A SAS/IML Macro for Computing Percentage Points of Pearson Distributions

The Pearson distribution family provides approximations to a wide variety of observed distributions using the ﬁrst four moments or the ﬁrst three moments with a left or right boundary. Curve ﬁtting utilizing Pearson distributions has been extensively applied in many ﬁelds. However, in practice, it is quite unwieldy to obtain percentage points of Pearson distributions when consulting the massive tables of Pearson and Hartley (1972) or using the out-of-date computer programs (Amos and Daniel 1971; Bouver and Bargmann 1974; Davis and Stephens 1983). The present study compiled a convenient computer program for computing the percentage points using the contemporary SAS/IML ( SAS Institute Inc. 2008) macro language.


Introduction
Pearson distributions (Pearson 1895) provide approximations to a wide variety of frequent distributions of empirical data using first four moments, or first three moments and left or right boundary.Curve fitting utilizing Pearson distributions has been extensively applied to statistical methodology as well as many practical fields, such as applied physics (Winterbon 1983), crystallography (Hall, Veeraraghavan, Rubin, and Winchell 1977), oceanography (Delignon, Garello, and Hillion 1997), water resources (US Water Resources Council 1967), and so on.
To find Pearson percentage points, researchers normally rely on Pearson and Hartley (1972)'s tables which give approximate percentage points in terms of skewness √ β 1 and kurtosis β 2 .However, the 28-page tables are unwieldy and often require second difference interpolation for both √ β 1 and β 2 .To deal with the cumbersome calculations, some computer programs were created for computing the percentage points (Amos and Daniel 1971;Bouver and Bargmann 1974;Davis and Stephens 1983).Unfortunately, the existing programs are over 25 years old and, therefore, not compatible with modern statistical software.Hence, it is imperative to have an updated and efficient computer program for computing percentage points of Pearson distributions, which constituted the purpose of the study.
The approximation (1) gives a standardized percentage point y α for a standardized Pearson variable Y .A transformation is needed to obtain an unstandardized percentage point from a standardized percentage point.For a random variable X of interest, its unstandardized percentage point can be calculated from x α = µ + σ × y α , where µ is the mean of X and σ is the standard deviation of X.
If the first four moments are known, the skewness √ β 1 and kurtosis β 2 can be calculated as follows: Note that σ = µ 2 .If only the first three moments and left boundary are available, the value of β 2 can be determined by using Müller and Vahl (1976)'s algorithm.For the right boundary, the value of β 2 can be determined for −X using the same algorithm.

SAS/IML macro program
Based on the numerical method presented in Section 2, a SAS/IML (SAS Institute Inc.When running the macro program, if the first four moments are used (i.e., type = 1), simply input a "." for bndry; if the first three moments and left or right boundary are used (i.e., type = 2 or type = 3), simply input a "." for b2.
The program also has user-friendly features to provide instant failure messages as clues for users to adjust parameter inputs.The failure messages are: Message 0: Successful parameter inputs; Message 1: σ is negative; Message 2: type is not 1, 2, or 3; Message 3: The absolute value of √ β 1 is larger than 2.0; Message 4: µ is impossible with the value entered for boundary; Message 5: β 2 cannot be computed for the first 3 moments and left (or right) boundary; Message 6: β 2 is out of the range, that is (Davis and Stephens 1983):

Evaluation of the program
To evaluate the efficiency of the SAS/IML (SAS Institute Inc. 2008) macro program, some Pearson curve percentage points approximated by the program were compared with the corresponding ones in Pearson and Hartley (1972)'s tables (Table 32, p. 276), taking µ = 0, σ = 1, √ β 1 = 1.3, and β 2 = 4.2 as an example.As can be seen in Table 1, the differences between the Pearson percentage points computed from the SAS/IML macro and those from Pearson and Hartley (1972)'s tables are all less than 0.0025 in term of the absolute value.
In addition, for comparing unstandardized Pearson curve percentage points, we took an example that was used in Pearson and Hartley (1972)  The examples above are for the case that the first four moments are known (i.e., type = 1).In order to verify that the SAS/IML macro is also effective for the case that the first three moments and left or right boundary are known (i.e., type = 2 or type = 3), two more examples are taken, one for type = 2: µ = 0, σ = 1, √ β 1 = 1, and the left boundary = −1; and the other for type = 3: µ = 0, σ = 1, √ β 1 = 0.5, and the right boundary = 2.932345.The SAS/IML macro returned β 2 = 3 for type = 2 and β 2 = 2.4 for type = 3.Table 2 and Table 3 show the the Pearson percentage points computed from the SAS/IML macro and those from Pearson and Hartley (1972)'s tables for type = 2 and type = 3, respectively.From the tables, we can see that the differences between the Pearson percentage points computed from the SAS/IML macro and those from Pearson and Hartley (1972)

Concluding remarks
The new program was written using the SAS/IML (SAS Institute Inc. 2008) macro language that can help researchers obtain Pearson percentage points much more efficiently than computing interpolation with Pearson and Hartley (1972)'s tables.It is worthy to note that this SAS/IML macro as well as other existing computer programs and Pearson and Hartley (1972)'s tables provide a means of finding percentage points of Pearson distributions other than percentiles or probability values of the distributions.In further study, it would be desirable to develop a computer program to obtain probability values of Pearson distributions to meet the needs of significance tests using Pearson distributions.