We present a first moment distribution-free bound on expected values of L-statistics as well as properties of some numerical characteristics of order statistics, in the case when the observations are possibly dependent symmetrically distributed about the common mean. An actuarial interpretation of the presented bound is indicated.

The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_i,Y_i)$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^1$[a,b]. The constructed estimators are of the form ${\widehat{f}}_n\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\widehat{c}}_k{e}_k\left(x\right)$, where the coefficients ${\widehat{c}}_0,{\widehat{c}}_1,...,{\widehat{c}}_N$ are determined by minimizing the empirical risk $n^{-1}{\sum }_{i=1}^n{(Y_i-{\sum }_{k=0}^N{c}_k{e}_k\left(X_i\right))}^2$. Sufficient conditions for...

Let $\left\{X_n\right\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set $𝒳$ and that $f\left(X\right)$ is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times ${\lambda }_n$ along which we will be able to estimate the conditional expectation $E\left(f\left(X_{{\lambda }_n+1}\right)\right|X_0,\cdots ,X_{{\lambda }_n})$ from the observations $(X_0,\cdots ,X_{{\lambda }_n})$ in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series...

In a previous paper we have studied the relevant analogies between the variance, applied to a compound scheme of probability and utility, and the measure which we had defined to evaluate the unquietness for such a compound scheme.The purpose of the present note is to display the advantage exhibited by the second measure, with respect to the first one, in quantifying the uncertainty corresponding to the utilities. This advantage consists of the larger ability to distinguish among the different compound...

Computationally attractive Fisher consistent robust estimation methods based on adaptive explanatory variables trimming are proposed for the logistic regression model. Results of a Monte Carlo experiment and a real data analysis show its good behavior for moderate sample sizes. The method is applicable when some distributional information about explanatory variables is available.

In this note we give an elementary proof of a characterization for stability of multivariate distributions by considering a functional equation.

Recently Balakrishnan and Iliopoulos [Ann. Inst. Statist. Math. 61 (2009)] gave sufficient conditions under which the maximum likelihood estimator (MLE) is stochastically increasing. In this paper we study test plans which are not considered there and we prove that the MLEs for those plans are also stochastically ordered. We also give some applications to the estimation of reliability.

In this note we consider the estimation of the differential entropy of a probability density function. We propose a new adaptive estimator based on a plug-in approach and wavelet methods. Under the mean ${𝕃}_p$ error, $p\ge 1$, this estimator attains fast rates of convergence for a wide class of functions. We present simulation results in order to support our theoretical findings.

In statistics of stochastic processes and random fields, a moment function or a cumulant of an estimate of either the correlation function or the spectral function can often contain an integral involving a cyclic product of kernels. We define and study this class of integrals and prove a Young-Hölder inequality. This inequality further enables us to study asymptotics of the above mentioned integrals in the situation where the kernels depend on a parameter. An application to the problem of estimation...