Distribution of the residual sum of squares in fitting inequalities

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Abstract
Consider a model for n observations, Yi = μi + σξi (i = 1,…, n), where the ξi are n independent unit normal variables and the μi are restrained by p linear inequalities. The maximum-likelihood estimate of {μi} is {Yoi} minimizing Σ(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 Σ(Yi − Yoi)2 which it is natural to use for inference about σ2. Using the Kuhn-Tucker conditions, which the Yoi must satisfy, upper and lower bounds are obtained for the percentage points of Σ(Yi − Yoi)2/σ2.The upper bound is Σ(Yi−Yoi)2/σ2≤ Xn−k2 which holds conditionally on exactly n−k independent linear inequalities being satisfied as equations by the Yoi A direct lower bound uγ, n, n−k is given with (n−k) regarded as a random variable. A more satisfactory Bayesian lower bound ¾X⅔(n−k)2≤Σ(Yi − Yoi)2/σ2 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.

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