Polynomial distribution functions on bounded closed intervals

The thesis explores several topics, related to polynomial distribution functions and their densities on [0,1]M, including polynomial copula functions and their densities. The contribution of this work can be subdivided into two areas. - Studying the characterization of the extreme sets of polynomial densities and copulas, which is possible due to the Choquet theorem. - Development of statistical methods that utilize the fact that the density is polynomial (which may or may not be an extreme density). With regard to the characterization of the extreme sets, we first establish that in all dimensions the density of an extreme distribution function is an extreme density. As a consequence, characterizing extreme distribution functions is equivalent to characterizing extreme densities, which is easier analytically. We provide the full constructive characterization of the Choquet-extreme polynomial densities in the univariate case, prove several necessary and sufficient conditions for the extremality of densities in arbitrary dimension, provide necessary conditions for extreme polynomial copulas, and prove characterizing duality relationships for polynomial copulas. We also introduce a special case of reflexive polynomial copulas. Most of the statistical methods we consider are restricted to the univariate case. We explore ways to construct univariate densities by mixing the extreme ones, propose non-parametric and ML estimators of polynomial densities. We introduce a new procedure to calibrate the mixing distribution and propose an extension of the standard method of moments to pinned density moment matching. As an application of the multivariate polynomial copulas, we introduce polynomial coupling and explore its application to convolution of coupled random variables. The introduction is followed by a summary of the contributions of this thesis and the sections, dedicated first to the univariate case, then to the general multivariate case, and then to polynomial copula densities. Each section first presents the main results, followed by the literature review

Polynomial distribution functions on bounded closed intervals

The thesis explores several topics, related to polynomial distribution functions and their densities on [0,1]M, including polynomial copula functions and their densities. The contribution of this work can be subdivided into two areas. - Studying the characterization of the extreme sets of polynomial densities and copulas, which is possible due to the Choquet theorem. - Development of statistical methods that utilize the fact that the density is polynomial (which may or may not be an extreme density). With regard to the characterization of the extreme sets, we first establish that in all dimensions the density of an extreme distribution function is an extreme density. As a consequence, characterizing extreme distribution functions is equivalent to characterizing extreme densities, which is easier analytically. We provide the full constructive characterization of the Choquet-extreme polynomial densities in the univariate case, prove several necessary and sufficient conditions for the extremality of densities in arbitrary dimension, provide necessary conditions for extreme polynomial copulas, and prove characterizing duality relationships for polynomial copulas. We also introduce a special case of reflexive polynomial copulas. Most of the statistical methods we consider are restricted to the univariate case. We explore ways to construct univariate densities by mixing the extreme ones, propose non-parametric and ML estimators of polynomial densities. We introduce a new procedure to calibrate the mixing distribution and propose an extension of the standard method of moments to pinned density moment matching. As an application of the multivariate polynomial copulas, we introduce polynomial coupling and explore its application to convolution of coupled random variables. The introduction is followed by a summary of the contributions of this thesis and the sections, dedicated first to the univariate case, then to the general multivariate case, and then to polynomial copula densities. Each section first presents the main results, followed by the literature review.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

In Vivo Assessment of Water Content, Trans-Epidermial Water Loss and Thickness in Human Facial Skin

Mapping facial skin in terms of its biophysical properties plays a fundamental role in many practical applications, including, among others, forensics, medical and beauty treatments, and cosmetic and restorative surgery. In this paper we present an in vivo evaluation of the water content, trans-epidermial water loss and skin thickness in six areas of the human face: cheeks, chin, forehead, lips, neck and nose. The experiments were performed on a population of healthy subjects through innovative sensing devices which enable fast yet accurate evaluations of the above parameters. A statistical analysis was carried out to determine significant differences between the facial areas investigated and clusters of statistically-indistinguishable areas. We found that water content was higher in the cheeks and neck and lower in the lips, whereas trans-epidermal water loss had higher values for the lips and lower ones for the neck. In terms of thickness the dermis exhibited three clusters, which, from thickest to thinnest were: chin and nose, cheek and forehead and lips and neck. The epidermis showed the same three clusters too, but with a different ordering in term of thickness. Finally, the stratum corneum presented two clusters: the thickest, formed by lips and neck, and the thinnest, formed by all the remaining areas. The results of this investigation can provide valuable guidelines for the evaluation of skin moisturisers and other cosmetic products, and can help guide choices in re-constructive/cosmetic surgery