caa a22 4500 47709225X CHVBK 20180405111518.0 cr unu---uuuuu 170330e19961201xx s 000 0 eng 10.1007/BF02589093 doi (NATIONALLICENCE)springer-10.1007/BF02589093 The multivariate Fermi-Dirac distribution and its applications in quality control [Elektronische Daten] [Mauro Gasparini, Peiming Ma] Summary: We investigate the multivariate elliptically contoured generalization of a parametric family of univariate distributions proposed by Ferreri (1964) for its potential applications in Quality Control. Such ap-variate Fermi-Dirac distribution has density $$f(x) = \frac{{\Gamma \left( {{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)\lambda ^p \left( \alpha \right)}}{{\pi ^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} F\left( {{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2} - 1,\alpha } \right)\left| \Sigma \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}\frac{1}{{1 + exp\left\{ {\alpha + \lambda ^2 \left( \alpha \right)\left( {x - \mu } \right)^\prime \Sigma ^{ - 1} \left( {x - \mu } \right)} \right\}}}$$ wherex, μ ∈R p, α ∈R, Σ is ap×p positive definite matrix of rankp and $$F(p,\alpha ): = \int_0^\infty {\frac{{u^p }}{{I + exp\left\{ {\alpha + u} \right\}}}du} $$ is the Fermi-Dirac integral used in statistical physics. The Fermi-Dirac family provides, through α, a one-dimensional continuous parametrization that joins the multivariate uniform distribution on an ellipsoid to the multivariate normal distribution. A discussion of maximum likelihood estimation of its parameters illustrates some interesting nonstandard phenomena. For example, as a by-product, a possible solution to the problem of circumscribing the smallest ellipsoid to a set of points inR p is obtained. The method is illustrated by a multivariate quality control example. Società Italiana di Statistica, 1996 Elliptically contoured distributions nationallicence platikurtosis nationallicence smallest enclosing ellipsoid nationallicence profile likelihood nationallicence Gasparini Mauro Biostatistics/MIPS, K-490.3.54, Novartis Pharma AG, CH-4002, Basel, Switzerland aut Ma Peiming University of Wisconsin-Stout, Wisconsin, USA aut Journal of the Italian Statistical Society Springer-Verlag 5/3(1996-12-01), 307-322 1121-9130 5:3<307 1996 5 10260 https://doi.org/10.1007/BF02589093 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02589093 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Gasparini Mauro Biostatistics/MIPS, K-490.3.54, Novartis Pharma AG, CH-4002, Basel, Switzerland aut NATIONALLICENCE 700 1- Ma Peiming University of Wisconsin-Stout, Wisconsin, USA aut NATIONALLICENCE 773 0- Journal of the Italian Statistical Society Springer-Verlag 5/3(1996-12-01), 307-322 1121-9130 5:3<307 1996 5 10260 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer