A Probability Distribution and Its Uses in Fitting Data

American Society for QualityA Probability Distribution and Its Uses in Fitting DataAuthor(s): John S. Ramberg, Pandu R. Tadikamalla, Edward J. Dudewicz, Edward F. MykytkaReviewed work(s):Source: Technometrics, Vol. 21, No. 2 (May, 1979), pp. 201-214Published by: American Statistical Association and American Society for QualityStable URL: http://www.jstor.org/stable/1268517 .Accessed: 31/01/2012 17:12Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jspJSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.American Statistical Association and American Society for Quality are collaborating with JSTOR to digitize,preserve and extend access to Technometrics.http://www.jstor.org

TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979A Probability Distribution and Its Uses in FittingDataJohn S. RambergSystems and Industrial EngineeringThe University of ArizonaTucson, AZ 85721Pandu R. TadikamallaGraduate School of BusinessThe University of PittsburghPittsburgh, PA 15260Edward J. DudewiczDepartment of StatisticsThe Ohio State UniversityColumbus, OH 43210Edward F. MykytkaSystems and Industrial EngineeringThe University of ArizonaTucson, AZ 85721A four-parameter probability distribution, which includes a wide variety of curve shapes, ispresented. Because of the flexibility, generality, and simplicity of the distribution, it is useful inthe representation of data when the underlying model is unknown. A table based on the firstfour moments, which simplifies parameter estimation, is given. Further important applicationsof the distribution include the modeling and subsequent generation of random variates forsimulation studies and Monte Carlo sampling studies of the robustness of statistical procedures.KEY WORDSData fittingProbability distributionSystems of probability distributionsMomentsRandom variate generationMonte CarloSimulation1. INTRODUCTIONReasons for fitting a distribution to a set of datahave been summarized by Hahn and Shapiro [6] (p.195) as: the desire for objectivity, the need for automatingthe data analysis, and interest in the values ofthe distribution parameters. Although various empiricaldistributions already exist, e.g., the Pearson systemand the Johnson system (see Chapter 7 of Hahnand Shapiro [6]) and the Burr distribution [1], we arepresenting another distribution because of its simplicity,flexibility, and generality.The new distribution is a generalization of Tukey's[16] lambda distribution. It was developed by Rambergand Schmeiser [9, 10] for the purpose of gener-ating random variates for Monte Carlo simulationstudies because of the simple form of the resultingalgorithm. (See (2).) A wide variety of curve shapesare possible with this distribution. Hence it is usefulfor the representation of data when the underlyingmodel is unknown. Silver [14], for example, showshow the distribution can be used to approximate thesafety factor in an inventory control model. It is alsouseful in Monte Carlo studies of the robustness ofstatistical procedures and for sensitivity analyses insimulation studies.To illustrate the distribution, consider the histogramfor 250 sample measurements of the coefficientof friction of a metal [6] (p. 219) given in Figure 1.The superimposed distribution was fitted by themethods described in this paper. This example isdiscussed further in Section 5.In Section 2 some of the properties of the distributionare described. Section 3 contains a discussion ofthe use of the method of moments for fitting thedistribution to data and a table to facilitate this procedure.Table construction and accuracy is describedin Section 4.Received August 1976; revised May 1978Winner of 1977 Shewell Award at ASQC Chemical DivisionTechnical Conference2012. THE PROPOSED DISTRIBUTION AND ITS PROPERTIESA continuous probability distribution is usuallydefined by its distribution function or by its densityfunction. Alternatively it can be defined by its per-

202RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKA2724021I.-> 18COal 150L 120Z 9LdILi3- Z Pi 1 1I I I I I I0.0 0.01 0.02 0.03 0.04 0.05COEFFICIENT OF FRICTION0.06 0.07FIGURE 1. Coefficient of friction relative frequency histogram and the fitted distribution.centile (or quantile) function, if the percentile functionexists. The percentile function is simply the inverseof the distribution function. This concept isparticularly useful in Monte Carlo simulation studiesbecause of the following result: If X is a continuousrandom variable with percentile function R, and U isa uniform random variable on the interval zero toone, then the transformation X = R(U) yields a randomvariable with the percentile function R.A specific example is Tukey's [16] lambda functionR(p) = [p - (l - p)X]/ (O?< p ?l1), (1)which is defined for all nonzero lambda values. (As X-- 0, the logistic distribution results.) Van Dyke [17]compared a normalized version of this function withStudent's t distribution. Filliben [5] used this distributionto approximate symmetric distributions with awide range of tail weights for studying location estimationproblems of symmetric distributions. He alsogave a very complete discussion of the properties ofthe percentile function. Joiner and Rosenblatt [7]studied the lambda distribution further and gave resultson the sample range. Ramberg and Schmeiser[9] showed how this distribution could be used toapproximate many of the well-known symmetric distributionsand explored its application to MonteCarlo simulation studies.Ramberg and Schmeiser [10] generalized (1) to afour-parameter distribution defined by the percentilefunctionR(p) = A + [pa - (1 - p4]/A2 (0 < p < 1), (2)where X, is a location parameter, ,A is a scale parameterand A3 and A4 are shape parameters. This distribution,which includes the original lambda distribution,also permits skewed curves to be represented. Althoughthe distribution function does not exist in"simple closed form," this should not be of concernto practitioners since the same is true of the normaldistribution (whose percentiles are not nearly so easilycomputed). Another asymmetric generalization of(1) was considered by Ramberg [11].The density function corresponding to (2) is givenby:f(x) = f[R(p)]- X2[X3pX3-1 + X4(1 -p)4-1]-i(0O

A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA2030.53 =0; Ca4=3,5,90.4> 0.3zLIJ0.20.1 -0.0 I I I I-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0XFIGURE 2a. Density plots for specified a, and a, values (a, = 0; a4 = 3, 5, 9).4.0ponential distribution. The non-truncated densitywas obtained using the parameter values given inTable 4; the other density demonstrates that positivelyskewed, J-shaped curves result when XA = 0.The latter curve provides a good approximation tothe exponential density. Indeed, Schmeiser [13] hasshown that the limiting distribution of this distributionis exponential with parameter 0 as X4 -- 0 when X,= XA = 0 and A2 = 4/0.The distribution can also provide good approximationsto other well-known densities. For example, thedistribution with X, = 0, X2 = 0.1975, and XA = A4 =0.1349 results in an approximation to the normaldistribution for which max, I ?(x) - R-(x)l .001,where I(x) is the normal distribution function.Although distributions are not necessarily deter-mined by their moments, the moments often do provideuseful information. In Figure 3 some distributionsare characterized by their skewness andkurtosis. The normal, the rectangular, and the exponentialdistribution are each represented by asingle point. The Student's t, the lognormal, thegamma, and the Weibull distributions are each representedby a line. The beta distribution is representedby a region of values. The proposed distribution coversthe screened area; refer to Table 4 for some of thevalues included.The Pearson and Johnson systems also cover largeregions of this diagram (see, for example, Hahn andShapiro [6]). Both of these systems incorporate anumber of functional forms whereas the proposeddistribution uses only one function and is computa-0.5 -0.4 -CQ3=1; O4=4,6,90.3 -zw0 r -U.c0.1-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0XFIGURE 2b. Density plots for specified as and a4 values (a(, = 1: (4y = 4, 6. 9).TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979

204RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKA0.51a3=0, .5,1; 0a4=40.4I-(I)zw0.30.20.10.0 I I I-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0XFIGURE 2c. Density plots for specified a3 and a4 values (a3 = 0,.5,1; a4 = 4).4.0tionally simpler. The Burr [1] distribution also coversa wide range of parameter values, but does not includesymmetric distributions.As originally indicated by Schmeiser [12], there arefour regions of parameter values where the distributionis a legitimate one, i.e., the density function isnonnegative for all x and integrates to one. Theseregions, which are numbered arbitrarily 1, 2, 3 and 4for reference purposes, are indicated in Figure 4. Ourinterest will center on Regions 3 and 4 since positivemoments do not exist for the distributions in Regions1 and 2. The values of X range from R(0) to R(1).These bounds, which depend on all of the lambdavalues, are given in Table 2. If A_ 2 1, f[R(0)] > 0 andthe distribution is truncated on the left. Similarly if A> 1, the distribution is truncated on the right.Ramberg and Schmeiser [10] showed that the kthmoment (XA = 0), of the proposed distribution, whenit exists, is given byE(Xk) =\2 -Z ()i (-1 ) (X3(k-i) + 1, X4i + 1),where d denotes the beta function, as defined, forexample, in [2]. (The kth moment does not exist whenany of the arguments of the beta function are negative.Thus, the kth moment exists if and only if - l/k< min(X3, A4).)They also derived the following expressions for themean, the variance, and the third (u3 = E(X - A)3)and fourth (z4 = E(X - )4) moments about themean for this distribution:A = XA + A/A2,a2 = (B-A2)/X2,0.6 - I: a3 = 0.0; a4= 1.75II: aQ3 0.2 a4 = 1.80III: 3 = 0.3; a4 = 1.850.5 ->-z*LJC)0.4 -Ii0.3 -III0.2-1.8 II I I I I L I I-1.2 -0.6 0.0 0.6 1.2 1.8 2.4XFIGURE 2d. Density plots for specified a, and a4 values resulting in U-shaped densities.TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979

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

A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA207TABLE 2-Lower and upper bounds of the distribution.RegionValue ofA2 _3 X4LowerBoundUpperBound2 > 3 >0 A4 > 03 X2 >0 O 30 X > 02 > 0 3 4> 04 = 0x1-1/X2A-1/X2xl+l/X2A1+1/X2A1and2

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

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

A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA211TABLE 4-Continued (see explanatory note, p. 210).a3 = 0.3003 = 0.35aL3 0. 40C04 LAP! 1 LA P!2 LAM83 LAM84a4 LAP! 1 LAN82 LAN83 LAN84a4 LAP! 1 LAN 2 LAN 3 LA B142. 0 -1. 550 .2660 .0000 .70202.2 -1.164 .2755 .0380 .55562.44 -.871 .2733 .0695 .4434482.6 -.642 .2586 .0911 .332142.8 -.4474 .2323 .0983 .24953.0 -.362 .1991 .0925 .18593.2 -.288 .1641 .0796 .13773.44 -.239 .1298 .06440 .10033.6 -.204 .0973 .0481 .07043.8 -.179 .0671 .0330 .0446044.0 -.160 .0389 .0190 .025544.2 -.144 .0127 .6175+ .8035'44.3 -.138 .0789+ .0380' .04489+44.4 -.131 -.0116 -.5554' -.7057+44.5 -.129 -.0231 -.0110 -.013944.6 -.121 -.0343 -.0163 -.02034.8 -.113 -.0554 -.0260 -.03195.0 -.105 -.0752 -.0350 -.04235.2 -.100 -.0939 -.04432 -.05175.4 -.0944 -.11144 -.0508 -.06015.6 -.089 -.1279 -.0578 -.06785.8 -.085 -.14435 -.0643 -.07486.0 -.081 -.1582 -.0703 -.08126.2 -.078 -.1722 -.0759 -.08726.44 -.075 -.1854 -.0811 -.09276.6 -.072 -.1979 -.0860 -.09776.8 -.069 -.2100 -.0906 -.10257.0 -.067 -.2214 -.09449 -.10697.2 -.065- -.2325 -.0990 -.11117.44 -.063 -.2427 -.1028 -.11497.6 -.061 -.2528 -.1064 -.11867.8 -.060 -.2623 -.1098 -.12208.0 -.058 -.2716 -.1131 -.12538.2 -.056 -.2805 -.1162 -.12848.44 -.055 -.2889 -.1191 -.13138.6 -.054 -.2971 -.1219 -.13418.8 -.053 -.3050 -.1246 -.13679.0 -.052 -.3125 -.1271 -.13929.2 -.051 -.3197 -.1295 -.14162.0 -1. 539 .2639 .0000+ . 6836+2. 2 -1. 