likelihoodratioStatistiken

Likelihood ratio statistics are a class of test statistics used to assess whether data support a null hypothesis that imposes constraints on model parameters, in comparison with a more general alternative. The statistic is based on the ratio of the maximum likelihood under the null hypothesis to the maximum likelihood under the full model.

Let L(θ) be the likelihood function for parameter θ. For hypotheses H0: θ ∈ Θ0 ⊂ Θ versus H1: θ ∈ Θ, the likelihood ratio is lambda = sup_{Θ0} L(θ) / sup_{Θ} L(θ), and the commonly used test statistic is W = -2 log lambda = -2 log sup_{Θ0} L(θ) / sup_{Θ} L(θ). Larger values of W indicate that the data are less compatible with H0.

Distribution and interpretation: Under regularity conditions and large samples, Wilks’ theorem states that W is approximately chi-square distributed under H0, with degrees of freedom equal to the number of constrained parameters (dim(Θ) − dim(Θ0)). This allows computation of p-values from the chi-square distribution. In small samples or non-regular models, the chi-square approximation may be inaccurate, and alternative methods or simulations may be used.

Applications: Likelihood ratio statistics are used for hypothesis testing in a wide range of models, including linear, generalized linear, and complex nonlinear models. They are also employed in model selection and in constructing confidence intervals by inversion of the test (profile likelihoods).

Extensions: The generalized likelihood ratio test (GLRT) handles composite hypotheses; profile likelihood approaches address nuisance parameters; and connections exist to information criteria such as AIC, which relate to the log-likelihood but are used for model comparison rather than formal hypothesis testing.