While Nadarajah (2005) may have used the term 'generalised Normal' to describe a density that nests this form, there are more suitable names that extend far further back in time, and which accordingly seem much more appropriate. 

In particular, I believe this should properly be referred to as a **Subbotin distribution** (Subbotin 1923). Other later references include: 

* Diananda (1949) 

* Turner (1960) 

* Zeckhauser and Thompson (1970) 

* McDonald and Newey (1988)

* Mineo and Ruggieri (2005)

The functional form given by Subbotin (1923) defines the pdf as:

$$f(x) = \frac{\alpha }{2 b \Gamma \left(\frac{1}{\alpha }\right)}\text{exp}\left[-\left|\frac{x}{b}\right|^{\alpha }\right]$$

Subbotin used parameter $b = 1/h$, but the functional form is otherwise identical to that given here. In this form: $$Var(X) = \frac{b^2 \Gamma \left(\frac{3}{\alpha}\right)}{\Gamma \left(\frac{1}{\alpha}\right)}$$

Here is a plot of the pdf with $b=2$, as parameter $\alpha>1$ varies:


![](https://i.sstatic.net/5DqE3.png)[(source)](http://www.tri.org.au/se/Subbotinpdfplot.png)

and again for parameter $0<\alpha<1$:

![](https://i.sstatic.net/tHyfS.png)[(source)](http://www.tri.org.au/se/Subbotinplotsmallalpha.png)

Other names include:  Box-Tiao distribution (McDonald and Newey 1988), and Power-Exponential (McDonald and Newey 1988, Johnson et al. 1995). Finally, it is worth noting that some economic papers inappropriately ascribe the name 'Subbotin distribution' to an Exponential-Power distribution that has a different functional form.


**References**

* Subbotin, M.T. (1923), On the law of frequency of error, _Matematicheskii Sbornik_, 31, 296-301.

* Diananda, P. H. (1949), Note on some properties of maximum likelihood estimates, _Proceedings of the Cambridge Philosophical Society_, 45, 536-544.

* Turner, M. E. (1960), On heuristic estimation methods, _Biometrics_, 16(2), 299-301.

* Zeckhauser, R. and Thompson, M. (1970), Linear regression with non-normal error terms, _The Review of Economics and Statistics_, 52, 280-286.

* McDonald, J. B. and Newey, W. K. (1988), Partially adaptive estimation of regression models via the generalized t distribution, _Econometric Theory_, 4, 428-457.

* Mineo, A. M. and Ruggieri, M. (2005), A software tool for the Exponential Power distribution: the normalp package, _Journal of Statistical Software_, 12(4), 1-21.