The AstroStat Slog

I wonder what Fisher, Neyman, and Pearson would say if they see “Technique” after “Likelihood Ratio” instead of “Test.” A presenter’s saying “Likelihood Ratio Technique” for source identification, I couldn’t resist checking it out not to offend founding fathers of the likelihood principle in statistics since “Technique” sounded derogatory to be attached with “Likelihood” to my ears. I thank, above all, the speaker who kindly gave me the reference about this likelihood ratio technique. Continue reading ‘Likelihood Ratio Technique’ »

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]’ »

From arxiv/math.st: 0708.0499v1
Inference for mixtures of symmetric distributions by Hunter, Wang, and Hettmansperger, Annals of Statistics, 2007, Vol.35(1), pp.224-251.
Continue reading ‘[ArXiv] Identifiability and mixtures of distributions, Aug. 3, 2007’ »