# Circular Likelihood Ratio Tests for the Main Problem

## Abstract

In the present chapter we discuss some ways in which the theory of Ch. 3 can be used to obtain simple and satisfactory procedures for testing problems with a polyhedral-cone-shaped alternative. While dealing with the main problem
\(
\left( {{H_0},{{\tilde H}_1}} \right)
\)
with r>2, in which the alternative is determined by the pointed polyhedral cone *K*, one may proceed as follows. First the circular cone *C* = *C*(a, *w*) is specified as a substitute for *K*, next a particular test is selected for the modified problem
\(
\left( {{H_0},{{\tilde H}_1}} \right)
\)
of testing against *C*\(0 _{r}), and finally this test is performed as a solution to the original problem (H_{0} , H_{1}). Assuming that the choice of test for (H_{0} ,
\(
{\tilde H_1}
\))
is made in advance, the gist is to choose the axis a and the axial angle *w* of the substitute *C* for *K*, cf. (3.0.1). The special case r=2, where *K* itself is a circular cone, can be considered a benchmark for the comparison of various approaches to the main problem.

### Keywords

Covariance Resid Hull Eter Gout## Preview

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