The Generalized Likelihood Ratio Test

There is no universally optimal test strategy for composite hypotheses (unknown parameters in the pdfs). The generalized likelihood ratio test (GLRT) is a likelihood ratio in which the unknown parameters are replaced by their maximum likelihood estimates. Example of the GLRT for detecting a signal of known shape but unknown amplitude in noise of unknown variance.

[kaiserhaz]

Thank you dear Sir for this video.

I would like to add something to the discussion.

The test statistic L'(x) was mentioned in (Kay1998, Chapter 9, pg342) as being a non-CFAR test. This was because the test distribution is in fact Gaussian, which depends on the unknown variance. The expression itself is correct, but the threshold, linked to the PFA and hence to the variance is non-constant as it depends on the unknown variance.

The MLE obtained by least squares as was done in the video is the correct approach, but for H0, the MLE needs to be restricted (Kay1998, Chapter 9, Appendix A). This can be understood better when the hypothesis test is reposed as a parameter test (i.e. H0: A=0, sigma0^2, H1: A!=0, sigma1^2). Since there is zero probability that A assumes the value 0, the MLE under H0 must be restricted in order to obtain the correct estimate for H0. If this is considered along with some scaling factors, then the test distribution indeed becomes an F distribution.

Alternatively, I would love to know how the performance of the tests are characterized (i.e. how the test distribution is obtained).

Thank you,

[allsignalprocessing]

The GLRT can be applied to the case where s is unknown - in that case you have to estimate s from the data (using the MLE) and then plug that estimate into the distribution in the likelihood ratio. Obviously, the less that is known about the signal s, the poorer the performance.

[allsignalprocessing]

Maximum likelihood estimation is normally applied to unknowns that are deterministic, not random. However, I'm guessing this is a hypothesis testing problem in which H_0 is s = 1, and H_1 is s = -1. In that case you can use the likelihood ratio (if everything else is known) or the GLRT by applying it separately to each hypothesis.

[sanariaz]

What is effect of type 1 error on GLRT

[toanvodai]

Can you give an easy-to-read reference to the GLRT, please? Thank you very much

[ilansch]

I have this homework where s is unknown, but we know is values are =1 and -1 with probability 0.5 each. I don't really know how to use that. can you give me a hint? thank you very much

[sdsachin]

hi sir, , I have derived GLRT for the hypothesis mentioned in the above video, , can i know how to plot P_d vs. P_fa curve?  How to obtain P_d and P_fa values with respect to varying threshold?

thanks in advance

[kiranpadakanti]

thank you sir, it is a great explanation. can we get MATLAB CODE for this example

[noramaliaab]

What is the differemce between lkelihood ratio test and generalized likelihood rato test??

[rakeshable]

Good explanation sir! do you have a matlab codes to implement these algorithms?

[mahdkhalid]

Can you please share Matlab code to implement this? It will be very helpful for me to visually understand this. Many thanks

[simritigoshal]

I was trying to derive L''(x) . But I am not able to get x^T.s in the numerator where as the denominator comes out exactly same. I just wanted to confirm if the final expression for L''(x) in the lecture is correct.My numerator is x^T.x

[noramaliaab]

Thank you so much!!the explanation is very helpful..thank you again :D