Evaluation of Gaussian orthant probabilities based on orthogonal projections to subspaces

In this paper, a new procedure is described for evaluating the probability that all elements of a normally distributed vector are non-negative, which is called the non-centered orthant probability. This probability is defined by a multivariate integral of the density function. The definition is simple, and this probability arises frequently in statistics because the normal distribution is prevalent. The method for evaluating this probability, however, is not obvious, because applying direct numerical integration is not practical except in low dimensional cases. In the procedure proposed in this paper, the problem is decomposed into sub-problems of lower dimension. Considering the projection onto subspaces, the solutions of the sub-problems can be shared in the evaluation of higher dimensional problems. Thus the sub-problems form a lattice structure. This reduces the computational time from a factorial order, where the interim results are not shared, to order \(p^{2}2^{p}\), which is faster than the procedures that have been reported in the literature.     

Partition of a query set into minimal number of subsets having consecutive retrieval property

This paper is concerned with information retrieval. The basic problem is how to store large masses of data in such a way that whenever information regarding some particular aspect of the data is needed, such information is easily and efficiently retrieved. Work in this field is thus very important for organizations dealing with large classes of data.The consecutive retrieval (C-R) property defined by S.P. Ghosh is an important relation between a set of queries and a set of records. Its existence enables the design of information retrieval system with a minimal search time and no redundant storage in that the records can be organized in such a way that those pertinent to any query are stored in consecutive storage locations. The C-R property, however, can not exist between every arbitrary query set and every record set.A subset of the query set Q having the C-R property is called a C-R subset and a C-R subset having the maximum cardinality is called the maximal C-R subset. A partition of Q is called the C-R partition if every subset has the C-R property. A C-R partition with minimum number of subsets is called the minimal C-R partition. With respect to the set of all binary queries and the set of all binary records, it is shown that the maximal cardinality of a C-R subset is 2l-1 where l is the number of attributes concerned. A combinatorial characterization of a maximal C-R subset is also given. A lower bound on the number of subsets in a C-R partition and several examples which attain the lower bound are given. A general procedure for obtaining a minimal C-R partition which attains the lower bound is given provided the number of attributes is even.     

Orthant probabilities of elliptical distributions from orthogonal projections to subspaces

Statistics and Computing - A new procedure is proposed for evaluating non-centred orthant probabilities of elliptical distributed vectors, which is the probabilities that all elements of a vector...