation of the Derivatives of a Function in a Convolution Regression Model with Random Design

“…Let us now present in detail the general result of [25,Theorem 6.1] used in the proof of Theorem 3. We consider the wavelet basis presented in Section 2 and a general form of the hard thresholding wavelet estimator denoted bŷfor estimating an unknown function ∈ L 2 ([ , ]) from independent random variables 1 , .…”

“…The proof of Theorem 3 is an application of a general result established by [25,Theorem 6.1]. Let us mention that (ln / ) 2 /(2 +2 +2 +1) corresponds to the rate of convergence obtained in the estimation of ( ) in the 1-periodic white noise convolution model with an adapted hard thresholding wavelet estimator (see, e.g., Chesneau [15]).…”

“…uses the double hard thresholding technique introduced by Delyon and Juditsky [14] and recently improved by Chaubey et al [25]. The main interest of the thresholding using is to makê( does not depend on the knowledge of the smoothness of ( ) .…”