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