The nonparametric estimation for the density and hazard rate functions for right-censored data using the kernel smoothing techniques is considered. The “classical” fixed symmetric kernel type estimator of these functions performs well in the interior region, but it suffers from the problem of bias in the boundary region. Here, we propose new estimators based on the gamma kernels for the density...

Let X1, . . . ,Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log-concave maximum likelihood estimator f̂n of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof ...

Let X1, . . . , Xn be i.i.d. observations, where Xi = Yi + Zi and Yi and Zi are independent. Assume that unobservable Y ’s are distributed as a random variable UV, where U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f and U and V are independent. Furthermore, let the random variables Zi have the standard normal distribution. ...

Copula modelling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily a↵ected by boundary bias issues. In addition, most common copulas admit unbounded ...

In this paper we consider a kernel estimator of a density in a convolution model and give a central limit theorem for its integrated square error (ISE). The kernel estimator is rather classical in minimax theory when the underlying density is recovered from noisy observations. The kernel is fixed and depends heavily on the distribution of the noise, supposed entirely known. The bandwidth is not...

Length-biased data are widely seen in applications. They are mostly applicable in epidemiological studies or survival analysis in medical researches. Here we aim to propose a Berry-Esseen type bound for the kernel density estimator of this kind of data.The rate of normal convergence in the proposed Berry-Esseen type theorem is shown to be O(n^(-1/6) ) modulo logarithmic term as n tends to infin...

Recent advances of kernel methods have yielded a framework for nonparametric statistical inference called RKHS embeddings, in which all probability distributions are represented as elements in a reproducing kernel Hilbert space, namely kernel means. In this paper, we consider the recovery of the information of a distribution from an estimate of the kernel mean, when a Gaussian kernel is used. T...

In this tutorial paper we give an overview of deconvolution problems in nonparametric statistics. First, we consider the problem of density estimation given a contaminated sample. We illustrate that the classical Rosenblatt-Parzen kernel density estimator is unable to capture the full shape of the density while the presented method experiences almost no problems. Second, we use the previous est...

• Boundary effects for kernel estimators of curves with compact supports are well known in regression and density estimation frameworks. In this paper we address the use of boundary kernels for distribution function estimation. We establish the ChungSmirnov law of iterated logarithm and an asymptotic expansion for the mean integrated squared error of the proposed estimator. These results show t...

We develop an integration by parts technique for point processes, with application to the computation of sensitivities via Monte Carlo simulations in stochastic models with jumps. The method is applied to density estimation and to the construction of a modified kernel estimator which is less sensitive to variations of the bandwidth parameter than standard kernel estimators. Simulations are pres...