What does log likelihood in dfittool in Matlab means?

Question

I have a data set (400 real valued numbers between zero and one) which I need to fit an appropriate distribution to this data set. I used "dfittool" of Matlab and tried different distribution. For each distribution "dfittool" gives me a "log likelihood" of that distribution. My question is what does this value means? Does it mean the distribution with the highest log likelihood fits better to the data? Here is a summary of results:

Distribution ---- Log Likelihood

Normal_fit ----  1.8272

Exponential_fit  ----  24.3249

Logistic_fit  ----  -1.8729

Generalized Extreme Value_fit  ----   6.95018

Beta_fit  ----  62.81

Gamma_fit  ----  42.1661

I appreciate any help.

-Eli

Answer 1

Yes, the higher value of log likelihood indicate better fit. See the Wikipedia article Maximum likelihood or these lecture notes on MLE.

In a nutshell: having observed the values $x_1,\dots,x_n$ you want to choose the parameters of your distribution that maximize the product $\prod_k P(X_k=x_k)$. But this product is maximal when its logarithm is maximal, and the logarithm is easier to work with, because it's a sum.

The above applies to a discrete distribution. For a continuous distribution, one multiplies the values of probability density function, not the actual probabilities.

Answer 2

The distribution fitting here is an estimation problem. Also you could do this using 'mle' function of MATLAB which is a Maximum Likelihood Estimation.

So each distribution estimation which yields higher log likelihood has less estimation error in Maximum likelihood sense.