Definition:Likelihood Ratio

Definition

Let $X$ be a continuous random variable belonging to a family $\FF$ of probability distributions, dependent on a parameter $\theta$.

Let $\map {\mathrm L} \theta$ be the likelihood function for $\theta$.

Let:

\(\ds \map {\mathrm L} {\theta_1}\)

\(=\)

\(\ds \mathrm L_1\)

\(\ds \map {\mathrm L} {\theta_2}\)

\(=\)

\(\ds \mathrm L_2\)

Then $\dfrac {\mathrm L_2} {\mathrm L_1}$ is known as the likelihood ratio.

This can then be used as the basis of a hypothesis test where:

the null hypothesis $H_0$ is that $\theta = \theta_1$

the alternative hypothesis $H_1$ is that $\theta = \theta_2$.

Also see

• Neyman-Pearson Lemma

• Results about likelihood ratios can be found here.

Historical Note

The concept of a likelihood ratio was raised by Jerzy Neyman and Egon Sharpe Pearson in $1928$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): likelihood ratio
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): likelihood ratio