An empirical Ω

Abstract

We hope to arrive at powerful tests using test statistics of the quadratic ratio type (2.1) with significance points that can be tabulated. For this purpose, we developed the BLUF estimator w of J′u; see (3.2). The estimator depends on the following matrices: the n × p matrix J, the p × p matrix Q, the p × p matrix Ω = KK ′, the n × n matrix Г, the n × k matrix X, and the n-element vector y. Both X and y are specified by observation and it is assumed that y ~n(Xβ, σ² Г), where Г = Г ₀ under the null hypothesis ℋ ₀ . Hence, Г in (3.2) is specified by ℋ ₀ . One is free to choose J and Q. The only practical specifications we know are J = I_(n) or J is an n × (n−k) submatrix of I_(n), like J in the BLUS vectors, and Q = I or Q = Г⁻¹. In this chapter we take J = I, so that p = n. Because most current tests have Г = I as the null hypothesis, the most important specifications for practical application would seem to be Г= Q = J = I_(n).