In my graduate-level "Themes in Econometrics" course we've been talking recently about the Neyman-Pearson Lemma. In 1933 Jerzy Neyman and Egon Pearson published one of the most important papers of modern statistics, referenced below. Specifically, they showed that we can use the likelihood ratio to construct the Most Powerful test (for a given significance level), when we are testing a point null hypothesis against a point alternative hypothesis. This set the scene for classical hypothesis testing as we practice it today.
Those of us with Bayesian inclinations find the two opening sentences of this masterpiece in frequentist statistics somewhat interesting! There, Neyman and Pearson begin:
"The problem of testing statistical hypotheses is an old one. Its origin is usually connected with the name of THOMAS BAYES, who gave the well-known theorem on the probabilities a posteriori of the possible "causes" of a given event."
In econometric applications we're usually concerned with testing a point null against a composite alternative. For example, we may wish to test H0: β1 = 0 against H1: β1 > 0 in a regression model setting. Once we move away from the "point null - point alternative" situation, there's no guarantee that the Likelihood Ratio Test (LRT) will be Uniformly Most Powerful (UMP). However, it often is, and it's certainly a good starting point for our testing in many cases.
To quote Neyman and Pearson (1993, 336-337) again:
"We give the solution of this problem for the case of testing a simple hypothesis.To solve the same problem in the case where the hypothesis tested is composite, the solution of a further problem is required; this consists in determining what has been called a region similar to the sample space with regard to a parameter.We have been able to solve this problem only under certain limiting conditions;.......
It has, however been shown that the critical region based on the principle of likelihood satisfies our intuitive requirements of "a good critical region"."
In the example given above, with Normal regression errors, the LRT test can be shown to be be equivalent to the usual t-test (by the monotone likelihood principle), and it is indeed UMP against one-sided alternatives.
About 3 years ago, Cosma Shalizi (Department of Statistics, Carnegie Mellon U.) had an interesting post on his Three-Toed Sloth. That post was titled "The Shadow Price of Power", and it gives an excellent "economic" interpretation and motivation for the Neyman-Person Lemma.
I liked it, and I think you will too!
Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.
Neyman, J. and E. S. Pearson (1933). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London, Series A, 231, 289-337.
© 2012, David E. Giles