# Beta Distribution for Quantile-Quantile Plots

The Beta distribution has the probability density function:

f(x) = G(n+w)/[G(n)G(w)] * [(x-q)n-1 * (s+q-x)w-1]/sn+w-1

0 < x < 1, n > 0, w > 0

where

 G (Gamma) is the Gamma function (of argument Alpha) n, w are the Shape parameters q is the Threshold (location) parameter s is the Scale parameter

The inverted Beta distribution is related to the B(n, w) variate by: IB(n, w)~B(n, w)/[1-B(n, w)]

The standardized Beta distribution with Shape parameters Shape1 and Shape2 will be used to find the best fitting distribution function. The Shape parameters can be specified in one of two ways:

1. On the Quantile-Quantile Plots - Advanced tab, enter user-defined values for Shape1 and Shape2 and clear the Compute parameters from check box.

2. Estimate the Shape1 and Shape2 parameters by selecting the Compute parameters from check box and entering user-defined Threshold and Scale parameters. The Shape parameters 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 (Beta distribution with the respective n and w parameters in this case) provides a good fit to the data; in that case, the intercept and slope of the fitted line can be interpreted as graphical estimates of the threshold (q) and scale (s) 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 parameters of the Beta distribution (see Evans, Hastings, & Peacock, 1993, for details). If the check box is cleared, then the method of matching moments is used.