We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannons entropy. As a consequence, an asymptotic 100% confidence interval of entropy is provided.
We derive asymptotic normality of kernel type deconvolution estimators of the density, the distribution function at a fixed point, and of the probability of an interval. We consider the so called super smooth case where the characteristic function of the known distribution decreases exponentially. It turns out that the limit behavior of the pointwise estimators of the density and distribution function is relatively straightforward while the asymptotics of the estimator of the probability of an interval depends in a complicated way on the sequence of bandwidths.
In the Gaussian white noise model, we study the estimation of an unknown multidimensional function $f$ in the uniform norm by using kernel methods. The performances of procedures are measured by using the maxiset point of view: we determine the set of functions which are well estimated (at a prescribed rate) by each procedure. So, in this paper, we determine the maxisets associated to kernel estimators and to the Lepski procedure for the rate of convergence of the form $(log n/n)^{be/(2be+d)}$. We characterize the maxisets in terms of Besov and Holder spaces of regularity $beta$.
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process of order 1.
We derive asymptotic normality of kernel type deconvolution density estimators. In particular we consider deconvolution problems where the known component of the convolution has a symmetric lambda-stable distribution, 0<lambda<= 2. It turns out that the limit behavior changes if the exponent parameter lambda passes the value one, the case of Cauchy deconvolution.
The paper discusses the estimation of a continuous density function of the target random field $X_{bf{i}}$, $bf{i}in mathbb {Z}^N$ which is contaminated by measurement errors. In particular, the observed random field $Y_{bf{i}}$, $bf{i}in mathbb {Z}^N$ is such that $Y_{bf{i}}=X_{bf{i}}+epsilon_{bf{i}}$, where the random error $epsilon_{bf{i}}$ is from a known distribution and independent of the target random field. Compared to the existing results, the paper is improved in two directions. First, the random vectors in contrast to univariate random variables are investigated. Second, a random field with a certain spatial interactions instead of i. i. d. random variables is studied. Asymptotic normality of the proposed estimator is established under appropriate conditions.