Applications of Neyman-Pearson-Lemma

Question

I studied the Neyman-Pearson-Lemma last semester, and now I was wondering what its applications are. It states that there exists a most-powerful randomized statistical test and a $c \in [0, \infty]$ such that the Null-Hypothesis is accepted iff the Likelhood-quotient is strictly bigger than $c$. But how would one calculate $c$ or even just the Likelhood-quotient? And how does one get that most-powerful randomized test?

Or: Why is this Lemma so important in mathematical statistics?

Thanks!

Answer 1

One application is in economics, believe it or not.

A classic dilemma in consumer theory is, when given prices, calculating a demand function for the consumer. More specifically, given a land-estate, a price measure for the land, and some formulated utility measure for the land, the consumer's problem is to calculate the land option with the largest utility value that he can purchase with his budget. This actually is similar to the problem of finding the most powerful statistical test, and the Neyman–Pearson lemma can be used, managing to relate likelihood ratios and test statistic values to marginal utility and cost.

Further reading on economic relation.

Answer 2

In a nutshell, it gives you the form of a very good test. You still have to set up the likelihood quotient, simplify it to get the right test statistic, and find its null distribution to determine $c$. The Neyman-Pearson Lemma gives you a procedure.

Think about the result from Calculus that a local interior maximum of a differentiable function is attained at a point where the derivative is zero. This does not tell you how to differentiate the function or how to find out where the derivative is zero. But it gives you a procedure. That is a similar situation.