19.3 The likelihood ratio test

\[ t_{LR} = 2[l(\hat{\theta})-l(\theta_0)] \sim \chi^2_v \]

where v is the degree of freedom.

Compare the height of the log-likelihood of the sample estimate in relation to the height of log-likelihood of the hypothesized population parameter

Alternatively,

This test considers a ratio of two maximizations,

\[ L_r = \text{maximized value of the likelihood under $H_0$ (the reduced model)} \\ L_f = \text{maximized value of the likelihood under $H_0 \cup H_a$ (the full model)} \]

Then, the likelihood ratio is:

\[ \Lambda = \frac{L_r}{L_f} \]

which can’t exceed 1 (since \(L_f\) is always at least as large as \(L-r\) because \(L_r\) is the result of a maximization under a restricted set of the parameter values).

The likelihood ratio statistic is:

\[ -2ln(\Lambda) = -2ln(L_r/L_f) = -2(l_r - l_f) \\ \lim_{n \to \infty}(-2ln(\Lambda)) \sim \chi^2_v \]

where v is the number of parameters in the full model minus the number of parameters in the reduced model.

If \(L_r\) is much smaller than \(L_f\) (the likelihood ratio exceeds \(\chi_{\alpha,v}^2\)), then we reject he reduced model and accept the full model at \(\alpha \times 100 \%\) significance level