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Kurtosis Vs. Skewness Vs. Coe Cient of Variation, Rotated Ball Bearings At 2.5 0 CV 0.5 2 1 Skewness 1.5 1.5 15 1 0.5 loglog 0 gamma Pareto 10 Kurtosis lognorm 5 Weibull 0 Figure Kurtosis vs Skewness vs Co ecient of Variation Rotated Ball Bearings at Skewness 0 0.5 1 1.5 2 2.5 0 CV 0.5 1 1.515 Pareto loglog gamma 10 lognorm Kurtosis 5 Weibull 0 Figure Kurtosis vs Skewness vs Co ecient of Variation Ball Bearings at Kurtosis 14 Pareto 12 Gamma 10 8 Lognormal 6 Log log 4 Weibull 2 0 Skewness 0.5 1 1.5 2 2.5 3 Figure Kurtosis vs Skewness Ball Bearings at Skewness 5 4 Lognormal 3 Pareto Log Logistic 2 Gamma 1 Weibull CV 0.2 0.4 0.6 0.8 1 1.2 Figure Skewness vs Co ecient of Variation Ball Bearings at 1 0.8 0.6 S(t) 0.4 0.2 0 0 50 100 150 200 t Figure MLE Survivor Function vs KaplanMeier Pro ductLimit Estimate Ball Bearings Weibull Fit 3 2.75 2.5 2.25 kappa 2 1.75 1.5 0.01 0.011 0.012 0.013 0.014 0.015 0.016 lambda Figure Condence Region for MLEs Ball Bearings Weibull Fit Log Likelihood Function -125 4 -150 -175 3 -200 0 2 kappa 0.005 0.01 1 lambda 0.015 0.02 0 Figure Log Likeliho o d Function with Maximum Value Identied Ball Bearings Weibull Fit log(-log(S(t))) log xi 3 3.5 4 4.5 5 -2 -4 y = -9.004 + 2.041 x -6 lambda0 = 0.012 kappa0 = 2.041 Figure Weibull Plot Ball Bearings Weibull Fit r(p) Reliability Bounds 1 0.8 0.6 0.4 0.2 p 0.2 0.4 0.6 0.8 1 Figure Reliability Bounds for a FourComp onent System 1 3 2 4 Figure Four Comp onent System AUTHORS Glen Hartless Department of Statistics GrinFloyd Hall University of Florida Gainesville FL USA Glen Hartless received his BS in statistics from Virginia Tech Blacksburg VA in He is currently pursuing his Masters degree in statistics at the University of Florida He is a student memb er of ASQC and ASA Lawrence Leemis Department of Mathematics College of William and Mary Williams burg VA USA Lawrence Leemis is a professor in the Mathematics Department at the College of William Mary He received his BS and MS degrees in Mathematics and his PhD in Industrial Engineering from Purdue University He has served as an Asso ciate Editor for the IEEE Transactions on Reliability and Bo ok Review Editor for the Journal of Quality Technology REFERENCES RE Barlow F Proschan Statistical Theory of Reliability and Life Testing Prob ability Models To Begin With Silver Springs MD DR Cox D Oakes Analysis of Survival Data Chapman and Hall New York JF Lawless Statistical Models and Methods for Lifetime Data John Wiley and Sons New York S Wolfram Mathematica A System for Doing Mathematics by Computer Second Edition AddisonWesley ACKNOWLEDGEMENT This research was carried out during a Research Exp erience for Undergraduates sp onsored by the National Science Foundation grant numb er at The College of William Mary during the summer of We thank the asso ciate editor and referees for their helpful comments Uniform NoShapeParameter Exponential NoShapeParameter Weibull k Gamma k LogLogistic k Pareto b LogNormal sigma Discussion It should b e kept in mind that the Euclidean distance technique should only b e used to determine which mo dels would b e obviously inferior It is not a metho d with which to cho ose a mo del For instance in testing this metho d using randomly generated samples from a log normal distribution the tted Weibull and gamma distributions would often b e closer to the sample moments the lognormal is not a clear cut winner based on distance to the sample moments for example the Kolmogorov Smirnov go o dnessoft test smaller for the MLE lognormal t than it is for the MLE Weibull t Hence For the ball b earing data set one should only conclude that the exp onential Pareto distributions are inferior in describing the distribution For the Weibull distribution the initial estimate of is close to the MLE which was see section The Mathematica implementation is similar to the Weibull maximum likeliho o d problem the ma jor step is p erformed by a ro otsolver Using Mathematica did lead us to a new discovery although the kurtosis vs skewness vs co ecient of variation plot was only marginally useful in and of itself the minimum Euclidean distance to the sample p oint provides us with some very useful exploratory results The moment curves given here are the function of at most a single shap e parameter The value of this parameter when the Euclidean distance is minimized can b e used as an initial estimate for obtaining its maximum likeliho o d estimate Hence not only can we see which distributions are p ossible mo dels we can also learn something ab out their parameter estimates Again due to Mathematicas ro otsolving ability this minimizer was not dicult to develop Unlike the likeliho o d function in section the