The likelihood ratio statistic evaluates the relative plausibility of two competing hypotheses on the basis of a collection of sample data. The favored hypothesis is determined by whether the ratio is greater than or less than one.

To introduce the likelihood ratio, suppose that yOBS denotes a vector of observed data. Assume that a parametric joint density is postulated for the random vector Y corresponding to the realization yOBS. Let f(y; θ) represent this density, with parameter vector θ. The likelihood of θ based on the data yOBS is defined as the joint density:

Although the likelihood and the density are the same function, they are viewed differently: The density f(y; θ) assigns probabilities to various outcomes for the random vector Y based on a fixed value ...