Weibull Distribution for Quantile-Quantile Plots

The Weibull distribution has the probability density function:

f(x) = c/b*[(x-q)/b]c-1 * e^ -[(x-q)/b]c

0 <= x < ∞, b > 0, c > 0, q > 0

where

b

is the Scale parameter

c

is the Shape parameter

q

is the Threshold (location) parameter

e

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

The inverse distribution function (of probability a) is (for q=0): b{log[1/(1-a)]}1/c

The standardized Weibull distribution 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 Shape1 and Shape2 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 (Weibull distribution with Shape parameter c 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 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 method to estimate the Shape and Scale parameters of the Weibull distribution (see Evans, Hastings, & Peacock, 1993, for details). If the check box is cleared, then the method of matching moments is used.