A Study on Volume of Difference of Two Joint pdf′s, Focused on the Relation to Normal Theory LR Tests

In this paper we explain that normal theory likelihood-ratio tests(z, t, $x^2$. F) for mean(s) or variance(s) can be geometrically related to volume of difference of two joint pdf's. One is an estimated joint pdf under null parameter space $\omega$ and the other is an estimated joint pdf under full parameter space $\Omega$. For explanations, ‘distance between two distributions’ is defined. We study properties of it, and derive some results on the distance between two multivariate normal distributions.