# Lagrange Multiplier Test

The Lagrange Multiplier (LM) test statistic is given by

$T_{LM}\left(X_{1}..X_{n}\right)=\frac{\left[\frac{\partial}{\partial\theta}l\left(\left.\theta_{0}\right|X_{1}..X_{n}\right)\right]^{2}}{-\frac{\partial^{2}}{\partial\theta^{2}}l\left(\left.\theta_{0}\right|X_{1}..X_{n}\right)}.$

The motivation for the LM test is that it is an approximation of the LRT. Unlike the LRT, though, the LM test does not require estimation! It suffices to evaluate the first and second derivative of the log-likelihood function at the parameter value of the null hypothesis.

This test is often known as the score test, because we often refer to $\frac{\partial}{\partial\theta}l\left(\left.\theta\right|X_{1}..X_{n}\right)$ as the score function.

## A Brief Note

• Notation $\frac{\partial}{\partial\theta}l\left(\left.\theta_{0}\right|X_{1}..X_{n}\right)$ means $\left.\frac{\partial}{\partial\theta}l\left(\left.\theta\right|X_{1}..X_{n}\right)\right|_{\theta=\theta_{0}}.$
• Notice that the expectation of the denominator of the LM test equals the Fisher information, $I\left(\theta_{0}\right)$.