# Interpreting the best-fit student t distribution (based on maximum likelihood estimation) for my datasets

I have fitted (through maximum likelihood estimation of parameters) a number of leptokurtic probability density functions (i.e.

• logistic
• hyperbolic-secant
• Laplace
• log-logistic
• student's t (with parameters $\nu, \mu$)
• gamma
• Weibull
• Cauchy
• Levy
• Gumbel)

to two datasets that are measured at different times and have different sample sizes.

All are 2-parameter PDFs. The Student's t distribution is the best fit PDF in terms of the highest log-likelihood values for both datasets. The second best-fitting PDF has the AIC difference of more than 6.

1. Intuitively speaking, what are the unique properties of the student's t distribution that differentiate it from the other leptokurtic PDFs (at least from the one listed here.)
2. How should I interpret/explain that why the student's t distribution is the best-fitting PDF and the others are not?
• Why do you think, that student t is a 2-parameter PDF? I would say, that it is a one-parameter PDF. – Ferdi Apr 30 '18 at 9:52
• @Fredi if parametrized by location $\mu$ and degrees of freedom $\nu$ it obviously has two parameters. – Tim Apr 30 '18 at 9:55
• The one-parameter student's t is extended with a location parameter because the mode of the datasets' bell-shaped histogram is away from zero. So it has two parameters now: the degrees-of-freedom $\nu$ and the location parameter $\mu$. – MM Khan Apr 30 '18 at 10:00