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...

We consider a multivariate regression (growth curve) model of the form $Y=XBZ+\epsilon$, $E\epsilon =0$, $var(vec\epsilon )=W\otimes \Sigma$, where $W={\sum }_{i=1}^k{\theta }_i{V}_i$ and ${\theta }_i$’s are unknown scalar covariance components. In the case of replicated observations, we derive the explicit form of the locally best estimators of the covariance components under normality and asymptotic confidence ellipsoids for certain linear functions of the first order parameters $\left\{B_{ij}\right\}$ estimating simultaneously the first and the second order parameters.

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...