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?
  • $\begingroup$ Why do you think, that student t is a 2-parameter PDF? I would say, that it is a one-parameter PDF. $\endgroup$ – Ferdi Apr 30 '18 at 9:52
  • 1
    $\begingroup$ @Fredi if parametrized by location $\mu$ and degrees of freedom $\nu$ it obviously has two parameters. $\endgroup$ – Tim Apr 30 '18 at 9:55
  • $\begingroup$ 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 $\mu$. $\endgroup$ – MM Khan Apr 30 '18 at 10:00

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