Volume 22, Issue 4 - 4

Summury For a probability measure P on R d and n ∊ N consider e n = inf ∫ min a∊α V (|| x − a ||) dP ( x ) where the infimum is taken over all subsets α of R d with card(α) ≤ n and V is a nondecreasing function. Under certain conditions on V , we derive the precise n -asymptotics of e n for nonsingular distributions P and we find the asymptotic performance of optimal quantizers using weighted empirical measures.

Summury In this note we consider the problem of confidence estimation of the covariance function of a stationary or locally stationary zero mean Gaussian process. The constructed confidence intervals are based on the usual empirical covariance estimate and a special estimate of its variance. The results about coverage probability are stated in a nonasymptotic way and apply for small and moderate sample size under mild conditions on the model. The presented numerical results are in agreement with the theoretical issues and demonstrate applicability of the method.

Summury In this paper we construct efficient estimators of linear functionals of a bivariate distribution with equal marginals. The proposed estimator generalizes the construction of efficient estimators given by Bickel, Ritov and Wellner (1991) for the case of known, but not necessarily equal, marginals.

Summury Let ( X i,j , Y i,j ), i = 1,…, n , j = 1,2, be a sample from two populations, where the X i,j are d -dimensional covariates which have an effect on the response variable Y i,j . It is assumed that the conditional distribution of Y i,j given X i,j = x is Q g(αj + βj T x ) where { Q ϑ | ϑ ∊ Θ}, Θ ⊆ R, is a parent family, g is the so-called link function and ϑ j = (α j ,β j ) the parameters of interest. Using the LAN theory, a sequence of locally asymptotically optimal tests φ ^ n for H 0 : ϑ 1 = ϑ 2 versus H A : ϑ 1 ≠ ϑ 2 is constructed for an unknown link function g . These tests are asymptotic maximin-tests and adaptive in the sense that the plugging-in of an estimator for the nuisance parameters g does not reduce the local asymptotic power compared to the situation of a known nuisance parameter g . To attain exact α-tests even for finite sample size a permutation test version is given with the same local asymptotic power as φ ^ n .

Summury We consider a two-phase random design nonlinear regression model, the regression function is discontinuous at the change-point. The errors ∊ are arbitrary, with E(∊) = 0 and E(∊ 2 ) < ∞. We prove that Koul and Qian’s results [12] for linear regression still hold true for the nonlinear case. Thus the maximum likelihood estimator r ^ n of the change-point r is n -consistent and the estimator θ ^ 1n of the regression parameters θ 1 is n 1/2 -consistent. The asymptotic distribution of n 1/2 (θ ^ 1n − θ 0 1 ) is Gaussian and n ( r ^ n − r ) converges to the left end point of the maximizing interval with respect to the change point. The likelihood process is asymptotically equivalent to a compound Poisson process.