A Probability Distribution and Its Uses in Fitting Data

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A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA209model fits the data quite well. However, since theparameters of the model are estimated by the methodof moments rather than by the maximum likelihoodmethod, the use of the x2 distribution is only approximate.6. SUMMARYA four-parameter distribution and a table facilitatingparameter estimation using the first four samplemoments have been presented. A wide variety ofcurve shapes are possible with this distribution asindicated by the figures in Section 2. Because of thisflexibility and the inherent simplicity of this distributionit is useful in fitting data when, as is often thecase, the underlying distribution is unknown. Thedefinition of the distribution leads to a simple algorithmfor generating random variates as is discussedin Section 2.7. ACKNOWLEDGMENTSThe comments of the referees were helpful in improvingthe presentation and are acknowledged withthanks. Support for John S. Ramberg's research wasprovided by National Institutes of Health Grant No.GM22271-02. Support for Edward F. Mykytka's researchwas provided by the Graduate College of theUniversity of Iowa.REFERENCES[1] BURR, I. W. (1973). Parameters for a general system ofdistributions to match a grid of a3 and a4. Comm. Statist., 2,1-21.[2] DUDEWICZ, E. J. (1976). Introduction to Statistics andProbability. New York: Holt, Rinehart and Winston.[3] DUDEWICZ, E. J., RAMBERG, J. S., and TADIKA-MALLA, P. R. (1974). A distribution for data fitting andsimulation. Annual Technical Conference Transactions of theAmerican Society for Quality Control, 28, 407-418.[4] DUDEWICZ, E. J., JOHNSON, M. E. and RAMBERG,J. S. (1976). Fitting distributions to data with moments:sampling variability effects. Annual Technical ConferenceTransactions of the American Society for Quality Control, 30,337-344.[5] FILLIBEN, J. J. (1969). Simple and robust linear estimationof the location parameters of a symmetric distribution. Ph.Dthesis, Princeton University.[6] HAHN, G. J. and SHAPIRO, S. S. (1967). Statistical Modelsin Engineering. New York: John Wiley & Sons, Inc.[7] JOINER, B. L. and ROSENBLATT, J. R. (1971). Someproperties of the range in samples from Tukey's symmetriclambda distribution. J. Amer. Statist. Assoc., 66, 394-399.[8] OLSSON, D. M. and NELSON, L. S. (1975). The Nelder-Mead simplex procedure for function minimization. Technometrics,17, 45-51.[9] RAMBERG, J. S. and SCHMEISER, B. W. (1972). Anapproximate method for generating symmetric random variables.Comm. ACM, 15, 987-990.[10] RAMBERG, J. S. and SCHMEISER, B. W. (1974). Anapproximate method for generating asymmetric random variables.Comm. ACM, 17, 78-82.[11] RAMBERG, J. S. (1975). A probability distribution withapplications to Monte Carlo simulation studies. StatisticalDistributions in Scientific Work: Vol. 2-Model Building andModel Selection. Edited by G. P. Patil, S. Kotz and J. K.Ord. Boston: D. Reidel Publishing Co.