Unit 6: Probability Plotting Ramón V. León Notes largely based on Statistical Methods for Reliability Data by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes. 9/12/2004 Stat 567: Unit 6 - Ramón V. León 1
Unit 6 Objectives Describe applications for probability plots Explain the basic concepts of probability plotting Show how to linearize a cdf on special plotting scales Explain how to plot a nonparametric estimate of the cdf to judge the adequacy of a particular parametric distribution Explain methods of separating useful information from noise when interpreting a probability plot Use a probability plot to obtain graphical estimates of reliability characteristics like failure probabilities and quantiles 9/12/2004 Stat 567: Unit 6 - Ramón V. León 2
Purpose of Probability Plots Probability plots are used to: Asses the adequacy of a particular distributional model To detect multiple failure modes or mixtures of different populations Obtain graphical estimates of model parameters (e.g., by fitting a straight line through the points on a probability plot) Displaying the results of a parametric likelihood fit along with the data Obtain, by drawing a smooth curve through the points, a nonparametric estimate of failure probabilities and distributional quantiles 9/12/2004 Stat 567: Unit 6 - Ramón V. León 3
Probability Plotting Scales: Linearizing a CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 4
Linearizing the Exponential CDF log(1 p) = t p γ θ 9/12/2004 Stat 567: Unit 6 - Ramón V. León 5
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Linearizing the Normal CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 7
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Linearizing the Lognormal CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 9
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Linearizing the Weibull CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 11
Linearizing the Weibull CDF - Continued 9/12/2004 Stat 567: Unit 6 - Ramón V. León 12
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Choosing Plotting Positions to Plot the Nonparametric Estimate of the CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 14
Criteria for Choosing Plotting Positions Criteria for choosing plotting positions should depend on the application or purpose for constructing the probability plot Some applications that suggest criteria: Checking distributional assumptions Estimation of parameters Display of maximum likelihood results with data. 9/12/2004 Stat 567: Unit 6 - Ramón V. León 15
Plotting Positions: Continuous Inspection Data and Single Censoring 9/12/2004 Stat 567: Unit 6 - Ramón V. León 16
Plotting Positions: Continuous Inspection Data and Multiple Censoring 9/12/2004 Stat 567: Unit 6 - Ramón V. León 17
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Six Distribution Probability Plots of the Shock Absorber Data 9/12/2004 Stat 567: Unit 6 - Ramón V. León 21
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Six Distribution Probability Plots Alloy T7987 Fatigue Life 9/12/2004 Stat 567: Unit 6 - Ramón V. León 23
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Plotting Positions: Interval Censored Inspection Data 9/12/2004 Stat 567: Unit 6 - Ramón V. León 27
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Probability Plots with Specified Shape Parameters The probability plotting techniques can be extended to construct probability plots for: Distributions that are not members of the location-scale family To help identify, graphically, the need for non-zero threshold parameters Estimate graphically a shape parameter 9/12/2004 Stat 567: Unit 6 - Ramón V. León 30
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Distributions with a Threshold Parameter 9/12/2004 Stat 567: Unit 6 - Ramón V. León 32
Linearizing the 3-Parameter Gamma CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 33
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Linearizing the 3-Parameter Weibull CDF Using Linear Time Axis and Specified Shape Parameter 9/12/2004 Stat 567: Unit 6 - Ramón V. León 35
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Linearizing the Generalized Gamma CDF 9/12/2004 Stat 567: Unit 6 - Ramón V. León 37
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Notes on the Application of Probability Plotting Using simulation to help interpret probability plots Try different assumed distributions and compare the results Assess linearity; allowing for more variability in the tails Use simultaneous nonparametric confidence bands Use simulation or bootstrap to calibrate Possible reason for a bend in a probability plot Sharp bend or change in slope generally indicates an abrupt change in a failure process 9/12/2004 Stat 567: Unit 6 - Ramón V. León 41
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