distance functions do not need a particularly accurate initial estimate to converge Example Consider again the ball b earing data from section Its moments are Figures are the skewness vs co ecient of variation 3 4 and kurtosis vs skewness plots for several lifetime distributions gamma log logistic log normal Pareto and Weibull The exp onential distribution is a sp ecial case of the gamma and Weibull and has moments Also the uniform 3 4 1 9 p distribution has moments The sample moments of the 3 4 5 3 ball b earing data are indicated by a p oint on each graph Figures show the dimensional moment curves from two p ersp ectives Figure lo oks down at the kurtosis vs skewness plot with depth provided by the co ecient of variation Figure is a rotated version with the kurtosis height scale reduced As explained previously the b ehavior is dicult to see What follows is the results of determining the minimum Euclidean distance b e tween and each of the moment curves in three dimensions 3 4 In MomentDistancedata OutMatrixForm DISTRIBUTION MIN DISTANCE INITIAL ESTIMATE This larger programming application revealed a downside of using Mathematica The program syntax can b e quite tedious Slight deviations from the desired command can pro duce a wealth of syntax errors and warnings andor incorrect results MODEL SELECTION We select a parametric mo del to describ e a set of lifetime data In order to limit the numb er of mo dels sample moments can b e compared to the theoretical moments of the distribution in question Two graphical metho ds used for this purp ose are plots of skewness vs co ef 3 cient of variation and kurtosis vs skewness see Cox and Oakes p 4 The rst plot is a visual representation of symmetry and spread while the second is of p eakedness and symmetry Some distributions such as the exp onential and uniform distributions reduce to a single p oint on these plots Close proximity of the curve to the corresp onding sample moments indicates that the distribution is not a bad p otential parametric mo del Mathematica Implementation Since Mathematica is capable of pro ducing a dimensional parametric plot it o ccurred to us that a plot of would b e of similar use Unfortunately the 3 4 simplicity of the curves makes their proximity to the sample moments dicult to discern without rotating the dimensional plot One solution to this problem is to nd the minimal Euclidean distance b etween and the dimensional 3 4 moment curves where is the ratio of the sample standard deviation to the sample mean and 3 4 n n X X s t t t t i i 4 3 n s n s t i=1 i=1 kappa lambda Log Likelihood at the MLE Observed Information Matrix Inverse Information Matrix VarianceCovariance Matrix of the MLEs Two comments with resp ect to the scondence region in Figure follow First Figure veries that a Weibull distribution is a b etter t than an exp onential dis tribution since the line is not interior to the scondence region Second adjusting an argument in ReliabilityFit gives smo other contours at the cost of larger computation times Discussion Mathematicas overall p erformance on this larger more complex problem was adequate We were pleased with the programs ability to overlay several graphic images such that the scaling app earance were appropriate We accomplished nearly everything we had hop ed to although much trial error was often necessary Another advantage of our Mathematica implementation is that our function has only three arguments the data set a survivor function and initial parameter es timates The log likeliho o d function score vector and information matrix are all determined by Mathematica As such the co de app ears much more like the mathe matical derivations than the corresp onding co de in an algorithmic language Other twoparameter distributions could b e tted in a similar fashion Example 6 The data b elow are ball b earing failure times in revolutions from Lawless The commands necessary to run our function ReliabilityFit are In data In SurvFunctWeibull NE LAM x K In ReliabilityFitdata WeibullSurvivor InitEstimWeibulldata The second element of each data pair gives its censoring status right censored observed and InitEstimWeibull is a function that returns initial estimates of the Weibull parameters The output of the program includes a scatterplot and tted regression line Figure a dimensional graph of the log likeliho o d function Figure and condence regions for the MLEs Figure and a comparison of the MLE survivor function to the corresp onding KaplanMeier pro ductlimit estimate Figure The function prints the survivor function maximum likeliho o d estimators the maximum value of the log likeliho o d function the observed information matrix and the
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