252 .2668 .0256 .55992.4 -.955 .2653 .0559 .444152.6 -.724 .2528 .0775 .34232.8 -.550 .2298 .0873 .26063.0 -.4427 .1 996 .08544 .19613.2 -.343 .1665 .0758 .14623.4 -.285 .1333 .0625 .10723.6 -.243 .1014 .0482 .07603.8 -.213 .0714 .03440 .05054.0 -.191 .0434 .0206 .02934.2 -.172 .0173 .8158+ .01124.3 -.163 .44870+ .2293+ .3090+4.4 -.156 -.7105+ -.3332+ -.44431+4.5 -.151 -.0187 -.8723+ -.01154.6 -.142 -.0298 -.0139 -.01804.8 -.132 -.0511 -.0236 -.03005.0 -.1244 -.0710 -.0325 -.04075.2 -.117 -.0898 -.0407 -.05035.4 -.110 -.1074 -.0483 -.05895.6 -.105 -.12440 -.0553 -.06685.8 -.100 -.1396 -.0618 -.07396.0 -.098 -.1545 -.0678 -.08056.2 -.091 -.1685 -.0735 -.08656.4 -.088 -.1818 -.0787 -.09216.6 -.085 -.1945 -.0836 -.09736.8 -.082 -.2067 -.0883 -.10217.0 -.079 -.2181 -.0926 -.10667.2 -.077 -.2291 -.0967 -.11087.4 -.074 -.2396 -.1006 -.11477.6 -.072 -.24496 -.10442 -.11847.8 -.070 -.2593 -.1077 -.12198.0 -.068 -.2685 -.1109 -.12528.2 -.066 -.2775 -.1141 -.12838.4 -.065 -.2860 -.1170 -.13138.6 -.064 -.2942 -.1198 -.134418.8 -.062 -.3020 -.1225 -.13679.0 -.060 -.3098 -.1251 -.13929.2 -.059 -.3172 -.1276 -.14172. 2 -1. 354 .2582 .0129 .56832.4 -1. 043 .2580 .0430 .45002.6 -.808 .24473 .06448 .35272.8 -.627 .2273 .0767 .27203.0 -.494 .2000 .0782 .20693.2 -.4400 .1690 .0718 .15553.44 -.333 .1371 .0609 .11493.6 -.284 .1060 .04482 .082443.8 -.248 .0764 .0351 .055844.0 -.222 .04485 .0223 .03374.2 -.200 .02244 .0103 .01494.3 -.190 .0100 .4597. .6521+4.4 -.182 -.0397+ -.0182+ -.02544+'44.5 -.1744 -.0136 -.6204+ -.8533'4.6 -.166 -.02448 -.0113 -.015344.8 -.155 -.0462 -.0209 -.02-775.0 -.146 -.0662 -.0297 -.03875.2 -.136 -.0850 -.0379 -.044855.4 -.129 -.1027 -.0455 -.057445.6 -.122 -.1194 -.0525 -.06545.8 -.115 -.1352 -.0591 -.07276.0 -.111 -.1501 -.0651 -.079446.2 -.106 -.16443 -.0708 -.08566.44 -.102 -.1778 -.0761 -.09136.6 -.098 -.1 906 -.0811 -.09666.8 -.094 -.2026 -.0857 -.10147.0 -.091 -.21442 -.0901 -.10607.2 -.089 -.2253 -.09442 -.11037.4 -.086 -.2359 -.0981 -.114437.6 -.083 -.2459 -.1018 -.11807.8 -.081 -.2558 -.1053 -.12168.0 -.079 -.2650 -.1086 -.li4498.2 -.077 -.2741 -.1118 -.12818.44 -.075 -.2827 -.1148 -.13118.6 -.073 -.2908 -.1176 -.13398.8 -.072 -.2988 -.1203 -.13669.0 -.070 -.3064 -.1229 -.13919.2 -.069 -.3139 -.12544 -.144169.4 -.067 - .3210 -.1278 -.1439Ca3 = 0. 45a44 LAP! I LAM 2 LAN83 LAN 4at3 -0. 50a4 LAM! 1 LAN82 LAN 3 LAN 4Ct3 = 0.55Ca4 LAP! 1 LAN 2 LAN 3 LAN842.2 -1.4471 .2500 .0000 .58122.4 -1.13e .2511 .0305 .46082.6 -.894 .24244 .0528 .36412.8 -.707 .22448 .0663 .284.03.0 -.565 .2003 .0707 .21843.2 -.460 .1716 .0674 .16573.44 -.3844 .1412 .0590 .12363.6 -.329 .1110 .0480 .08973.8 -.287 .0818 .0361 .061944.0 -.255 .0542 .0241 .038844.2 -.230 .0282 .0126 .01934.3 -.221 .0158 .7045+ .010644.44 -.208 .44102+ .1833+ .2691+4.5 -.200 -.7861+ -.3505' -.5065+44.6 -.192 -.0191 -.8511+ -.01214.8 -.178 -.0406 -.0180 -.02495.0 -.165 -.0607 -.0268 -.03625.2 -.157 v=.0796 -.03449 -.0446445.44 -.147 -.0975 -.0425 -.05555.6 -.140 -.1142 -.0495 -.06375.8 -.132 -.1302 -.0561 -.07126.0 -.127 -.1453 -.0622 -.07816.2 -.121 -.1595 -.0679 -.08446.44 -.116 -.1731 -.0733 -.09026.6 -.112 -.1860 -.0783 -.09566.8 -.108 -.1983 -.0830 -.10067.0 -.104 -.2098 -.08744 -.10527.2 -.101 - .2211 -.0916 -.10967.44 -.097 -.2316 -.0955 -.11367.6 -.095 -.2419 -.0992 -.11757.8 -.092 -.2518 -.1028 -.12118.0 -.090 -.2611 -.1061 -.12458.2 -.088 -.2702 -.1093 -.12778.4 -.085 -.2789 -.11244 -.13078.6 -.084 -.2871 -.1152 -.13368.8 -.081 -.2952 -.1180 -.13639.0 -.080 - .3029 -.1206 -.13899.2 -.078 -.3102 -.1231 -.14139.4 -.076 -.3176 -.1256 -.144372.4 -1.2445 .2445 .0178 .47482.6 -.987 .2376 .0410 .37702.8 -.790 .2225 .0561 .29693.0 -.639 .2006 .0630 .23073.2 -.525 .1742 .0625 .17683.4 -.440 .1454 .0566 .13323.6 -.376 .1163 .04476 .09793.8 -.329 .0877 .0369 .06894.0 -.290 .06044 .0259 .044474.2 -.262 .0345 .0149 .02434.3 -.248 .0221 .9582+ .01524.4 -.238 .0101 .44383+ .6815+44.5 -.228 -.1612+ -.0700+ -.1066+4.6 -.219 -.0128 -.5570' -.8334+4.8 -.202 -.0344 -.0149 -.02165.0 -.188 -.0546 -.0236 -.03335.2 -.177 -.0737 -.0317 -.04385.4 -.167 -.0917 -.0393 -.05325.6 -. 