Likelihood Ratio

The likliehood ratio statistics is used as a benchmark for m odel solution. The likelihood ratio is defined as the difference between the log-likelihood of the null \(L_0\) and alternative hypothesis \(L\).

\[ \Lambda \equiv L - L_0 \]

In most cases, the likelihood ratio refers to the maximum likelihoods of each hypothesis.

\[ \Lambda^* \equiv \mathop{\arg\max}_{H} L - \mathop{\arg\max}_{H_0} L_0 \]

Relation to Chi-Squared Distribution

By Wilks’ theorem, as the sample size grows large assuminng the null hypothesis is true, the following test statistics \(D\) approaches the \(\chi^2_k\) distribution

\[\begin{split} \begin{gather*} D = -2\log(\Lambda)&\\ D \sim \chi^2_k\,. \tag{as $N \to \infty$} \end{gather*} \end{split}\]

• \(k\): The degrees of freedom. It equals the difference in size of the free parameter space,

  \[ k = \dim{\Theta} - \dim{\Theta_0} \,. \]