LikelihoodRatioStatistik

LikelihoodRatioStatistik, in statistics often referred to as the likelihood ratio statistic, is a measure used to test nested statistical models by comparing how well different parameterizations explain observed data. It is based on the likelihood function and the maximum likelihood estimates under both a full model and a constrained model.

Let L(θ) denote the likelihood of the data given parameters θ. Suppose H0 specifies a subset Θ0 of permissible parameters. Denote θ̂ as the maximum likelihood estimate under the full model Θ, and θ̂0 as the maximum likelihood estimate under Θ0. The likelihood ratio is Lambda = L(θ̂0)/L(θ̂), and the likelihood ratio statistic is D = -2 log Lambda = -2 [log L(θ̂0) - log L(θ̂)]. Large values of D indicate that the full model provides a substantially better fit than the constrained model.

Under regularity conditions and for large samples, Wilks’ theorem states that D is approximately chi-square distributed with k degrees of freedom, where k is the number of free parameters constrained by H0 (the difference in dimensionality between Θ and Θ0). This allows the computation of p-values to assess the strength of evidence against H0.

Variants include the profile likelihood ratio, which handles nuisance parameters by maximizing over them for each value of the parameter of interest. In practice, the statistic relies on model correctness and asymptotic approximations; for small samples or complex models, bootstrap or Monte Carlo methods may be used to obtain more accurate distributions.