Some Applications of Interval Analysis to Statistical Problems

Abstract. This paper contribution is about guaranteed numerical methods based on interval analysis for approximating sets, and about the application of these methods to vast classes of statistical problems. ’Guaranteed ’ means here the inner and outer approximations of the sets of interest are obtained, which can be made as precise as desired, at the cost of increasing the computational effort. It thus becomes possible to archieve tasks still thought by many to be out of the reach of numerical methods, such as finding all solutions of sets of non-linear equations and inequalities or a global optimizer of possible multi-modal criteria. Notations Throughout the text, the following conventions are used: lower case letters in italics such as x or yi denote scalar variables and elements of vectors. Vectors are printed in bold. A row vector is denoted by the transpose operator, i.e. xT. Uppercase bold characters denote matrices, for instance X.