Recurrence Relations for Equi-Coordinate and Orthant Probabilities

• Published in: Sankhyā. Series B 1983-04, Vol.45 (1), p.30-49
• Main Author: Six, F. B.
• Format: Article
• Language: English
• Subjects: Approximation; Gaussian distributions; Mathematical moments; Probabilities; Recurrence relations; Sample mean; Sampling distributions; Statistical variance; Statistics
• ISSN: 0581-5738

The equi-coordinate probability is the probability that a particular r of a collection of n variates are less than or equal to a constant, t, with the remaining n-r variates greater than t. The orthant probability,$p_{r\colon n}$, which gives the distribution of signs of the variates, is the special case t = 0. The purpose of this paper is to present several new recurrence relations among equi-coordinate probabilities and also among the distributions of signs. These permit us to calculate the$p_{r\colon n}$given the values of$p_{2m\colon 2m}$for integral m, 0 < m ≤ n/2. For equi-correlated normal variates with correlation ρ, -1/(n-1)< ρ < 1, these relations, together with calculating the$p_{2m\colon 2m}$directly from simple integrals, provide a feasible method for calculating the$p_{r\colon n}$. For ρ = -1/(n-1), Steck's (1962) recurrence relation can be used with our results to calculate the$p_{r\colon n}$. Exact values for n less than and equal to ten are obtained in this case. This constitutes the small sample solution of Youden's 'Demon Problem' which is to find the probability that the sample mean lies between the rth and (r+1)-st order statistic in a sample of n normally distributed variates.