157 -.1 087 -.0464 -.06175.8 -.150 -.1246 -.0529 -.06946.0 -.1442 -.1398 -.0591 -.07646.2 -.137 -.1542 -.0648 -.08296.4 -.131 -.1679 -.0702 -.08896.6 -.126 -.1809 -.0753 -.09446.8 -.122 -.1 933 -.0800 -.09957.0 -.117 -.2050 -.0845 -.10427.2 -.114 -.2163 -.0887 -.10877.4 -.110 -.2270 -.0927 -.11287.6 -.107 -.2374 -.0965 -.11677.8 -.104 -.2473 -.1001 -.12048.0 -.101 -.2567 -.1035 -.12388.2 -.098 -.2659 -.1067 -.12718.4 -.095 -.2745 -.1098 -.13018.6 -.094 -.2830 -.1127 -.13318.8 -.091 -.2910 -.1155 -.13589.0 -.089 -.2986 -.1181 -.13849.2 -.088 -.3064 -.1207 -.14109.4 -.086 -.3134 -.1231 -.144339.6 -.084 -.3206 -.1255 -.14562.44 -1.370 .2379 .44463+ .49312.6 -1.087 .2331 .0292 .39202.8 -.878 .2202 .04459 .31093.0 -.716 .2009 .0551 .24403.2 -.593 .1767 .0572 .18893.4 -.4499 .1497 .0538 .14383.6 -.4428 .1217 .0467 .10703.8 -.372 .0940 .0376 .076744.0 -.330 .0670 .0275 .0514444.2 -.298 .0413 .0172 .030144.4 -.269 .0170 .71449+ .011844.5 -.257 .5355+ .2258+ .36444+44.6 -.247 -.5954+ -.2515. -.3975+44.7 -.237 -.0169 -.7160+ -.011144.8 -.227 -.0276 -.0117 -.01785.0 -.213 -.0480 -.0203 -.03005.2 -.200 -.0671 -.0283 -.044085.44 -.187 -.0852 -.0359 -.05055.6 -.177 -.1024 -.04430 -.05935.8 -.16S -.1184 -.04495 -.06726.0 -.161 -.1338 -.0557 -.074456.2 -.153 -.14483 -.0615 -.08116.4 -.147 -.1620 -.0669 -.08726.6 -.1441 -.1753 -.0721 -.09296.8 -.136 -.1878 -.0769 -.09817.0 -.131 -.1 997 -.08144 -.10307.2 -.127 -.2111 -.0857 -.10757.4 -.123 -.2218 -.0897 -.11177.6 -.119 -.2322 -.0935 -.11577.8 -.115 -.2422 -.0972 -.119448.0 -.113 -.2519 -.1006 -.12308.2 -.110 -.2610 -.1039 -.12638.4 -.107 -.2698 -.1070 -.129448.6 -.1044 -.27844 -.1100 -.13248.8 -.102 -.28644 -.1128 -.13529.0 -.100 -.29443 -.1155 -.13799.2 -.097 -.3019 -.1181 -.1440449.4 -.095 -.3092 -.1206 -.144289.6 -.0944 -.31644 -.1230 -.1452TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979

212 RAMBERG, TADIKAMALLA, DUDEWICZ AND MYKYTKATABLE 4-Continued (see explanatory note, p. 210).03 = 0.60C03=0.650 3 -0.70a4 LAN 1 LAM 2 LAN 3 LAN 14014 LAMK 1 LANM2 LAM 3 LAN 4014 LkAM 1 LAM 2 LAM 3 LAM 142.14 -1.1411 .23147 .0000* .4951*2.6 -1.198 .2286 .0171 .140982.8 -.972 .2180 .0355 .32653.0 -.800 .2009 .01467 .25833.2 -.665 .1791 .05114 .20203.14 -.562 .1539 .05014 .155143.6 -.1482 .1273 .014514 .11713.8 -.1420 .1005 .0379 .0851414.0 -.372 .07140 .0289 .05894.2 -.335 .0486 .01914 .036614.14 -.302 .02144 .9911. .017514.5 -.289 .0128 .5215. .8965*14.6 -.277 .1492* .0611. .10 25.14.7 -.266 -.9531* -.3916* -.61425*14.8 -.256 -.0202 -.8326* -.013145.0 -.238 -.01407 -.0168 -.02615.2 -.222 -.0600 -.0248 -.03735.4 -.209 -.0782 -.0323 -.0147145.6 -.197 -.0956 -.03914 -.05655.8 -.187 -.1118 -.01460 -.06476.0 -.179 -.1273 -.0522 -.07226.2 -.171 -.11419 -.0580 -.07906.4 -.163 - .1559 -.0635 -.08536.6 -.157 -.1691 -.0686 -.09116.8 -.151 -.1818 -.0735 -.09657.0 -.1146 -.1938 -.0781 -.10157.2 -.1141 -.2052 -.0824 -..10617.14 -.137 -.2163 -.0865 -.11057.6 -.132 -.2267 -.09014 -.111457.8 -.128 -.2368 -.09141 -.11838.0 -.1214 -.21465 -.0976 -.12198.2 -.121 -.2557 -.1009 -.12538.14 -.118 -.26147 -.10141 -.12858.6 -.115 -.2732 -.1071 -.13158.8 -.113 -.2815 -.1100 -.134149.0 -.110 -.28914 -.1127 -.13719.2 -.108 -.2970 -.1153 -.13979.4 -.105 -.30145 -.1179 -.114229.6 -.103 -.3116 -.1203 -.1141452.6 -1.329 .22140 .3908. .143182.8 -1.076 .2157 .02146 .314433.0 -.889 .2010 .0380 .271423.2 -.7144 .1812 .014149 .21623.14 -.630 .1582 .014614 .16823.6 -.5142 .1330 .01435 .12833.8 -.1472 .1072 .0377 .095214.0 -.141e .0813 .0300 .0671414.2 -.3714 .05614 .0215 .01414014.14 -.338 .0323 .0126 .023914.5 -.3214 .0207 .8137* .015014.6 -.310 .9399, .3719. .6660*14.7 -.297 -.1593* -.06314. -.1106*14.8 -.285 -.0123 -.14921. -.8391.5.0 -.265 -.0328 -.0132 -.02165.2 -.2148 -.05214 -.0211 -.033145.14 -.231 -.0707 -.0286 -.014385.6 -.219 -.0880 -.0356 -.05325.8 -.209 -.10146 -.01422 -.06186.0 -.198 -.1201 -.014814 -.06956.2 -.189 -.1350 -.05143 -.07666.14 -.181 -.11491 -.0598 -.08316.6 -.1714 -.1625 -.0650 -.08916.8 -.167 - .1753 -.0700 -.091467.0 -.161 -.1 8714 -.07146 -.09977.2 -.155 -.1 991 -.0790 -.101457.14 -.150 -.2100 -.0831 -.10897.6 -.1145 -.2208 -.0871 -.11317.8 -.1141 -.2309 -.0908 -.11708.0 -.137 -.21407 -.091414 -.12.078.2 -.1314 -.2501 -.0977 -.12428.14 -.130 -.2591 -.1010 -.127148.6 -.127 -.2677 -.10140 -.13058.8 -.1214 -.2761 -.1069 -.13359.0 -.121 -.28140 -.1097 -.13629.2 -.119 -.2919 -.11214 -.13899.14 -.116 -.2 9914 -.1150 -.1141149.6 -.1114 -.3065 -.11714 -.114389.8 -.112 -.3136 -.1198 -.114612. 6 -1. 