Non-Normal Distributions - Fitting Distributions by Moments
In addition to the specific continuous distributions described in the other Overviews, Statistica allows you to fit general families of distributions - the so-called Johnson and Pearson curves - with the goal to match the first four moments of the observed distribution.Statistica
General approach
The shapes of most continuous distributions can be sufficiently summarized in the first four moments.
Put another way, if you fit to a histogram of observed data a distribution that has the same mean (first moment), variance (second moment), skewness (third moment), and kurtosis (fourth moment) as the observed data, then you can usually approximate the overall shape of the distribution very well. Once a distribution has been fitted, you can then calculate the expected percentile values under the (standardized) fitted curve, and estimate the proportion of items produced by the process that fall within the specification limits.
Johnson curves
Johnson (1949) described a system of frequency curves that represents transformations of the standard normal curve (see Hahn and Shapiro, 1967, for details).
By applying these transformations to a standard normal variable, a wide variety of non-normal distributions can be approximated, including distributions which are bounded on either one or both sides (e.g., U-shaped distributions). The advantage of this approach is that once a particular Johnson curve has been fit, the normal integral can be used to compute the expected percentage points under the respective curve. Methods for fitting Johnson curves, so as to approximate the first four moments of an empirical distribution, are described in detail in Hahn and Shapiro, 1967, pages 199-220; and Hill, Hill, and Holder, 1976.
Pearson curves
Another system of distributions was proposed by Karl Pearson (e.g., see Hahn and Shapiro, 1967, pages 220-224).
The system consists of seven solutions (of 12 originally enumerated by Pearson) to a differential equation that also approximate a wide range of distributions of different shapes. Gruska, Mirkhani, and Lamberson (1989) describe in detail how the different Pearson curves can be fit to an empirical distribution. A method for computing specific Pearson percentiles is also described in Davis and Stephens (1983).