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How can I get the probability density function (PDF) of standard logistic distribution from the PDF of symmetric student's t-distribution with location and scale parameters. In the $(\beta_1, \beta_2)$ graph on this link, the logistic distribution lies on the straight line of symmetric student's t-distribution. If it is 'complex', please share the reference.

In addition, if possible, how can I get the PDF of "skew-logistic" distribution (a.k.a. generalized logistic distribution type I) from the PDF of skewed t distribution proposed by [i] and mentioned on this wiki page.

[i] Hansen, B (1994). "Autoregressive Conditional Density Estimation". International Economic Review. 35: 705–730. doi:10.2307/2527081.

MM Khan
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Forgive me for contradicting you, but the student's t pdf is not the same thing as the logistic pdf. It is possible for them to have the same skew and kurtosis, as indicated in the link provided; however, that doesn't imply that they can have the same pdf. From Wikipedia, we can see that the logistic distribution has skewness of 0 and kurtosis of 4.2, and the student's t has skewness of 0 and kurtosis of $\frac{6}{\nu - 4}$. Basic calculations show that the t and logistic distributions will have the same kurtosis when $\nu = \frac{11}{4}$ (if my working is accurate!) This completely determines the t distribution, since $\nu$ is its only parameter. The logistic distribution, on the other hand, still has 2 parameters, $\mu$ and $s$; moreover, the first moment of the logistic distribution (the mean) is $\mu$, whereas it is always 0 (or not defined) for the t distribution. To summarise: just because two distributions might have the same third, second, and fourth moments, they may not have the same first moment, meaning that they may not have the same pdf. Let me know if this makes sense.

notebook
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  • You are right. The logistic PDF being covered on the skewness-kurtosis plot by student's t PDF does not imply that logistic is a special case of student's t. PS: I was talking about the [non-standardized student's t distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution#Generalized_Student's_t-distribution) which has three parameters instead of only $\nu$. – MM Khan May 03 '18 at 07:22