36 8 .2217 .0000* .14353*2.8 -1.1914 .2132 .0130 .36513.0 -.987 .2008 .0286 .29183.2 -.828 .1833 .0378 .23193.14 -.7014 .1621 .01416 .18213.6 -.606 .1385 .01409 .114063.8 -.529 .1139 .0369 .106014.0 -.1467 .0889 .0307 .076814.2 -.1419 .06143 .0232 .052214.14 -.379 .0406 .0151 .031214.6 -.3144 .0178 .6767* .013014.7 -.331 .6799. .2607. .14872*14.8 -.317 -.3 917. -.1512. -.2750.14.9 -.305 -.011414 -.55714. -.9893*5.0 -.2914 -.02145 -.9565. -.01665.2 -.276 -.014141 -.0173 -.02895.14 -.257 -.0626 -.02147 -.03985.6 -.2143 -.0802 -.0317 -.014965.8 -.229 -.0 967 -.0383 -.058146.0 -.219 -.1125 -.014145 -.06656.2 -.209 -.1275 -.05014 -.07386.14 -.199 -.11417 -.0560 -.08056.6 -.191 -.15514 -.0613 -.08676.8 -.1814 -.1682 -.0662 -.092147.0 -.177 -.1805 -.0709 -.09777.2 -.170 -.1923 -.07514 -.10267.14 -.165 -.2036 -.0796 -.10727.6 -.160 -.21144 -.0836 -.11157.8 -.155 -.22146 -.08714 -.11558.0 -.151 -.23146 -.0910 -.11938.2 -.1147 -.21439 -.091414 -.12288.14 -.1143 -.2532 -.0977 -.12628.6 -.139 -.2618 -.1008 -.12938.8 -.136 -.2703 -.1038 -.13239.0 -.133 -.27814 -.1066 -.13529.2 -.13C -.2862 -.1093 -.13799.14 -.127 -.2937 -.1119 -.1140149.6 -.125 -.3011 -.11414 -.114299.8 -.122 -.3081 -.1168 -.11452C03 = 0.75C04 LAM 1 LAN 2 LkM 3 LAM 1403 - 0.80C04 LANMi LANM2 LANM3 LAN 403 0.8504 LAMi1 LANM2 LANM3 LAN 42 . 8-1.3 314 .21014 .0000 .39033.0 -1. 097 .2 003 .0183 . 31193. 2 -. 92 1 .1 850 . 0299 .214923.14 -.785 .1658 .0360 .19743.6 -.677 .114140 .0375 .15423.8 -.590 .1206 .0355 .117914.0 -.521 .0966 .0309 .087314.2 -.1466 .0726 .02146 .0611414.14 -.1419 .01492 .01714 .039214.6 -.3814 .0266 .9663. .020214.7 -.367 .0156 .57149. .011614.8 -.352 .149140. .1833. .3583*14.9 -.339 -.5509* -.2061. -.3916*5.0 -.3214 -.0157 -.5915* -.01095.2 -.306 -0353 -.01314 -.02385.14 -.2814 -.0539 -.0207 -.03525.6 -.268 -.0716 -.0276 -.0145145.8 -.2514 -.08814 -.03142 -.051476.0 -.2140 -.101414 -.01405 -.06306.2 -.229 -.1195 -.0464 -.07066.14 -.219 -.1339 -.0520 -.07766.6 -.209 -.11476 -.0573 -.081406.8 -.201 -.1607 -.0623 -.08997.0 -.1914 -. 1731 -.0670 -.095147.2 -.188 -.1851 -.0715 -.10057.14 -.181 -.19614 -.0758 -.10527.6 -.175 -.20714 -.0799 -.10967.8 -.170 -.2177 -.0837 -.11378.0 -.165 -.2278 -.08714 -.11768.2 -.160 -.2375 -.0909 -.12138.14 -.156 -.21466 -.09142 -.121478.6 -.152 -.25514 -.0974 -.12798.8 -.1148 -.26140 -.10014 -. 13109.0 -.1145 -.2722 -.1033 -.13399.2 -.1142 -.2802 -.1061 -.13679.14 -.13e -.2879 -.1088 -.13939.6 -.135 -2952 -.1113 -.114189.8 -.133 -.3023 -.1137 -.11414210.0 -.130 -.3093 -.1161 -.114653. 0 -1. 22 5 .1 996 .6847* . 33563. 2 -1. 02 5 .1 8614 .021 1 .26873.14 -8714 .1692 .0295 .211433. 6 -7514 .11492 .0333 .16913. 8 -65 7 .1272 .033 3 . 131014.0 -582 .10142 .0303 .098914.2 -.519 .0810 .0254 .071614.14 -.468 .0580 .0192 .048214.6 -.1425 .0357 .0123 .028114.8 -.392 .01142 .5035. .010714.9 -.375 .3770* .1352. .2770*5.0 -.361 -.6291* -.2278* -.14531.5.1 -.3149 -.01614 -.5981. -.01165.2 -.335 -.0261 -.9598. -.01815.14 -.313 -.014149 -.0167- -.03015.6 -.295 -.0626 -.0235 -.04085.8 -.279 -.0795 -.0300 -.050146.0 -.2614 -.0958 -.0363 -.05926.2 -.251 -.1110 -.01422 -.06716.14 -.2140 -.1255 -.01478 -.071436.6 -.230 -.13914 -.0531 -.08106.8 -.220 -.1527 -.0582 -.08717.0 -.212 -.1653 -.0630 -.09287.2 -.2014 -.17714 -.0676 -.09807.14 -.197 -.1889 -.0719 -.10297.6 -.191 -.2000 -.0760 -.10757.8 -.185 -.21014 -.0799 -.11178.0 -.180 -.2205 -.0836 -.11578.2 -.1714 -.23014 -.0872 -.11958.4 -.169 -.2397 -.0906 -.1230le. 6 -.166 -.21488 -.0938 -.12614~8.8 -.161 -.25714 -.0969 -.12959.0 -.157 -.2658 -.0999 -.13259.2 -.1514 -.2737 -.1027 -.13539.14 -.150 -.2815 -. 1054 -.13809.6 -.1147 -.2890 -.1080 -.114069.8 -.1144 -.2962 -.1105 -.1143010.0 -.1141 -.3033 -.1129 -.1451410.2 -.139 -.3100 -.1152 -.114763.0 -1. 30 3 .1 985 .0000* .34883. 2 -1.114 5 .1875 .0110 .29123.14 -97 3 .1723 .0220 .23323. 