A Probability Distribution and Its Uses in Fitting Data

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A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA2051.00.75III: approximation using tabledparameter valuesII: approximation with X3 = 0Cnz0.50.25-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0XFIGURE 2e. Two densities with approximately the same first four moments as the exponentialdistribution.whereA = 1/(1 + A)-,/3 = (C- 3AB + 2A3)/A23,4, = (D - 4AC + 6A2B -1/(1 + X4),B = 1/(1 + 2X3)+ 1/(1 + 2A4)- 2j(1 + A3, 1 + A4),C = 1/(1 + 3A3)- 3/(1 + 23X, 1 + A4)3A')/24,+ 3/(1 + A3, 1 + 2A) - 1/(1 + 3X4),D = 1/(1 + 4X3)-4/(1+ 6/(1 + 2A3, I + 2X4)+ 3A3, 1 + X,)- 4f(1 + A3, 1 + 3A4) + 1/(1 + 4X4).The skewness and kurtosis, as given byanda3 =a4 =a3/a3A4/0'4,are functions of A3 and A4, but do not depend upon XAand X2.3. PARAMETER ESTIMATION ANDDISTRIBUTION FITTING(4)(5)In this section we show how to determine the parametersof the distribution using the first four momentsand how to fit the resulting distribution. Itshould be recognized that sample moments are sensitiveto extreme observations and that the samplingvariability of the third and fourth moments can belarge. (See, for example, Dudewicz, Johnson andRamberg [4].) However, we elected to use momentsbecause of their widespread use in practice; for someproperties of moments, see pp. 174ff of [2].The values of XA, A2, ,A and 34 are given in Table 4for selected values of a3 and a4 with Au = 0 and a = 1.(The construction and accuracy of this table is discussedin Section 4.) If the values of u, a, as and a4are known, the lambda values are determined fromTable 4 using as entry points the a3 and a4 values.One simply picks the values of A3 and A4 in Table 4 forwhich the a3 and a4 are closest to the desired values.If a3 is negative, one uses its absolute value, and afterfinding the values of 3, and A4, interchanges theirvalues and changes the sign of A1. (The density with askewness of -a3 is the mirror image of the densitywith a skewness of a3.)Since the XA and A2 values given in Table 4 are for avariate with a mean of zero and a variance of one,multiplying the resulting variate by a and adding ,u toit achieves the desired result. This reduces to computingTABLE I-Lambda values for Figure 2 densities (see Table 4 fordefinition of + and $ symbols).a3 Ia40 1.750 90.2 1.800.3 1.850.5 4X1 X2 3.5943 1.4501 1.4501.1974 .1349.0262 .01480 -.0870 - .04430 -.3203 -.1359.1349.0148-.0443-.1359.166 .5901 1.7680 1.1773.246 .5852 1.934-.290 .0604 .0259-.886 .1333 .01931.062.0447.1588-.379 -.0562 -.0187 -.0388-.215 -.2356 -.0844 -.1249.007 -.1081+ -.0407$ -.1076+-.0580+ 0 -.0580+TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979
206RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKAQ 230 123 4EXPONENTIAL/FIGURE 3. Characterization of distributions by their third and fourth moments (the proposeddistribution covers the screened region).REGION 1(X31)No positivemoments existREGION 3 (X3>0, X4>0)All positive moments existpositiveskewness1 21/3I/positiveskewnessthird third moment moment exists existsREGION 4 (X30O, X4l, X4< -1)No positivemoments existFIGURE 4. Some properties of the distribution which are dependent on the values of X,and ,4.TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979
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