A modified Prekopa's approach is considered for the problem of optimum
allocation in multivariate stratified random sampling. An example is solved by
applying the proposed methodology.

The allocation problem for multivariate stratified random sampling as a
problem of stochastic matrix integer mathematical programming is considered.
With these aims the asymptotic normality of sample covariance matrices for each
strata is established. Some alternative approaches are suggested for its
solution. An example is solved by applying the proposed techniques.

This work sets the statistical affine shape theory in the context of real
normed division algebras. The general densities apply for every field: real,
complex, quaternion, octonion, and for any noncentral and non-isotropic
elliptical distribution; then the separated published works about real and
complex shape distributions can be obtained as corollaries by a suitable
selection of the field parameter and univariate integrals involving the
generator elliptical function.

This paper proposes a unified approach to enable the study of diverse
distributions in the real, complex, quaternion and octonion cases,
simultaneously. In particular, the central, nonsingular matricvariate and
matrix multivariate Pearson type II distribution, beta type I distributions and
the joint density of the singular values are obtained for real normed division
algebras.

In this work it is propose an alterative proof of one of basic properties of
the zonal polynomials. This identity is generalised for the Jack polynomials.

The non isotropic and non central elliptical shape distributions via the Le
and Kendall SVD decomposition approach are derived in this paper in the context
of invariant polynomials and zonal polynomials. The so termed cone and disk
densities here obtained generalise some results of the literature. Finally,
some particular densities are applied in a classical data of Biology, and the
inference is performed after choosing the best model by using a modified BIC
criterion.

This work sets the non isotropic noncentral elliptical shape distributions
via QR decomposition in the context of zonal polynomials, avoiding the
invariant polynomials and the open problems for their computation. The new
shape distributions are easily computable and then the inference procedure can
be studied under exact densities instead under the published approximations and
asymptotic densities under isotropic models. An application in Biology is
studied under the classical gaussian approach and a two non gaussian models.