6 -8386 .15141 .0281 .18553.8 -73 2 .1336 .0301 .1145514.0 -64 5 .1 119 .0291 .111714.2 -.577 .0895 .0256 .082914.14 -.519 .0671 .0206 .058214.6 -.1472 .01451 .0146 .037014.8 -.1430 .0238 .8001* .01854.9 -.413 .01314 .14581* .01025.0 -.398 .3503. .1211* .2612*5.1 -.383 -.6701* -.23145. -.4896*5.2 -.37C -.0165 -.5808. -.01185.14 -.3144 -.0353 -.0127 -.0214145.6 -.324 -.0531 -.0193 -.03565.8 -.305 -.0703 -.0258 -.04576.0 -.290 -.08614 -.0319 -.051486.2 -.275 -.1019 -.0378 -.06316.4 -.262 -.1168 -.01435 -.07076.6 -.251 -.1307 -.01488 -.07766.8 -.2141 -.114142 -.0539 -.081407.0 -.231 -.1 570 -.0588 -.08997.2 -.223 -.1692 -.06314 -.09537.4 -.215 -.1809 -.0678 -.100147.6 -.207 -.1921 -.0720 -.10517.8 -.201 -.2028 -.0759 -.10958.0 -.195 -.2130 -.0797 -.11368.2 -.190 -.2229 -.0833 -.11758.14 -.1814 -.23214 -.0868 -.12118.64 -.179 -.21416 -.0901 -.121468.8 -.175 -.2503 -.0932 -.12789.0 -.171 -.2587 -.0962 -.13099.2 -.167 -.2669 -.0991 -.13389.14 -.163 -.2748 -.1019 -.13669.6 -.159 -.2823 -.10145 -.13929.8 -.156 -.2897 -.1071 -.1141710.0 -.153 -.2967 -.1095 -.144110.2 -.150 -.3037 -.1119 -.14614TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979

A PROBABILITY DISTRIBUTION AND ITS USES IN FITTING DATA213TABLE 4-Continued (see explanatory note, p. 210).Ca3 = 0. 90a4 LAM 1 LAM 2 LkAM3 LAM 4=f 1. 00Ca 4 LAN 1 LkM 2 LAN 3 LkM 4Ct3 = 1.10Ca4 LAM 1 LkM 2 LAM 3 LANM43. 2 -1.2 77 .1 880 .0000 .31 603. 4 -1. 08 5 .1 751 .0133 . 25483. 6 -93 3 .1 586 . 021 8 .20 393. 8 -81 4 .1397 .0260 .16 154. 0 -71 7 .1 19 3 .0269 .12 584. 2 -6 3 .0979 .0251 .09 534. 4 -57 5 .0 762 .0214 .06 934. 6 -5 22 .0 547 .016 4 .04 684. 8 -4786 .0 337 . 0106 .02735. 0 -43 9 .0 132 . 4328+ .01025. 1 -42 2 .333 9+ . 111 1+ .2526+5. 2 -40 7 - .6388+ -2154+ -. 4735.5. 3 -39 4 -.0159 -5428. -.01 165.4 -.375 -.0252 -.8694+ -.01805.6 -.353 -.0432 -.0152 -.0298*5.8 -.334 -.0605 -.0215 -.04056.0 -.317 -.0768 -.0275 -.05006.2 -.301 -.0924 -.0334 -.05876.4 -.287 -.1073 -.0390 -.06666.6 -.273 - .1215 -.0444 -.07386.8 -.262 -.1352 -.0495 -.08057.0 -.252 -.1481 -.0544 -.08667.2 -.242 -.1606 -.0591 -.09237.4 -.233 -.1 72 3 -.0635 -.09757.6 -.225 -.1838 -.0678 -.10247.8 -.21e -.1947 -.0718 -.10708.0 -.212 -.2051 -.0756 -.11138.2 -.205 -.2151 -.0793 -.11538.4 -.199 -.2246 -.0828 -.11908.6 -.194 -.2340 -.0862 -.12268.8 -.189 -.2428 -.0894 -.12599.0 -.185 -.2514 -.0924 -.12919.2 -.180 -.2597 -.0954 -.13219.4 -.176 -.2676 -.0982 -.13499.6 -.172 -.2753 -.1009 -.13769.8 -.168 - .2827 -.1035 -.140210.0 -. 165 -.2900 -.1060 -.142710.2 -.162 - .2969 -.1084 -.145010.4 -.159 -.3035 -.1107 -.14723. 4 -1. 253 .1 772 .0000* .2854*3. 6 -1. 169 .1 664 .4828. .24903. 8 -1. 01 0 .1509 .0141 .19964. 0 -886E .1 333 .0193 . 15 884. 2 -7 87 .1142 .021 2 .12444.4 -.706 .0943 .0206 .09504.6 -.63e .0741 .0182 .06974.8 -.581 .0539 .0144 .04775.0 -.533 .0340 .9695+ .02855.2 -.492 .0146 .4383+ .01175.3 -.474 .5192+ .1584+ .4061+5.4 -.445 -.0317+ -.0101+* -.0242+*5.5 -.442 -.0132 -.4176. -.9946+5.6 -.429 -.0222 -.7097+ -.01645.8 -.403 -.0395 -.0129 -.0282*6.0 -.379 -.0562 -.0187 -.03886.2 -.358 -.0721 -.0244 -.04846.4 -.341 -.0873 -.0299 -.05716.6 -.325 -. 1019 -.0352 -.06516.8 -.309 -.1158 -.0404 -.07237.0 -.297 -.1291 -.0453 -.07907.2 -.285 -.1419 -.0500 -.08527.4 -.275 - .1 540 -.0545 -.09097.6 -.265 -.1658 -.0589 -.09627.8 -.256 - .1769 -.0630 -.10118.0 -.24e -.1878 -.0670 -. 10 588.2 -.241 -.1980 -.0707 -.11018.4 -.233 -.2079 -.0744 -.11418.6 -.227 -.2174 -.0778 -.11798.8 -.220 -.2267 -.0812 -.12159.0 -.215 -.2356 -.0844 -.12499.2 -.210 -.2440 -.0874 -.12819.4 -.204 -.2522 -.0904 -.13119.6 -.200 - .2602 -.0932 -.13409.8 -.195 -.2678 -.0959 -.136710.0 -.191 -.2752 -.0985 -.139310.2 -.187 -.2824 -.1010 -.141810.4 -.184 -.2893 -.1034 -.144210.6 -.180 -.2959 -.1057 -.14643.8 -1.215 .1582 .0000* .2.3794.0 -1.108 .1459 .6035+ .20134.2 -.974 .1294 .0125 .16074.4 -.869 .1117 .0157 .12674.6 -.781 .0932 .0165 .09774.8 -.708 .0743 .0154 .07275.0 -.647 '.0552 .0128 .05085.2 -.596 .0365 .9168. .03185.4 -.552 .0181 .4839+ .01505.5 -.532 .9038+ .2484+ .7342+5.6 -.517 .0997+ .0279+ .0795.5.7 -.497 -.8629+ -.2479+ -.6726+5.8 -.481 -.0173 -.5046+ -.01326.0 -.451 -.0340 -.0103 -.0i5l6.2 -.427 -.0501 -.0155 -.03586.4 -.403 - .0656 -.0208 -.04556.6 -.384 -.0805 -.0259 -.05446.8 -.366 -.0947 -.0309 -.06247.0 -.350 -.1084 -.0358 -.06987.2 -.335 -.1214 -.0405 -.07667.4 -.322 -.1341 -.0451 -.08297.6 -.311 -.1460 -.0494 -.08877.8 -.299 -.1577 -.0537 -.09418.0 -.289 -.1687 -.0577 -.09918.2 -.280 -.1794 -.0616 -.10388.4 -.271 -.1896 -.0653 -.10828.6 -.263 -.1994 -.0689 -.11238.8 -.256 -.2090 -.0724 -.11629.0 -.249 -.2180 -.0757 -.11989.2 -.242 -.2267 -.0788 -. 