After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure...

We present new goodness-of-fit tests for the exponential distribution based on equidistribution type characterizations. For the construction of the test statistics, we employ an $L^2$-distance between the corresponding V-empirical distribution functions. The resulting test statistics are V-statistics, free of the scale parameter. The quality of the tests is assessed through local Bahadur efficiencies as well as the empirical power for small and moderate sample sizes. According to both criteria, for...

Highly robust statistical and econometric methods have been developed not only as a diagnostic tool for standard methods, but they can be also used as self-standing methods for valid inference. Therefore the robust methods need to be equipped by their own diagnostic tools. This paper describes diagnostics for robust estimation of parameters in two econometric models derived from the linear regression. Both methods are special cases of the generalized method of moments estimator based on implicit...

In the present paper, we establish the moderate and large deviations for the linear combinations of uniform order statistics. As applications, the moderate and large deviations for the $k$-th order statistics from uniform distribution, Gini mean difference statistics and the $k$-th order statistics from general continuous distribution are obtained.

The paper discusses applications of permutation arguments in testing problems in linear models. Particular attention will be paid to the application in L₁-test procedures. Theoretical results will beaccompanied by a simulation study.

We investigate controlling false discovery rate (FDR) under dependence. Our main result is a generalization of the results obtained by Genovese and Wasserman (2004) and Farcomeni (2007).

Let (X, Y) be a random couple in S×T with unknown distribution P. Let (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y), Pn being their empirical distribution. Let h1, …, hN:S↦[−1, 1] be a dictionary consisting of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let ℓ:T×ℝ↦ℝ be a given loss function, which is convex with respect to the second variable. Denote (ℓ•f)(x, y):=ℓ(y; f(x)). We study the following penalized empirical risk minimization problem $${\widehat{\lambda }}^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P_n(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right],$$ which is an empirical version of the problem $${\lambda }^{\epsilon }:=\underset{\lambda \in {ℝ}^N}{argmin}\left[P(\ell •f_{\lambda })+\epsilon {\parallel \lambda \parallel }_{{\ell }_p}^p\right]$$ (hereɛ≥0...

We discuss the prediction of a spatial variable of a multivariate mark composed of both dependent and explanatory variables. The marks are location-dependent and they are attached to a point process. We assume that the marks are assigned independently, conditionally on an unknown underlying parametric field. We compare (i) the classical non-parametric Nadaraya-Watson kernel estimator based on the dependent variable (ii) estimators obtained under an assumption of local parametric model where explanatory...

Given a random sample from some unknown density $f_0:ℝ\to [0,\infty )$ we devise Haar wavelet estimators for $f_0$ with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen and Spokoiny (Ann. Statist.25(1997) 927–947)). We show that these estimators satisfy an oracle inequality that adapts to heterogeneous smoothness of $f_0$, simultaneously for every point $x$ in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under...

We consider a stationary symmetric stable bidimensional process with discrete time, having the spectral representation (1.1). We consider a general case where the spectral measure is assumed to be the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. We estimate the density of the absolutely continuous measure and the density on the lines.

When we apply stacked regression to classification we need only discriminant indices which can be negative. In many situations, we want these indices to be positive, e.g., if we want to use them to count posterior probabilities, when we want to use stacked regression to combining classification. In such situation, we have to use leastsquares regression under the constraint βₖ ≥ 0, k = 1,2,...,K. In their earlier work [5], LeBlanc and Tibshirani used an algorithm given in [4]. However, in this paper...