Comments on: Likelihood Ratio Test Statistic [Equation of the Week] http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-lrt-statistic/ Weaving together Astronomy+Statistics+Computer Science+Engineering+Intrumentation, far beyond the growing borders Fri, 01 Jun 2012 18:47:52 +0000 hourly 1 http://wordpress.org/?v=3.4 By: vlk http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-lrt-statistic/comment-page-1/#comment-253 vlk Thu, 19 Jun 2008 02:40:30 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=319#comment-253 Thanks for those useful clarifications, Alex! Thanks for those useful clarifications, Alex!

]]> By: Alex http://hea-www.harvard.edu/AstroStat/slog/2008/eotw-lrt-statistic/comment-page-1/#comment-252 Alex Thu, 19 Jun 2008 02:20:51 +0000 http://hea-www.harvard.edu/AstroStat/slog/?p=319#comment-252 A couple of points. First, the LR test statistic is applicable for dependent samples (ie time series). In such cases, the likelihood is typically written as the product of densities conditional on the previous observations, ie: 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 <a href="http://en.wikipedia.org/wiki/Neyman-Pearson_lemma" rel="nofollow">Neyman-Pearson lemma</a>. A couple of points. First, the LR test statistic is applicable for dependent samples (ie time series). In such cases, the likelihood is typically written as the product of densities conditional on the previous observations, ie:
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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