© 1967 by Biometrika Trust
Distribution of the residual sum of squares in fitting inequalities
Princeton University
Consider a model for n observations, Yi = µi + 
i (i = 1,..., n), where the i are n independent unit normal variables and the µi are restrained by p linear inequalities. The maximum-likelihood estimate of {µi} is {Yoi} minimizing 
(Yi – Yoi)2 subject to the linear inequalities; the computation of {Yoi} requires quadratic programming. This paper is concerned with the distribution of the residual sum of squares (Yi – Yoi)2 which it is natural to use for inference about 2. Using the Kuhn-Tucker conditions, which the Yoi must satisfy, upper and lower bounds are obtained for the percentage points of (Yi – Yoi)2/2.The upper bound is (Yi–Yoi)2/2
Xn–k2 which holds conditionally on exactly n–k independent linear inequalities being satisfied as equations by the Yoi A direct lower bound u
, n, n–k is given with (n–k) regarded as a random variable. A more satisfactory Bayesian lower bound ¾X
(n–k)2(Yi – Yoi)2/2 holds if .µ. is a priori uniformly distributed over its possible values, and under restrictive conditions on the linear inequalities. This bound holds conditionally, given n–k. Some suggestions for further developments are given, and an applica tion to paired comparisons considered.