Unit 10: Planning Life Tests

Transcription

1 Unit 10: Planning Life Tests 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. 11/2/2004 Unit 10 - Stat Ramón León 1 Unit 10 Objectives Explain the basic ideas behind planning a life test Use simulation to anticipate the results, analysis, and precision for a proposed test plan Explain large-sample approximate methods to assess precision of future results from a reliability study Compute sample size needed to achieve a degree of precision Assess tradeoffs between sample size and length of a study. Illustrate the use of simulation to calibrate the easier-touse large-sample approximate methods 11/2/2004 Unit 10 - Stat Ramón León 2

2 Basic Ideas in Test Planning The enormous cost of reliability studies makes it essential to do careful planning. Frequently asked questions include: How many units do I need to test in order to estimate the.1 quantile of life? How long do I need to run the life test? Clearly, more test units and more time will buy more information and thus more precision in estimation To anticipate the results from a test plan and to respond to the questions above, it is necessary to have some planning information about the life distribution to be estimated 11/2/2004 Unit 10 - Stat Ramón León 3 Engineering Planning Values and Assumed Distribution for Planning an Insulation Life Test 11/2/2004 Unit 10 - Stat Ramón León 4

3 11/2/2004 Unit 10 - Stat Ramón León 5 Simulation as a Tool for Test Planning Use assumed model and planning values of model parameters to simulate data from the proposed study Analyze the data perhaps under different assumed models Assess precision provided Simulate many times to assess actual sample-to-sample differences Repeat with different sample sizes to gauge needs Repeat with different input planning values to assess sensitivity to these inputs. Any surprises? 11/2/2004 Unit 10 - Stat Ramón León 6

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6 Simulations of Insulation Life Tests 11/2/2004 Unit 10 - Stat Ramón León 11 Trade-offs Between Test Length and Sample Size 11/2/2004 Unit 10 - Stat Ramón León 12

7 Assessing the Variability of the Estimates 11/2/2004 Unit 10 - Stat Ramón León 13 Simulations of Insulation Life Test-Continued 11/2/2004 Unit 10 - Stat Ramón León 14

8 11/2/2004 Unit 10 - Stat Ramón León 15 Motivation for Use of Large-Sample Approximations of Test Plan Properties Asymptotic methods provide: Simple expressions giving precision of a specified estimator as a function of sample size Simple expressions giving needed sample size as a function of specified precision of a specified estimator Simple tables and graphs that will allow easy assessments of tradeoffs in test planning decisions like sample size and test length Can be fine tuned with simulation evaluation 11/2/2004 Unit 10 - Stat Ramón León 16

9 Asymptotic Variances 11/2/2004 Unit 10 - Stat Ramón León 17 Delta Method for Two Parameters g( θ ) g( θ) g( θ) v v θ , θ1 θ 2 v12 v = 2 g( θ ) θ 2 g( θ ) g( θ) g( θ) g( θ) g( θ) θ v v, v v θ θ θ θ θ = g( θ ) g( θ) g( θ) g( θ) g( θ) g( θ) g( θ) v1+ v12 + v12 + v2 = θ1 θ2 θ1 θ1 θ2 θ2 2 2 g( θ) g( θ) g( θ) g( θ) v + 2 v + v θ θ θ θ /2/2004 Unit 10 - Stat Ramón León 18

10 11/2/2004 Unit 10 - Stat Ramón León 19 Example 11/2/2004 Unit 10 - Stat Ramón León 20

11 Sample Size Determination for Positive Functions of the Parameters 11/2/2004 Unit 10 - Stat Ramón León 21 Sample Size Determination for Positive Functions of the Parameters-Continued 11/2/2004 Unit 10 - Stat Ramón León 22

12 Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation Life 11/2/2004 Unit 10 - Stat Ramón León 23 Sample Size Needed to Estimate the Mean of an Exponential Distribution Used to Describe Insulation Life- Continued 11/2/2004 Unit 10 - Stat Ramón León 24

13 Derivation.1 r e L= e θ e i= 1 θ = i= r+ 1 l = rlogθ TTTθ ti θ n ti TTT θ r l = rθ + TTTθ θ 11/2/2004 Unit 10 - Stat Ramón León l = rθ 2TTTθ 2 θ 2 3 θ Derivation.2 l E E TTT R θ = 2 θ θ ( E[ TTT ]) θ E[ R] 3 2 ( E[ R] ) E[ R] ( E[ R] ) ( n P[ T tc] ) θ ( npc) = 2θ = 2θ θ θ = θ = θ = 2 2 = nθ 1 tc 2 e θ 11/2/2004 Unit 10 - Stat Ramón León 26

14 Location-Scale Distributions and Single Right Censoring Asymptotic Variance-Covariance 11/2/2004 Unit 10 - Stat Ramón León 27 Location-Scale Distributions and Single Right Censoring Fisher Information Elements 11/2/2004 Unit 10 - Stat Ramón León 28

15 Table of Information Matrix Elements and Variance Factors 11/2/2004 Unit 10 - Stat Ramón León 29 11/2/2004 Unit 10 - Stat Ramón León 30

16 Large-Sample Asymptotic Variance for Estimators of Functions of Location-Scale Parameters 11/2/2004 Unit 10 - Stat Ramón León 31 Proportion failing by censoring time t c 11/2/2004 Unit 10 - Stat Ramón León 32

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19 Figures for Sample Sizes to Estimate Weibull, Lognormal, and Loglogistic Quantiles 11/2/2004 Unit 10 - Stat Ramón León 37 Generalization: Location-Scale Parameters and Multiple Censoring 11/2/2004 Unit 10 - Stat Ramón León 38

20 Test Plans to Demonstrate Conformity with a Reliability Standard 11/2/2004 Unit 10 - Stat Ramón León 39 Minimum Sample Size Reliability Demonstration Test Plans 11/2/2004 Unit 10 - Stat Ramón León 40

21 11/2/2004 Unit 10 - Stat Ramón León 41 Justification for the Weibull Zero-Failures Test Plan 11/2/2004 Unit 10 - Stat Ramón León 42

22 Justification for the Weibull Zero-Failures Test Plan (Continued) 11/2/2004 Unit 10 - Stat Ramón León 43 Additional Comments on Zero Failure Test Plans 11/2/2004 Unit 10 - Stat Ramón León 44

23 Other Topics in Chapter 10 Uncertainty in planning values and sensitivity analysis Sample size to estimate unrestricted functions of the parameters, the mean of an exponential, the hazard function of a location-scale distribution Test planning for non-location-scale distributions 11/2/2004 Unit 10 - Stat Ramón León 45