A Probability Distribution and Its Uses in Fitting Data

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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?ined 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 and 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
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