# What are the ''desirable'' statistical properties of the likelihood ratio test?

I am reading an article whose method is fully based on the likelihood ratio test. The author says that the LR test against one sided alternatives is UMP. He proceeds by claiming that

"...even when it [the LR test] can not be shown to be uniformly most powerful, the LR test often has desirable statistical properties."

I am wondering what statistical properties are meant here. Given that the author refers to those in passing, I assume they are common knowledge among statisticians.

The only desirable property I have managed to find so far is the asymptotic chi-squared distribution of $-2 \log \lambda$ (under some regularity conditions), where $\lambda$ is the LR ratio.

I would also be thankful for a reference to a classical text where one can read about those desired properties.

• You could have a look at (chap 15 & 16) of van Der Waart : "Asymptotic Statistics". – kjetil b halvorsen Aug 31 '15 at 18:10

## 1 Answer

It might be good to read What follows if we fail to reject the null hypothesis? before the explanation below.

## Desirable properties: power

In hypothesis testing, the goal is to find 'statistical evidence' for $H_1$. Thereby we can make type I errors, i.e. we reject $H_0$ (and decide that there is evidence in favour of $H_1$) while $H_0$ was true (i.e. $H_1$ is false). So a type I error is 'finding false evidence' for $H_1$.

A type II error is made when $H_0$ can not be rejected while it is false in reality, i.e. we ''accept $H_0$'' and we 'miss' the evidence for $H_1$.

The probability of a type I error is denoted by $\alpha$, the choosen significance level. The probability of a type II error is denoted as $\beta$ and $1-\beta$ is called the power of the test, it is the probability to find evidence in favour of $H_1$ when $H_1$ is true.

In statitistical hypothesis testing the scientist fixes an upper threshold for the probability of a type I error and under that constraint tries to find a test with maximum power, given $\alpha$.

The desirable properties of likelihood ratio tests have to do with power

In a hypothesis test $H_0: \theta=\theta_0$ versus $H_1: \theta = \theta_1$ the null hypothesis and the alternative hypothesis are called ''simple'', i.e. the parameter is fixed to one value, just as well under $H_0$ as under $H_1$ (more precisely; the distributions are fully determined).

The Neyman-Pearson Lemma states that, for hypothesis tests with simple hypothesises, and for given type I error probability, a likelihood ratio test has the highest power. Obviously, high power given $\alpha$ is a desirable property: power is a measure of 'how easy it is to find evidence for $H_1$'.

When the hypothesis is composite; like e.g. $H_0: \theta = \theta_1$ versus $H_1: \theta > \theta_1$ then the Neyman-Pearson lemma can not be applied because there are 'multiple values in $H_1$'. If one can find a test such that it is most powerfull for every value 'under $H_1$' then that test is said to be 'uniformly most powerfull' (UMP) (i.e. most powerfull for every value under $H_1$).

There is a theorem by Karlin and Rubin that gives the necessary conditions for a likelihood ratio test to be uniformly most powerfull. These conditions are fullfilled for many one-sided (univariate) tests.

So the desirable property of the likelihood ratio test lies in the fact that in several cases it has the highest power (although not in all cases).

In most cases the existence of an UMP test can not be shown and in many cases (especially the multivariate) it can be shown that an UMP test does not exist. Nevertheless, in some of these cases likelihood ratio tests are applied because of their desirable properties (in the above context), because they are relatively easy to apply, and sometimes because no other tests can be defined.

As an example, the one-sided test based on the standard normal distribution is UMP.

## Intuition behind the likelihood ratio test:

If I want to test $H_0: \theta=\theta_0$ versus $H_1: \theta = \theta_1$ then we need an observation $o$ derived from a sample. Note that this is one single value.

We know that either $H_0$ is true or $H_1$ is true, so one can compute the probability of $o$ when $H_0$ is true (lets call it $L_0$) and also the probability of observing $o$ when $H_1$ is true (call it $L_1$).

If $L_1 > L_0$ then we are inclined to believe that ''probably $H_1$ is true''. So if the ration $\frac{L_1}{L_0} > 1$ we have reasons to believe that $H_1$ is more realistic than $H_0$.

If $\frac{L_1}{L_0}$ would be something like $1.001$ then we might conclude that it could be due to chance, so to decide we need a test and thus the distribution of $\frac{L_1}{L_0}$ which is ... a ratio of two likelihoods.

I found this pdf on the internet.

• I think this misses the OP's question: the quote states that even when it can't be shown that the LRT is UMP, it still has other attractive features. So what are the attractive features that aren't that it is UMP? – Cliff AB Sep 2 '15 at 1:13
• @Cliff AB: I think that is there at the end of the first section and the second section tells intuitively why it makes sense to use LRT. Note that in most cases there is no UMP and if there is no 'best test' or no alternative then it's not unreasonable to take something that 'makes sense' I think? But if you have additional elements then you are invited to post these in your own answer. That's the idea behind SE I think. – user83346 Sep 2 '15 at 5:09
• Perhaps it's just me reading the original quote slightly different: I read it as "LRT has other attractive features, besides just power". – Cliff AB Sep 2 '15 at 5:24
• @CliffAB I agree with your comment, apparently the author of the article I referred in my question meant that LRT is for some reason good even if it is not a UMP test, and I hope that this reason is not just the ease of implementation or the lack of other alternatives. I suspect (hope) that the LRT has some good asymptotic properties (e.g. it is consistent, i.e. its power for any $H1$ goes to $1$ if we increase the number of observations). – Sergey Zykov Sep 2 '15 at 13:39
• don't under estimate ease of implementation! – Cliff AB Sep 2 '15 at 15:06