[12] SCHMEISER, B. W. (1971). A general algorithm for generatingrandom variables. Master's thesis, The University ofIowa.[13] SCHMEISER, B. W. (1977). Methods for modelling andgenerating probabilisticomponents in digital computer simulationwhen the standard distributions are not adequate: asurvey. Proceedings of the Winter Simulation Conference, 51-57.[14] SILVER, E. A. (1977). A safety factor approximation basedupon Tukey's lambda distribution. Operational ResearchQuarterly, 28, 743-46.[15] TADIKAMALLA, P. R. (1975). Modeling and generatingstochastic inputs for simulation studies. Ph.D. thesis, TheUniversity of Iowa.[16] TUKEY, J. W. (1960). The Practical Relationship Between theCommon Transformations of Percentages of Counts and ofAmounts. Technical Report 36, Statistical Techniques ResearchGroup, Princeton University.[17] VAN DYKE, J. (1961). Numerical Investigation of the RandomVariable y = C (uX- (I - ut). Unpublished workingpaper, National Bureau of Standards Statistical EngineeringLaboratory.TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979
210RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKATABLE 4-Lambda parameters for given values of skewness (a,) and kurtosis (a.) when ji = 0 and a = I.a3 - 0=0a4 LAN 1 LAN2 LAN 3 LAN 4aL3 = 0.05O4 LAM 1 LA5 2 LAM 3 LIN 4a3 = 0.10a4 LAN 1 LAN 2 Lill 3 LAN 41.8 .02.0 .02.2 .02.4 .02.6 .02.8 .03.0 .03.2 .03.4 .03.6 .0.5774 1.0000 1.0000.4952 .5843 .5843.4197 .4092 .4092.3533 .3032 .3032.2949 .2303 .2303.2433 .1765 .1765.1974 .1349 .1349.1563 .1016 .1016.1191 .0742 .0742.0852 .0512 .05121.8 -1.703 .2861 .0000 .9502*2.0 -1.229 .3122 .0505 .76032.2 -.802 .3314 .1128 .58022.4 -.375 .3328 .1876 .39412.6 -.143 .2924 .1973 .26052.8 -.083 .2429 .1625 .19033.0 -.059 .1975 .1276 .14253.2 -0 4 6 .1 56 5 .0974 .10613.4 -.038 .1194 .0718 .07703.6 -.033 .0856 .0499 .05301.8 -1.678 .2835 .0000* .9071*2.0 -1.271 .3028 .0412 .73732.2 -.872 .3177 .0941 .57002.4 -.515 .3164 .1477 .41162.6 -.269 .2863 .1678 .28312.8 -.164 .2417 .1486 .20333.0 -.117 .1977 .1205 .15033.2 -.092 .1572 .0936 .11113.4 -.076 .1203 .0698 .08033.6 -.065 .0866 .0490 .05523.8 .04.0 .04.1 .0U.2 .04.3 .04.4 .04.6 .04.8 .05.0 .05. 2 .0.0545 .0317 .0317.0262 .0148 .0148.0128 .7140+ .7140*-.0659. -.0363. -.0363*-.0123 -.6706* -.6706.-.0241 -.0130 -.0130-.0466 -.0246 -.0246-.0676 -.0350 -.0350-.0870 -.0443 -.0443-.1053 -.0528 -.05283.8 -.027 .0548 .0311 .03274.0 -.026 .0264 .0146 .01534.1 -.024 .0132 .7184+ .7504.4.2 -.024 .0704. .0380. .0397*4.3 -.022 -.0120 -.6386+ -.6643.4.4 -.022 -.0238 -.0126 -.01314.6 -.018 -.0462 -.0240 -.02484.8 -.019 -.0671 -.0342 -.03545.0 -.016 -.0867 -.0435 -.04485.2 -.016 -.1050 -.0519 -.05343.8 -.057 .0558 .0308 .03424.0 -.049 .0276 .0149 .01634.1 -.048 .0142 .7606+ .8302.4.2 -.046 .1440+ .0762* .0828*4,3 -.044 -.0109 -.5703+ -.6174.4.4 -.041 -.0227 -.0118 -.01274.6 -.037 -.0452 -.0231 -.02474.8 -.036 -.0661 -.0332 -.03545.0 -.033 -.0857 -.0424 -.04505.2 -.032 -.1040 -.0507 -.05375.4 .05.6 .05.8 .06.0 .0?.2 .06.4 .06.6 .06.8 .07.0 .07.2 .0-.1227 -.0606 -.0606-.1389 -.0677 -.0677-.1541 -.0742 -.0742-.1686 -.0802 -.0802-.1823 -.0858 -.0858-.1954 -.0910 -.0910-.2077 -.0958 -.0958-.2194 -.1003 -.1003-.2306 -.1045 -.1045-.2414 -.1085 -.10855.4 -.015 -.1222 -.0596 -.06125.6 -.014 -.1386 -.0667 -.06845.8 -.014 -.1538 -.0731 -.07506.0 -.013 -.1682 -.0791 -.08106.2 -.012 -.1820 -.0847 -.08666.4 -.012 -.1950 -.0899 -.09186.6 -.012 -.2074 -.0947 -.09676.8 -.011 -.2192 -.0992 -.10127.0 -.011 -.2303 -.1034 -.10547.2 -.010 -.2411 -.1074 -.10945.4 -.030 -.1213 -.0584 -.06165.6 -.028 -.1375 -.0654 -.06885.8 -.027 -.1530 -.0719 -.07556.0 -.027 -.1674 -.0778 -.08166.2 -.025 -.1811 -.0834 -.08726.4 -.024 -.1943 -.0886 -.09256.6 -.023 -.2066 -.0934 -.09736.8 -.023 -.2184 -.0979 -.10197.0 -.022 -.2297 -.1021 -.10627.2 -.021 -.2405 -.1061 -.11027.4 .07.6 .07.8 .08.0 .08.2 .08.0 .08.6 .08.8 .09.0 .0-.2518 -.1123 -.1123-.2615 -.1158 -.1158-.2709 -.1191 -.1191-.2800 -.1223 -.1223-.2887 -.1253 -.1253-.2969 -.1281 -.1281-.3050 -.1308 -.1308-.3128 -.1334 -.1334-.3203 -.1359 -.13597.4 -.010 -.2515 -.1112 -.11327.6 -.979. -.2613 -.1147 -.11677.8 -.999, -.2707 -.1180 -.12018.0 -.928+ -.2797 -.1212 -.12328.2 -.906+ -.2884 -.1242 -.12628.4 -.931+ -.2968 -.1270 -.12918.6 -.912. -.3048 -.1297 -.13188.8 -.852. -.3125 -.1323 -.13439.0 -.837. -.3201 -.1348 -.13687.4 -.020 -.2507 -.1099 -.11397.6 -.020 -.2606 -.1134 -.11757.8 -.020 -.2699 -.1167 -.12088.0 -.019 -.2791 -.1199 -.12408.2 -.019 -.2878 -.1229 -.12708.4 -.018 -.2961 -.1258 -.12988.6 -.017 -.3041 -.1285 -.13258.8 -.017 -.3119 -.1311 -.13519.0 -.017 -.3193 -.1335 -.1376a73 - 0.15a'3 = 0.20a3 = 0.25Ca4 LAI 1 LAn 2 LAN 3 LAM 4a4 LAMI 1 LAM2 LAM 3 LAM 4a4 LAM 1 LAN 2 LAM 3 LAN 41.8 -1.655 .2811 .0000* .8700*2.0 -1.323 .2934 .0314 .72042.2 -.940 .3056 .0782 .56232.4 -.617 .3031 .1215 .41942.6 -.376 .2791 .1435 .29942.8 -.244 .2397 .1350 .21563.0 -.177 .1980 .1135 .15863.2 -.138 .1584 .0901 .11673.4 -.114 .1219 .0682 .08433.6 -.098 .0884 .0485 .05812.0 -1.387 .2841 .0212 .70902.2 -1.011 .2947 .0638 .55712.4 -.706 .2919 .1013 .42462.6 -.471 .2718 .1233 .31202.8 -.322 .2374 .1221 .22733.0 -.237 .1983 .1065 .16723.2 -.187 .1599 .0866 .12303.4 -.154 .1240 .0667 .08893.6 -.132 .0908 .0482 .06153.8 -.116 .0601 .0314 .03892.0 -1.465 .2748 .0105 .70342.2 -1.084 .2847 .0506 .55482.4 -.790 .2820 .0843 .42942.6 -.558 .2650 .1062 .32262.8 -.398 .2349 .1099 .23853.0 -.298 .1987 .0996 .17633.2 -.237 .1619 .0831 .13003.4 -.196 .1266 .0653 .09423.6 -.167 .0937 .0481 .06563.8 -.147 .0632 .0321 .04213.8 -.086 .0577 .0310 .03634.0 -.076 .0294 .0155 .01784.1 -.073 .0160 .8378+ .9564.4.2 -.069 .3217. .1667+ .1890.4.3 -.066 -.9113. -.4680* -.5278.4.4 -.063 -.0210 -.0107 -.01204.6 -.056 -.0435 -.0218 -.02424.8 -.055 -.0644 -.0318 -.03515.0 -.051 -.0842 -.0410 -.04495.2 -.048 -.1025 -.0493 -.05375.4 -.045 -.1198 -.0569 -.06175.6 -.043 -.1361 -.0639 -.06905.8 -.042 -.1514 -.0703 -.07576.0 -.040 -.1660 -.0763 -.08196.2 -.038 -.1798 -.0819 -.08766.4 -.037 -.1928 -.0870 -.09296.6 -.035 -.2053 -.0919 -.09786.8 -.034 -.2172 -.0964 -.10247.0 -.033 -.2284 -.1006 -.10677.2 -.032 -.2392 -.1046 -.11077.4 -.031 -.2496 -.1084 -.11457.6 -.030 - .2 593 -.1119 -.11807.8 -.029 -.2688 -.1153 -.12148.0 -.028 -.2780 -.1185 -.12468.2 -.028 -.2866 -.1215 -.12768.4 -.027 -.2948 -.1243 -.13048.6 -.027 -.3031 -.1271 -.13328.8 -.026 -.3108 -.1297 -.13579.0 -.025 -.3183 -.1322 -.13824.0 -.103 .0318 .0164 .01984.1 -.097 .0185 .9467+ .01134.2 -.093 .5707. .2894* .3429*4.3 -.089 -.6641. -.3342. -.3929*4.4 -.085 -.0185 -.9261* -.01084.6 -.079 -.0410 -.0202 -.02334.8 -.074 -.0622 -.0302 -.03455.0 -.069 -.0818 -.0392 -.04445.2 -.065 -.1003 -.0475 -.05345.4 -.061 -.1176 -.0551 -.06155.6 -.o5e -.1339 -.0621 -.06895.8 -.055 -.1494 -.0686 -.07576.0 -.053 -.1639 -.0745 -.08196.2 -.051 -.1778 -.0801 -.08776.4 -.049 -.1909 -.0853 -.09306.6 -.047 -.2034 -.0901 -.09806.8 -.045 -.2153 -.0947 -.10267.0 -.044 -.2265 -.0989 -.10697.2 -.043 -.2374 -.1029 -.11107.4 -.041 -.2477 -.1067 -.11487.6 -.040 -.2577 -.1103 -.11847.8 -.039 -.2671 -.1136 -.12188.0 -.038 -.2762 -.1168 -.12508.2 -.037 -.2850 -.1199 -.12808.4 -.036 -.2935 -.1228 -.13098.6 -.035 -.3014 -.1255 -.13368.8 -.035 -.3092 -.1281 -.13629.0 -.034 -.3168 -.1306 -.13879.2 -.034 -.3241 -.1330 -.14114.0 -.131 .0351 .0176 .02244.1 -.126 .0217 .0108 .01364.2 -.118 .8889. .4408* .5467.4.3 -.113 -.3476. -.1713* -.2103*4.4 -.108 -.0154 -.7540. -.9175.4.6 -.099 -.0380 -.0184 -.02204.8 -.094 -.0591 -.0282 -.03345.0 -.087 -.0790 -.0373 -.04365.2 -.082 -.0974 -.0455 -.05275.4 -.077 -.1149 -.0531 -.06105.6 -.073 -.1312 -.0601 -.06855.8 -.070 -.1467 -.0665 -.07546.0 -.067 -.1613 -.0725 -.08176.2 -.064 -.1753 -.0781 -.08766.4 -.062 - .1 885 -.0833 -.09306.6 -.059 -.2010 -.0882 -.09806.8 -.058 -.2129 -.0927 -.10277.0 -.055 -.2242 -.0970 -.10707.2 -.054 -.2350 -.1010 -.11117.4 -.052 -.2455 -.1048 -.11507.6 -.051 -.2554 -.1084 -.11867.8 -.049 -.2649 -.1118 -.12208.0 -.048 -.2742 -.1151 -.12528.2 -.047 -.2829 -.1181 -.12838.4 -.046 -.2914 -.1210 -.13128.6 -.044 -.2995 -.1238 -.13398.8 -.044 -.3072 -.1264 -.13659.0 -.043 -.3147 -.1289 -.13909.2 -.042 -.3220 -.1313 -.1414The parameter values given in this table are for a variate with zero mean and unit variance. The procedure for adjusting the parameters toreflect a different mean or variance is given in Section 3. A plus sign (+) next to a tabled value indicates that the value has two leading zeroesand should be multiplied by 10-2. Similarly, a dollar sign ($) next to a tabled value indicates that the value should be multiplied by 10-4. Anasterisk (*) next to a tabled value of Xj indicates that the difference between the calculated and specified values of aj, i.e. I ai(Xq,X4) - ai I, issomewhat greater than 0.01. See Section 4 for a discussion of the construction and accuracy of this table.TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979
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A Probability Distribution and Its Uses in Fitting Data

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