p(D|theta) = p(y_t|theta,y_t-1,…,y_1) * p(y_t-1|theta,y_t-2,…,y_1)*…*p(y_1|theta)

Second, even given the regularity conditions, the LR statistic is not necessarily distributed as a chi-square random variable for finite samples. This distribution holds asymptotically given regularity conditions (as n -> infinity), but it need not hold in finite samples. Likelihood-ratio type tests are actually the basis of many common tests; for example, the one-sided Z test is equivalent to a likelihood ratio test. The use of likelihood ratios tests is largely based on an important result in frequentist testing theory known as the Neyman-Pearson lemma.

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