Anomalies of the likelihood ratio test for testing restricted hypotheses (Q811061)

Let \(x\) be a \(k\)-dimensional random vector having normal distribution \(N_ k(\theta,\Gamma)\) with unknown mean vector \(\theta =(\theta_ 1,...,\theta_ k)\) and a known covariance matrix \(\Gamma\). Consider the hypotheses \(H_ 0: a'_ j\theta =0\), \(j=1,...,r\); \(a'_ j\theta \geq 0\), \(j=r+1,...,n\), and \(H_ A: a'_ j\theta \geq 0\), \(j=1,...,n\), where \(a_ 1,...,a_ n\) are \(k\)-dimensional fixed vectors such that \(H_ A\) defines a polyhedric closed convex cone in \(\mathbb R^ k\). The author shows an anomaly in the L.R.T. (likelihood ratio test) by proving that the L.R.T. for \(H_ 0\) versus \(H_ A-H_ 0\) is dominated by another suitably constructed test.