The documentation for Python's SciPy package provides a table that concisely summarizes functions that are associated with continuous probability distributions. This article provides a similar table for SAS functions.
For more information on the CDF, PDF, quantile, and random-variate functions, see "Four essential functions for statistical programmers."
SAS functions for probability distributions
The following table shows the names and mathematical definitions for common probability distribution functions. The second column links to the SAS documentation. The fourth column explains how to apply a location parameter (L) and a scale parameter (S) to obtain a shifted and scaled quantity. The standard parameterization for the distribution assume L=0 and S=1.
| Function Name | SAS Function | Notation or Definition | Shift and Scale |
|---|---|---|---|
| Cumulative Distribution Function | CDF | \(F\left(x\right)\) | \(F\left(x;L,S\right)=F\left(\frac{x-L}{S}\right)\) |
| Quantile Function | QUANTILE | \(G\left(\alpha\right)=F^{-1}\left(\alpha\right)\) | \(G\left(\alpha;L,S\right)=L+SG\left(\alpha\right)\) |
| Survival Function | SDF | \(S\left(x\right)=1-F\left(x\right)\) | \(S\left(x;L,S\right)=S\left(\frac{x-L}{S}\right)\) |
| Inverse Survival Function | SQUANTILE | \(Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)\) | \(Z\left(\alpha;L,S\right)=L+SZ\left(\alpha\right)\) |
| Probability Density Function | \(f\left(x\right)=F^{\prime}\left(x\right)\) | \(f\left(x;L,S\right)=\frac{1}{S}f\left(\frac{x-L}{S}\right)\) | |
| Random Variates | RAND | \(Y=G\left(U\right)\), where \(U \sim U(0,1)\) | \(X=L+SY\) |
| Log CDF | LOGCDF | \(\log\left( F\left(x\right) \right)\) | \(\log\left( F\left(x;L,S\right) \right)\) |
| Log PDF | LOGPDF | \(\log\left( f\left(x\right) \right)\) | \(\log\left( f\left(x;L,S\right)\right)\) |
| Log SDF | LOGSDF | \(\log\left( S\left(x\right) \right)\) | \(\log\left( S\left(x;L,S\right)\right)\) |
Some authors refer to the quantile function as the percent-point function (PPF) or the inverse distribution function.
Related functions for probability distributions
The following functions are related to the mathematical functions in the previous section. Although there is no built-in SAS function to evaluate these function, you can use SAS language to evaluate them by using the CDF and SDF function in the previous section.
| Function Name | Definition | Shift and Scale |
|---|---|---|
| Cumulative Hazard Function | \(H\left(x\right)=\) \(-\log\left(1-F\left(x\right)\right)\) | \(H\left(x;L,S\right)=H\left(\frac{x-L}{S}\right)\) |
| Hazard Function | \(h\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}\) | \(h\left(x;L,S\right)=\frac{1}{S}h\left(\frac{x-L}{S}\right)\) |
1 Comment
Nice summary. Thanx.