The AstroStat Slog

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

T_LRT = -2 ln R(D,Θ₀,Θ)

R(D,Θ₀,Θ) = [ sup_θεΘ0 p(D|Θ₀) ] / [ sup_θεΘ p(D|Θ) ]

where D are an independent data sample, Θ are model parameters {θ_i, i=1,..M,M+1,..N}, and Θ₀ form a subset of the model where θ_i = θ_i⁰, 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.
Continue reading ‘Likelihood Ratio Test Statistic [Equation of the Week]’ »

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’ »

One of the speakers from the google talk series exemplified model based clustering and mentioned the likelihood ratio test (LRT) for defining the number of clusters. Since I’ve seen the examples of ill-mannerly practiced LRTs from astronomical journals, like testing two clusters vs three, or a higher number of components, I could not resist indicating that the LRT is improperly used from his illustration. As a reply, the citation regarding the LRT was different from his plot and the test was carried out to test one component vs. two, which closely observes the regularity conditions. I was relieved not to find another example of the ill-used LRT. Continue reading ‘The LRT is worthless for …’ »

Seven preprints were chosen this week and two mentioned model selection. Continue reading ‘[ArXiv] 3rd week, Jan. 2008’ »