Gamma Distribution for Quantile-Quantile Plots

The Gamma distribution has the probability density function:

f(x) = {1/[G(c)b]}*[(x-q)/b]c-1 *e[-(x-q)/b]

0 <= x, c > 0, b > 0

where

G

(Gamma) is the Gamma function (of argument Alpha)

c

is the Shape parameter

q

is the Threshold (location) parameter

b

is the Scale parameter

e

is the base of the natural logarithm, sometimes called Euler's e (2.71...)

The inverted Gamma variate has probability distribution function (with quantile y) is:

[exp(-l/y)*lc * (1-y)c+1]/G(c)

where

l = 1/b

The standardized Gamma distribution (Scale = 1, Threshold = 0) with Shape parameter will be used to find the best fitting distribution function. The Shape parameter can be specified in one of two ways:

  1. On the Quantile-Quantile Plots - Advanced tab, enter user-defined values for the Shape parameter, and clear the Compute parameters from: check box.

  2. Estimate the Shape parameter by selecting the Compute parameters from: check box and entering a user-defined Threshold parameter. The Shape parameter will be estimated using either the maximum likelihood or matching moments approximation (see below).

In general, if the points in the Q-Q plot form a straight line, then the respective family of distributions (Gamma distribution with the Shape parameter c in this case) provides a good fit to the data; in that case, the intercept and slope of the of the fitted line can be interpreted as graphical estimates of the threshold (q) and scale (b) parameters, respectively.

Use Max. Likelihood. The Use Max. Likelihood check box is available on the Quantile-Quantile Plots - Advanced tab when the Compute parameters from: check box is selected. When Use Max. Likelihood is selected, STATISTICA uses the maximum likelihood parameter method to estimate the Shape parameter of the Gamma distribution (see Evans, Hastings, & Peacock, 1993, for details). If the check box is cleared, then the method of matching moments is used.