I'm studying the distributional properties of a laplace distribution, and I'm trying to get some intuition beyond plotting the distribution of what it means to have an undefined moment.
In wikipedia you can see that the mgf is only defined for $|t| < 1/b$ so as the variance of the laplace distribution increases to 1, you lose all moments including the mean. Does this matter? What is the intuition? For example the fourth moment can blow up, but the distribution will still look generally okay. What is the benefit of having a defined moment if you have the distribution?
If I have some data that fits the laplace distribution fits well with a very high b, should I be concerned? If for two data sets where b is close to 1 in one data set, but smaller in another, am I more confident in the fit to the data set with a b that generates more moments?
Any thoughts would be greatly appreciated. And if I'm thinking about this the wrong way let me know.