Posts tagged ‘Protassov’

#### Likelihood Ratio Test Statistic [Equation of the Week]

From Protassov et al. (2002, ApJ, 571, 545), here is a formal expression for the Likelihood Ratio Test Statistic,

TLRT = -2 ln R(D,Θ0,Θ)

R(D,Θ0,Θ) = [ supθεΘ0 p(D|Θ0) ] / [ supθεΘ p(D|Θ) ]

where D are an independent data sample, Θ are model parameters {θi, i=1,..M,M+1,..N}, and Θ0 form a subset of the model where θi = θi0, i=1..M are held fixed at their nominal values. That is, Θ represents the full model and Θ0 represents the simpler model, which is a subset of Θ. R(D,Θ0,Θ) is the ratio of the maximal (technically, supremal) likelihoods of the simpler model to that of the full model.
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#### The Flip Test

Why is it that detection of emission lines is more reliable than that of absorption lines?

That was one of the questions that came up during the recent AstroStat Special Session at HEAD2008. When you look at the iconic Figure 1 from Protassov et al (2002), which shows how the null distribution of the Likelihood Ratio Test (LRT) and how it holds up for testing the existence of emission and absorption lines. The thin vertical lines are the nominal F-test cutoffs for a 5% false positive rate. The nominal F-test is too conservative in the former case (figures a and b; i.e., actual existing lines will not be recognized as such), and is too anti-conservative in the latter case (figure c; i.e., non-existent lines will be flagged as real). Continue reading ‘The Flip Test’ »