We study the estimation of a linear integral functional of a distribution F, using i.i.d. observations which density is a mixture of a family of densities k(.,y) under F. We examine the asymptotic distribution of the estimator obtained by plugging the non parametric maximum likelihood estimator (NPMLE) of F in the functional. A problem here is that usually, the NPMLE does not dominate F.
Our main aim here is to show that this can be overcome by considering a convex combination of F and the...

Let $X_j,1\le j\le N$, be independent random $p$-vectors with respective continuous cumulative distribution functions $F_{j}1\le j\le N$. Define the $p$-vectors $R_j$ by setting $R_{ij}$ equal to the rank of $X_{ij}$ among $X_{ij},...,X_{iN},1\le i\le p,1\le j\le N$. Let $a^{\left(N\right)}(.)$ denote a multivariate score function in $R_p$, and put $S={\sum }_{j=1}^N{c}_j{a}^{\left(N\right)}\left(R_j\right)$, the $c_j$ being arbitrary regression constants. In this paper the asymptotic distribution of $S$ is investigated under various sets of conditions on the constants, the score functions, and the underlying distribution functions. In particular, asymptotic normality of $S$ is established...

Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T(X₁,...,Xₙ)=n^{-1}{\sum }_{1\le iLet{q}_i\left(i\ge 1\right)beeigenvaluesoftheHilbert-Schmidtoperatorassociatedwiththekernelh(x,y),andq₁bethelargestinabsolutevalueone.Weprovethat}$Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12)$,$where $G_i$, 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and ${\beta }^{\text{'}}:=E\left|h(X,X₁)\right|³+{E\left|h(X,X₁)\right|}^{18/5}+E\left|g\left(X\right)\right|³+{E\left|g\left(X\right)\right|}^{18/5}+1<\infty$.

Asymptotic study of canonical correlation analysis gives the opportunity to present the different steps of an asymptotic study and to show the interest of an operator and tensor approach of multidimensional asymptotic statistics rather than the classical, matrix and analytic approach. Using the last approach, Anderson (1999) assumes the random vectors to have a normal distribution and the non zero canonical correlation coefficients to be distinct. The new approach we use, Fine (2000), is coordinate-free,...

By using three theorems (Oktaba and Kieloch [3]) and Theorem 2.2 (Srivastava and Khatri [4]) three results are given in formulas (2.1), (2.8) and (2.11). They present asymptotically normal confidence intervals for the determinant $|{\sigma }^2\sum |$ in the MGM model $(U,XB,{\sigma }^2\sum \otimes V)$, $\sum >0$, scalar ${\sigma }^2>0$, with a matrix $V\ge 0$. A known $n\times p$ random matrix $U$ has the expected value $E\left(U\right)=XB$, where the $n\times d$ matrix $X$ is a known matrix of an experimental design, $B$ is an unknown $d\times p$ matrix of parameters and ${\sigma }^2\sum \otimes V$ is the covariance matrix of $U,\otimes$ being the symbol of the Kronecker...

Linear relations, containing measurement errors in input and output data, are taken into account in this paper. Parameters of these so-called errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances. Such an estimate is highly non-linear. Moreover in some realistic situations, the errors cannot be considered as independent by nature. Weakly dependent ($\alpha$- and $\varphi$-mixing) disturbances, which are not necessarily stationary nor identically...

We consider the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions. A diagnostic procedure for assessing homogeneity of the conditional densities is also proposed.

The purpose of this paper is to study Bayesian like R- and M-estimators of change point(s). These estimators have smaller variance than the related argmax type estimators. Confidence intervals for the change point based on the exchangeability arguments are constructed. Finally, theoretical results are illustrated on the real data set.

We introduce the function $Z(x;\xi ,\nu ):={\int }_{-\infty }^x\varphi (t-\xi )·\Phi \left(\nu t\right)\text{d}t$, where $\varphi$ and $\Phi$ are the pdf and cdf of $N(0,1)$, respectively. We derive two recurrence formulas for the effective computation of its values. We show that with an algorithm for this function, we can efficiently compute the second-order terms of Bonferroni-type inequalities yielding the upper and lower bounds for the distribution of a max-type binary segmentation statistic in the case of small samples (where asymptotic results do not work), and in general for max-type random variables...