It occurred to me today that the distribution $$ f(x)\propto\exp\left(-\frac{|x-\mu|^p}{\beta}\right) $$ could be viewed as a compromise between the Gaussian and Laplace distributions, for $x\in\mathbb{R}, p\in[1,2]$ and $\beta>0.$ Does such a distribution have a name? And does it have an expression for its normalization constant? The calculus stumps me, because I don't know how to even begin solving for $C$ in the integral $$ 1=C\cdot \int_{-\infty}^\infty \exp\left(-\frac{|x-\mu|^p}{\beta}\right) dx $$

up vote 31 down vote accepted
+100

Short Answer

The pdf you describe is most appropriately known as a Subbotin distribution ... see the paper in 1923 by Subbotin which has exactly the same functional form, with say $Y = X-\mu$.

  • Subbotin, M. T. (1923), On the law of frequency of error, Matematicheskii Sbornik, 31, 296-301.

who enters the pdf at his equation 5, of form:

$$f(y) = K \exp\left[-\left(\frac{|y|}{\sigma}\right)^p\right]$$

with constant of integration: $K = \large\frac{p}{2 \sigma \Gamma \left(\frac{1}{p}\right)}$, as per Xian's derivation where $\beta = \sigma^p$

Longer answer

Wikipedia is unfortunately not always 'up to date', or accurate, or sometimes just 80 years behind the times. After Subbotin (1923), the distribution has been widely used in the literature, including:

  • 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.

  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995), Continuous Univariate Distributions, volume 2, 2nd edition, Wiley: New York (1995, p.422)

  • 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.

... all before the paper referenced on Wiki. Aside from being 80 years out of date, the name used on Wiki 'a Generalised Normal' also seems inappropriate because there are an infinity of distributions that are generalisations of the Normal, and the name is, in any event, ambiguous to the literature. It also fails to acknowledge the original author.

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    (+1) great references. Are you sure about the $\beta^{-1}$ in the constant? My change of scale argument $$y=\frac{x-\mu}{\beta^{1/p}}$$ rather suggests a $\beta^{-1/p}$ term. – Xi'an Mar 11 '16 at 9:38
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    @Xi'an Thanks for picking that up. Fixed notation to be consistent. I was using the form as defined in Subbotin i.e. $(\frac{|x|}{\sigma})^p$, rather than $\frac{{|x|}^p}{\beta}$. All fixed. Sorry for any confusion. – wolfies Mar 11 '16 at 15:07

For obvious reasons, you can get rid of μ and β so all that remains is $$\int_0^\infty \exp\{−x^p\}\text{d}x\stackrel{y=x^p}{=}\int_0^\infty \exp\{−y\}\left|\dfrac{\text{d}x}{\text{d}y}\right|\text{d}y\stackrel{x=y^{1/p}}{=}\int_0^\infty \exp\{−y\}\frac{1}{p}y^{\frac{1}{p}-1}\text{d}y=\Gamma(1/p)\frac{1}{p} $$ Hence $$\int_{-\infty}^\infty \exp\{−\beta^{-1}|x-\mu|^p\}\text{d}x=\dfrac{2\Gamma(1/p)}{p}\beta^{1/p}$$

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    D'oh. Of course. And chance you happen to know if it has a name? – Sycorax Mar 10 '16 at 19:10
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    It is somewhat connected with the[ Weibull and Fréchet distributions](en.wikipedia.org/wiki/…), however those have a power term in front of the exponential. It is thus more of a Gaussian distribution for another metric that the quadratic distance. – Xi'an Mar 10 '16 at 19:13

According to Wikipedia, this is known as Generalized normal distribution (version 1 in the article), and the restriction $p\in [1,2]$ is not required but any positive value is fine.

The reference given in Wikipedia is Saralees Nadarajah (2005) A generalized normal distribution, Journal of Applied Statistics, 32:7, 685-694, DOI: 10.1080/02664760500079464. This article mentions that the normalization constant is found by 'simple integration' - I presume following Xi'an's answer.

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