Computation of parameters of T-student distribution

To compute the parameters of a T-student distribution we have to take in account the fact that some moments of the distribution might not exist, because they diverge, so we use fractional moments to avoid problems. If we consider:

$$<S^{F\mp1}>=\frac{1}{T}\sum_{t=1}^{T}\left\vert R(t)\right\vert^{F\mp1}$$ (A-1)

as a fractional moment of the distribution of the returns $R(t)$ because $F$ is a fractional number, and we compute the rate of the moments as:

$$r_{F}=\frac{<S^{F-1}>}{<S^{F+1}>}=\frac{1}{(2 k -3)\sigma^2}\left[ (k-1) \frac{2}{F}\right] - \frac{1}{(2 k -3) \sigma^2}$$ (A-2)

for different exponents in the interval: $\frac{2}{3} < F < 1$, we can see that $r_{F}=a \frac{1}{F}+b$ is a linear function of the parameters, so we can take the values of $\sigma$ and $k$ from the linear regression:

$$k=\frac{2a - b}{2 a}$$ (A-3)

and

$$\sigma^2 = \frac{1}{a+b}$$ (A-4)