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Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a means to let us know how 'good' the estimate for a particular parameter is.

But the maximum likelihood estimator that takes account of the entire parameter space will always be the best estimator of a statistic what is the point of just considering subsets of the parameter space and likelihood ratios based on those subsets?

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  • $\begingroup$ There is a difference between an estimator and a hypothesis test $\endgroup$ – Henry Jul 30 '14 at 8:42
  • $\begingroup$ What is the likelihood ratio test for then? Can you explain it as clearly as possible, I've read so many convoluted explanations at this stage that I feel I have a worse understanding of them now than I did a few days ago. $\endgroup$ – sonicboom Jul 30 '14 at 8:45
  • $\begingroup$ An estimator gives us a good guess of what the true parameter value is, but it says nothing about which other values are likely to be the true parameter. The likelihood ratio test "tests" whether it is likely that the true parameter lies within some region or subset of parameter values. The simplest case is when this region or subset contains just one point. $\endgroup$ – Stefan Hansen Jul 30 '14 at 8:55
  • $\begingroup$ Maximum likelihood is for estimating parameters whilst the ratio test is for testing the hypothesis of two fits based on the ratio of the null fit. The wiki page is quite good. $\endgroup$ – Chinny84 Jul 30 '14 at 8:56
  • $\begingroup$ @StefanHansen I'm still not getting it. We already have the maximum likelihood estimator. This is the most likely value of the true parameter given the observed data. Why then do we care about other values that may be the true parameter? $\endgroup$ – sonicboom Jul 30 '14 at 9:23
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The short answer is that the likelihood ratio test is just an inference scheme and is different than the maximum likelihood estimator. For example, in a normal population, the sample mean is the MLE of the population mean. Now lets say you want to determine if the population mean is zero vs not zero. One way is the usual "Z-test" using the sample mean, but you could also formulate it as a likelihood ratio test (using Wilks' likelihood ratio statistic) and get the same answer.

In general, a major use of the likelihood ratio test wrt MLE's is to form approximate confidence intervals. You approximate the distribution of the "normalized" likelihood ratio (that is, likelihood divided by the maximum likelihood, so the function is bounded above by 1) by a Chi-squared distribution (per Wilks' likelihood ratio statistic) and proceed to find what values of the population mean are consistent with that statistics to a certain degree of statistical significance.

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