The multivariate Fermi-Dirac distribution and its applications in quality control

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.