Gamma distribution

Sometimes, the Gaussian distribution curve is not a good description of the data because these are not symmetrically distributed with respect to their values. In such cases, non-symmetric distribution functions may be used to describe the data, such as the Gamma distribution:

$$f(x)= \frac{(x/\beta)^{\alpha-1}exp \left[ - x/\beta \right]}{\beta \Gamma(\alpha)} .$$ (3.6)

$x, \alpha, \beta > 0$.

The estimated ``shape parameter'' $\hat{\alpha}=\frac{\overline{x}^2}{s^2}$ (Note: $\alpha \neq \frac{\overline{x^2}}{s^2}$; p. 85 in Wilks (1995) [] may give the impression that the estimator uses the mean of the square) and the ``scale parameter'' $\hat{\beta}=\frac{s^2}{\overline{x}}$, so that $\hat{\alpha} \hat{\beta} = \overline{x}$. Another method to estimate the gamma-parameters is the maximum-likelihood fitting described by Wilks (1995) [] on p. 89, but this method doesn't allow negative and zero values (can be avoided by using only non-zero values).

$\Gamma(\alpha)$ denotes the gamma-function defined as:

$$\Gamma(\alpha)=\int_0^{\infty} t^{\alpha-1}e^{-t} dt$$

The gamma function has a useful property which is that:

$$\Gamma(\alpha + 1)=\alpha \Gamma(\alpha)$$ (3.7)

If the $\Gamma(\alpha)$ is known for any value $\alpha < 1$, then it is easy to calculate the corresponding value for and number with similar decimal points.

  

Figure 3.4: An example of a non-symmetric distribution and the best-fit Gamma distribution. The plot is based on the daily precipitation measurements made in Oslo (St. Hanshaugen), 01-Jan-1883 to 31-Jul-1964, after which the measurements were made at Blindern. [stats_uib_3_4.m]

Fig. 3.4 shows the distribution function for daily precipitation in Oslo between 1883 and 1964.

An alternative to the Gamma distribution is the Weibull function. Stephenson et al (1999) [] gives a discussion on the distribution of the Indian rainfall data.