Continuous Distributions

You can test the fit of many different continuous distributions to your data using the UNIVARIATE procedure in Base SAS^®, the CAPABILITY procedure in SAS/QC^® (see the distribution options in the various plotting statements of these procedures), the SEVERITY and HPSEVERITY procedures in SAS/ETS^®, or the RELIABILITY procedure in SAS/QC.

The following lists the distribution families available and the procedures that can be used to estimate them:

• Beta (also known as Pearson Type I or Type II distributions and includes Uniform, power-function and arc-sine distributions as special cases) - UNIVARIATE, CAPABILITY
• Burr - SEVERITY, HPSEVERITY
• Exponential (special case of gamma, Weibull, Pareto distributions) - UNIVARIATE, CAPABILITY, SEVERITY, HPSEVERITY, RELIABILITY
• Extreme Value - RELIABILITY
• Gamma (also known as Pearson Type III distributions and includes chi-square, Erlang, and Rayleigh distributions as special cases) - UNIVARIATE, CAPABILITY, SEVERITY, HPSEVERITY
• Pareto and Generalized Pareto (Uniform and exponential are special cases) - UNIVARIATE, SEVERITY, HPSEVERITY
• Gumbel (also known as Type I extreme value distribution) - UNIVARIATE, CAPABILITY
• Inverse Gaussian (the Wald distribution is a special case) - UNIVARIATE, SEVERITY, HPSEVERITY
• Logistic and Log Logistic - RELIABILITY
• Lognormal - UNIVARIATE, CAPABILITY, SEVERITY, HPSEVERITY, RELIABILITY
• Normal/Gaussian - UNIVARIATE, CAPABILITY, RELIABILITY
• Power Function - UNIVARIATE, CAPABILITY
• Rayleigh - UNIVARIATE, CAPABILITY
• Tweedie - SEVERITY (beginning in SAS 9.3 TS1M0), HPSEVERITY
• Johnson S_U and S_B - UNIVARIATE, CAPABILITY (parameters can be estimated but tests of fit are not available)
• Weibull - UNIVARIATE, CAPABILITY, SEVERITY, HPSEVERITY, RELIABILITY

Modeling procedures, such as SAS/STAT^® procedures GENMOD, GLIMMIX, and FMM, can also be used to estimate the parameters of specified distributions. See this note illustrating the use of GENMOD to estimate parameters of several distributions.

Kernel density estimation is also available for fitting distributions of unspecified or more general types such as multimodal distributions.

The SEVERITY and HPSEVERITY procedures can automatically fit all of its predefined distributions to the data and identify the best fitting distribution using several criteria such as AIC, BIC, and others.

The RELIABILITY procedure can estimate the parameters for the common life distributions when the data are complete, right censored, or interval censored.

For details, see these sections of procedure documentation:

• "Details: Formulas for Fitted Continuous Distributions" in the UNIVARIATE procedure documentation,
• "Histogram Statement Details: Formulas for Fitted Curves" in the CAPABILITY documentation,
• "Details: Predefined Distributions" in the SEVERITY or HPSEVERITY documentation, and
• "Detail: Probability Distributions" in the RELIABILITY documentation.

To compare the distributions (of unspecified type) from two or more samples, use the EDF option in the NPAR1WAY procedure in SAS/STAT.

Discrete Distributions

You can use features in the FREQ procedure to test the fit of many discrete distributions. Use the TABLES statement to specify a one-way table of observed frequencies or probabilities. In the TABLES statement, specify the CHISQ option to request a Pearson chi-square test of fit, and the TESTF= or TESTP= option to specify the expected frequencies or probabilities of the hypothesized distribution. If you estimated distribution parameters in order to determine the expected values, you should also specify the DF= option in order to properly adjust the degrees of freedom of the test.

Testing the fit of discrete distributions is further discussed and illustrated in this note.