## Asymptotic Tests and Related Topics

4.5.1 Likelihood Ratio and Related Tests

Let Ux, 0) be the joint density of a Г-vector of random variables x = (Xi, x2,. . . , xTY characterized by a ЛГ-vector of parameters 6. We assume all the conditions used to prove the asymptotic normality (4.2.23) of the maximum likelihood estimator 6. In this section we shall discuss the asymptotic tests of the hypothesis

h(0) = O, (4.5.1)

where h is a ^-vector valued differentiable function with q<K. We assume that (4.5.1) can be equivalently written as

В = r(a),

where a is a p-vector of parameters such that p = K— q. We denote the constrained maximum likelihood estimator subject to (4.5.1) or (4.5.2) as в = Ha).

Three asymptotic tests of (4.5...

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