12329.4 -.236 -.2353 -.0819 -.12659.6 -.231 -.2435 -.0848 -.12969.8 -.226 -.2513 -.0876 -.132510.0 -.221 -.2590 -.0903 -.135310.2 -.216 -.2664 -.0930 -.137910.4 -.211 -.2735 -.0955 -.140410.6 -.207 -.2804 -.0479 -. 142810.8 -.203 -.2870 -.1002 -.145111.0 -.199 -.2936 -.1025 -.1473at3 = 1. 20a4 LANMi LAkM 2 LAM 3 LAMi4a3 = 1. 30a4 LANMi LAN 2 LkAM3 LkAM4C3=1.40a4 LAMi1 LAN 2 LAM 3 LAM44. 2 -1. 1 83 .1 407 .0000* . 19974.4 -1.083 .1278 .5096. .16754.6 -.965 .1113 .9968+ .13294.8 -.870 .0941 .0122 .10365.0 -.792 .0764 .0124 .07845.2 -.723 .0586 .0112 .05655.4 -.668 .0408 .8705+ .03725.6 -.615 .0233 .5411+ .02025.7 -.597 .0146 .3525+ .01245.8 -.577 .6 088+ .1515+ .5050.5.9 -.55e -.2319+ -.0594+ -.1884.6.0 -.562 -.0962+ -.0245+ -.0784+6.2 -.50e -.0268 -.7343+ -.02066.4 -.481 -.0424 -.0120 -.03156.6 -.454 - .0575 -.0168 -.04146.8 -.432 -.0719 -.0215 -.05047.0 -.412 -.0860 -.0262 -.05877.2 -.394 -.0993 -.0308 -.06627.4 -.378 -.1123 -.0353 -.07327.6 -.362 -.1247 -.0397 -.07967.8 -.349 -.1366 -.0439 -.08568.0 -.337 -.1480 -.0480 -.09118.2 -.325 -.1589 -.0519 -.09628.4 -.314 -.1695 -.0558 -.10108.6 -.305 -.1796 -.0594 -.10558.8 -.296 -.1896 -.0630 -.10989.0 -.287 -. 1990 -.0664 -.11379.2 -.280 -.2082 -.0697 -.11759.4 -.273 -.2168 -.0728 -.12109.6 -.265 - .2253 -.0759 -.12439.8 -.259 -.2335 -.0788 -.127510.0 -.254 -.2414 -.0816 -.130510.2 -.248 -.2490 -.0843 -.133310.4 -.242 -.2564 -.0870 -.136010.6 -.237 -.2636 -.0895 -.138610.8 -.233 -.2704 -.0919 -.141011.0 -.228 -.2772 -.0943 -.143411.2 -.224 -.2837 -.0966 -.145611.4 -.220 -.2901 -.0988 -.14784. 6 -1. 15E .1244 .0000* .16794. 8 -1. 08 4 .1129 .3174. .14355.0 -97 5 .0968 .722 5+ . 11305. 2 -886 .0802 . 903 5. .08705. 4 -81 2 .0634 . 914 8+ .06455.6 -.749 .0466 .7959. .04475.8 -.695 .0300 .5783. .02736.0 -.604 .0286. .6619S* .0239.6.1 -.617 .0446+ .0100+ .0375+*6.2 -.616 -.0526+ -.0118. -.0442+6.3 -.585 -.0104 -.2450. -.8504+6.4 -.572 -.0182 -.4399+ -.01466.6 -.535 -.0333 -.8469+ -.02586.8 -.510 -.0480 -.0127 -.03607.0 -.485 -.0622 -.0170 -.04537.2 -.463 -.0758 -.0213 -.05387.4 -.4442 -.0890 -.0256 -.06167.6 -.424 -.1017 -.0298 -.06887.8 -.407 -.1140 -.0340 -.07548.0 -.392 -. 1258 -.0380 -.08168.2 -.378 -.1372 -.0420 -.08738.4 -.365 -.1480 -.0458 -.09268.6 -.353 -. 1584 -.0495 -.09758.8 -.342 -.1687 -.0531 -.10229.0 -.332 -.1784 -.0566 -.10659.2 -.322 -.1878 -.0600 -.11069.4 -.314 -. 1969 -.0632 -.11459.6 -.305 - .2057 -.0664 -.11819.8 -.298 -.2141 -.0694 -.121510.0 -.291 -.2223 -.0723 -.124810.2 -.284 -.2304 -.0752 -.127910.4 -.277 -.2379 -.0779 -.130810.6 -.272 -.2453 -.0805 -.133610.8 -.266 -.2525 -.0831 -.136211.0 -.261 -.2595 -.0855 -.1388*11.2 -.256 -.2662 -.0879 -.141211.4 -.251 -.2728 -.0902 -.143511.6 -.246 -.2792 -.0925 -.145711.8 -.242 -.2852 -.0946 -.14785.0 -1.132 .1092 .0000* .14115.4 -1.001 .0855 .4546+ .09915.6 -.916 .0697 .6296+ .07545.8 -.844 .0538 .6530+ .05476.0 -.782 .0379 .5603+ .03656.2 -.729 .0222 .3785+ .02046.3 -.706 .0145 .2611+ ..01306.4 -.683 .6822+ .1292. .5987+6.5 -.66C -.1226+ -.0244+ -.1052+6.6 -.643 -.8266+ -.1702+ -.6968+6.8 -.607 -.0230 -.5060+ -.01877.0 -.575 -.0373 -.8670+ -.02937.2 -.547 -.0510 -.0124 -.03897.4 -.521 - .0645 -.0163 -.04787.6 -.498 -.0775 -.0202 -.05597.8 -.475 -.0900 -.0242 -.0633*8.0 -.458 -.1020 -.0280 -.07028.2 -. 44C -.1137 -.0319 -.07668.4 -.423 -.1250 -.0357 -.0825*8.6 -.410 -.1358 -.0393 -.0881*8.8 -.395 -.1463 -.0430 -.09329.0 -.383 -.1564 -.0465 -.09809.2 -.372 -.1662 -.0499 -.10269.4 -.361 -.1756 -.0532 -.10689.6 -.351 -.1846 -.0564 -.11089.8 -.342 -.1935 -.0595 -.114610.0 -.333 -.2018 -.0625 -.118110.2 -.325 -.2102 -.0655 -.121510.4 -.317 -.2181 -.0683 -.124710.6 -.310 -.2257 -.0710 -.127710.8 -.303 -.2332 -.0737 -.130611.0 -.297 -.2405 -.0762 -.1334*11.2 -.291 -.2475 -.0787 -.136011.4 -.285 -.2 542 -.0811 -.1385*11.6 -.279 -.2609 -.0835 -.140911.8 -.274 -.2671 -.0857 -.143112.0 -.269 -.2734 -.0879 -.145312.2 -.265 -.2794 -.0900 -.1474TECHNOMETRICS ?, VOL. 21, NO. 2, MAY 1979