📂Mathematical Statistics

Definition of Likelihood Ratio Test in Mathematical Statistics

Definition ¹

$$ \begin{align*} H_{0} :& \theta \in \Theta_{0} \\ H_{1} :& \theta \in \Theta_{0}^{c} \end{align*} $$

For the hypothesis test described above, the statistic $\lambda$ is called the Likelihood Ratio test statistic. $$ \lambda \left( \mathbf{x} \right) := {{ \sup_{\Theta_{0}} L \left( \theta \mid \mathbf{x} \right) } \over { \sup_{\Theta} L \left( \theta \mid \mathbf{x} \right) }} $$

A hypothesis test that has a rejection region $\left\{ \mathbf{x} : \lambda \left( \mathbf{x} \right) \le c \right\}$ for a given $c \in [0,1]$ is called a Likelihood Ratio Test and is often abbreviated as LRT.

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• $L$ is a likelihood function.

Explanation

In the definition of $\lambda$, the numerator involves finding the supremum in $\sup_{\Theta_{0}}$, and the denominator involves finding the supremum in $\sup_{\Theta}$. The parameter space under the null hypothesis $\Theta_{0}$ is a subset of the entire parameter space $\Theta_{0} \subseteq \Theta$, and naturally, $0 \le \lambda \left( \mathbf{x} \right) \le 1$ holds. The closer this ratio is to $0$, the less plausible the parameters are under the null hypothesis.

Reflecting back on when we first encountered statistics, starting from basic probability distribution theory, studying test statistics like the t-distribution, F-distribution, chi-squared distribution separately seemed much cleaner. Of course, there is a certain motivation behind the Likelihood Ratio Test, but it makes sense without any buildup unlike the other tests mentioned.

Example: Normal Distribution

For practical applications of LRT, the supremum $\sup$ must be reflected. Since the denominator becomes largest over the entire parameter space $\Theta$, the maximum likelihood estimator is used, and the numerator is set to be maximized under the null hypothesis. $$ \begin{align*} H_{0} :& \theta = \theta_{0} \\ H_{1} :& \theta \ne \theta_{0} \end{align*} $$ Consider a hypothesis test for a known-variance normal distribution $N \left( \theta , \sigma^{2} \right)$ with a random sample $X_{1} , \cdots , X_{n}$. In this case, the denominator should use the sample mean $\bar{\mathbf{x}}$, which is the maximum likelihood estimator for the population mean $\theta$, and the numerator should directly use $\theta_{0}$ since the parameter space of the null hypothesis is a singleton set $\Theta_{0} = \left\{ \theta_{0} \right\}$. It can be mathematically derived as follows. $$ \begin{align*} \lambda \left( \mathbf{x} \right) =& {{ \sup_{\Theta_{0}} L \left( \theta \mid \mathbf{x} \right) } \over { \sup_{\Theta} L \left( \theta \mid \mathbf{x} \right) }} \\ =& {{ L \left( \theta_{0} \mid \mathbf{x} \right) } \over { L \left( \bar{\mathbf{x}} \mid \mathbf{x} \right) }} \\ =& {{ (2\pi)^{-n/2} \exp \left( - \sum \left( x_{k} - \theta_{0} \right)^{2} / 2 \right) } \over { (2\pi)^{-n/2} \exp \left( - \sum \left( x_{k} - \bar{\mathbf{x}} \right)^{2} / 2 \right) }} \\ =& \exp \left( -n \left( \bar{\mathbf{x}} - \theta_{0} \right)^{2} / 2 \right) \end{align*} $$ Note that $\lambda (\mathbf{x})$ is exactly $\bar{\mathbf{x}} = \theta_{0}$ when $1$, and the larger the difference, the closer it gets to $0$. Of course, it is understood that in the mathematical definition of LRT, it ranges from $0$ to $1$, but seeing the calculated result intuitively allows one to perform hypothesis testing based on how similar the sample variance is to the population mean under the null hypothesis.