The Gaussian rank correlation equals the usual correlation coefficient computed from the normal scores of the data. Although its influence function is unbounded, it still has attractive robustness properties. In particular, its breakdown point is above 12%. Moreover, the estimator is consistent and asymptotically efficient at the normal distribution. The correlation matrix obtained from pairwis...

The rank-based fit of a linear model is based on minimizing a norm. A score function needs to be selected for the fit and the proper choice leads to asymptotically efficient regression estimators, i.e., fits equivalent to the maximum likelihood estimators (mle). In this paper, we present the family of optimal scores functions for the skew-normal family of distributions. We show the easy computa...

Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic informa...

We show that the probability to be of rank 2 for a 2× 2× 2 tensor with elements from a standard normal distribution is π/4, and that the probability to be of rank 3 for a 3× 3× 2 tensor is 1/2. In the proof results on the expected number of real generalized eigenvalues of random matrices are applied. For n × n × 2 tensors with n ≥ 4 we also present some new aspects of their rank.

We use this theorem to derive the same corollaries for the theories covered by the theorem as were derived for the strongly minimal case in [2]. We also note that the theorem is in some senses optimal. Specifically we can easily construct trivial Morley Rank 1 theories which are not categorical and for which the conclusion of the theorem fails. Also Marker in [3] constructs trivial totally cate...

and Applied Analysis 3 Problem 1 into the fixed rank solution of a matrix equation. Finally, we establish an algorithm for solving Problem 1. Lemma 2. AmatrixX ∈ Rm×n is a solution of Problem 1 if and only if it is a solution of the following matrix equation: XBB T = AB T , rank (X) = d. (12) Proof. It is easy to verify that a matrixX ∈ Rm×n is a solution of Problem 1 if and only if X satisfies...

Sufficient conditions are given for groups of finite Morley rank having non-trivial torsion-free nilpotent normal subgroups to have linear representations with small kernels. In particular, centreless connected soluble groups of finite Morley rank with torsion-free Fitting subgroups have faithful linear representations. On the way, using a notion of definable weight space, we prove that certain...