A Probability Distribution and Its Uses in Fitting Data

208RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKAalso be calculated. For a specified frequency histogramthis requires the solution of (2) for p at each ofthe interval endpoints. This is straightforward numerically,since R(p) is a well-behaved increasingfunction of p. Alternatively, intervals can be determinedusing the estimated percentile function withspecified (perhaps equal) probabilities. Then one simplycounts the number of observations falling in eachinterval and computes the approximate X2 statistic inthe usual manner. (See, for example, p. 302 of [6].)One might wish to use some criterion other thanmoments for estimating the parameters. For example,nonlinear least squares can be used to obtainthe values of the parameters which minimize thesquared distance between the percentile function (2)and the empirical percentile function.4. TABLE CONSTRUCTION AND ACCURACYTable 4 gives the values of X,, XA, X3 and A4 correspondingto values of a3 and a4 with zero mean andunit variance. (A shorter preliminary version of Table4 appeared in [3].) The lambda values are given for a3ranging from 0.0 and 0.90 in steps of 0.05 and from0.90 to 2.0 in steps of 0.1. For each a3, lambda valuesare given for 39 values of a4 in steps of 0.2, exceptwhere SA and X4 are near zero and are given in steps of0.1, starting from the lower limit indicated in Figure3. The grid of a3 and a4 values is sufficiently fine sothat interpolation is generally unnecessary. (The densityplots of adjacent entries in the table will differonly slightly, even though the lambda values maydiffer substantially.)The values of Table 4 have been rounded to fourdecimal digits, except where indicated by a plus sign(+) or a dollar sign ($). The plus sign (+) indicatesthat the parameter value has two leading zeroes andconsequently has been rounded to six decimal digitsand multiplied by 100. Similarly, those values indexedby a dollar sign ($) have four leading zeroesand have been rounded to eight digits and multipliedby 10,000. This was done to provide as many significantdigits as possible in order to enhance the accuracyof the tabled value. Tabled values marked by aplus sign (+) should thus be multiplied by 10-2 andthose marked by a dollar sign ($) should be multipliedby 10-4.The vast majority of the tabled values yield differencesbetween the calculated and specified values ofa3 and a4 of less than 0.01, i.e. |la(X3, A4) - ajl 1.0. The values of XAand X2 were checked in a similar fashion and yielded amean within 0.01 of zero and a variance within 0.01of one. These values result from closed form calculationsand can be calculated by the user exactly.TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979For specified values of a3 and a4, the 3, and A4values were originally computed by Tadikamalla [15]by solving the nonlinear equationsa3(A3, A4) = a3a4(X3, A4) = a4,using the subroutine ZSYSTM in the IMSL system(available from International Mathematical & StatisticalLibraries, GNB Bldg., 7500 Bellaire, Houston,TX 77036). The expressions for a3(X3, X4) and a4(X3,A4) were obtained from equations (4) and (5).The table given here is an extended and correctedversion of Tadikamalla's. New values were obtainedusing the Nelder-Mead simplex procedure for functionminimization where the objective functionf(A3, X4) = (a3(A3, A4) - a3)2 + (a4(X3, X4) - a4)2was minimized over A3 and X4, subject to the constraintthat X3X4 > 0, which insures that X3 and X4 areof the same sign. (See Olsson and Nelson [8] for adiscussion and applications of the Nelder-Mead simplexprocedure.)Solutions other than those given in the table mayexist for certain values of a3 and a4, usually with X3 >1 and X4 > 1, for which the corresponding densityfunctions are truncated.Either of the two procedures just described can beused to obtain the lambda parameters correspondingto a3 and a4 values not in the table.5. NUMERICAL EXAMPLEWe illustrate our method with the following datataken from Hahn and Shapiro [6]. Measurements ofthe coefficient of friction for a metal were obtained on250 samples. The resulting values are summarized inthe first two columns of Table 3. Hahn and Shapiro[6] give the following values for the moments whichwere calculated from the original data: x = 0.0345,/m2 = 0.0098, &8 = 0.87 and a4 = 4.92. Rounding a3and a4 to the nearest tabular values (0.85 and 4.9respectively), the lambda values from Table 4 are X,= -.413, A2 = .0134, X3 = .004581, X4 = .0102. Thevalues for A, and A2 are calculated asandi = -.413 (.0098) + .0345 = .0305A2 = .0134/.0098 = 1.3673.Figure 1 shows the relative frequency histogramfor this data and the probability density curve correspondingto the above lambda values. The observedand expected frequencies are given in Table 3. Acomparison of the computed value of x2 (2.04) withthe tabulated x2 values (see Table 3) indicates that the