## Anomalies of the likelihood ratio test for testing restricted hypotheses.(English)Zbl 0734.62058

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.

### MSC:

 62H15 Hypothesis testing in multivariate analysis 62F03 Parametric hypothesis testing 62F99 Parametric inference
Full Text: