In which scenarios is the exponential power distribution better than the t-student distibution? I know there are some differences between these two distributions (exponential power distribution does have a cusp at the origin and t-student doesn't), but there are also similarities (heavy tails).
I'm wondering for example when dealing with time series and when the normality assumption is violated, which distribution is a better option and why. The situation doesn't have to be strictly connected with time series analysis. I am curious to see other examples too.
 A: A big difference between the exponential power distribution and the t-distribution is the tail behaviour.

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*The t-distribution has polynomial tails. The heaviness of the tails (order) is dictated by the degrees of freedom.


*The exponential power distribution has exponential tails, and the power parameter also dictates how heavy the tails are, but these are always exponential $\sim \exp(-\vert x\vert^p)$.
Thus, the t-distribution has heavier tails than those of the exponential power distribution for any values of the shape parameters (degrees of freedom and power parameter).
Which one is better? It depends on the tails of the true generating mechanism: whether this is more similar to a distribution with exponential tails or with polynomial tails.
In time series, tail behaviour is important, but also skewness, in order to capture departures from normality. For instance:

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*This paper studies "A generalized asymmetric Student- distribution with application to financial econometrics". Thus, polynomial behavior + asymmetry.

*This paper studies "Properties and estimation of asymmetric exponential power distribution". Thus, exponential tail behavior + asymmetry.

*This paper studies another distribution (DTP SAS) which has tails heavier than exponential but lighter than polynomial + asymmetry, thus a trade-off between the previous ones.

