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CGandC

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- TL;DR Summary
- In both Akaike Information Criterion and Likelihood Ratio Test we compare likelihoods to understand the better fit to the empirical data, but what's the difference between Akaike Information Criterion and Likelihood Ratio Test?

Hello,

I want to understand the difference between both goodness-of-fit tests, I would be glad if you could help me:

## AIC_i = - 2log( L_i ) + 2K_i ##

Where ##L_i## is the likelihood function defined for distribution model ##i## .

##K_i## is the number of parameters of the distribution model. for example, for exponential distribution we have only lambda so ##K_{exponential} = 1##

So if I want to know which distribution better fits the empirical data, I see which AIC is higher and choose the representative distribution for that high AIC.

" The basic idea behind the likelihood ratio test is to compute the likelihood of

the data under two competing distributions. The one with the higher likelihood is

then the better fit. Alternatively, one can calculate the ratio of the two likelihoods,

or equivalently the logarithm R of the ratio, which is positive or negative depending

on which distribution is better, or zero in the event of a tie. "

We can see that both in the Akaike and Likelihood ratio test, essentially I compare the likelihood functions for different distribution models and I choose the bigger one which is representative of the better distribution fit. using both methods as described above yielded me very similar results in MATLAB.

So I don't really understand the difference between both methods, maybe I'm understanding these methods wrong?

I want to understand the difference between both goodness-of-fit tests, I would be glad if you could help me:

**Akaike Information criterion is defined as:**## AIC_i = - 2log( L_i ) + 2K_i ##

Where ##L_i## is the likelihood function defined for distribution model ##i## .

##K_i## is the number of parameters of the distribution model. for example, for exponential distribution we have only lambda so ##K_{exponential} = 1##

So if I want to know which distribution better fits the empirical data, I see which AIC is higher and choose the representative distribution for that high AIC.

**Likelihood Ratio Test is defined as ( According to " Clauset , et al . Power law distributions in empirical data" ):**" The basic idea behind the likelihood ratio test is to compute the likelihood of

the data under two competing distributions. The one with the higher likelihood is

then the better fit. Alternatively, one can calculate the ratio of the two likelihoods,

or equivalently the logarithm R of the ratio, which is positive or negative depending

on which distribution is better, or zero in the event of a tie. "

**Bottom line:**We can see that both in the Akaike and Likelihood ratio test, essentially I compare the likelihood functions for different distribution models and I choose the bigger one which is representative of the better distribution fit. using both methods as described above yielded me very similar results in MATLAB.

So I don't really understand the difference between both methods, maybe I'm understanding these methods wrong?