unit 3: nonparametric estimationweb.utk.edu/~leon/rel/fall04pdfs/567unit3handout.pdf · 9/3/2009...

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Unit 3: Nonparametric Estimation Unit 3: Nonparametric Estimation

Notes largely based on “Statistical Methods for Reliability Data” by W.Q. Meeker and L. A. Escobar, Wiley, 1998 and on their class notes.

9/3/2009 Stat 567: Unit 3 - Ramón V. León 1

Ramón V. León

Unit 3 ObjectivesUnit 3 Objectives

Show the use of the binomial distribution to estimate F(t) from interval and singly right censored data, without assumptions on F(t) This is called without assumptions on F(t). This is called nonparametric estimation

Explain and illustrate how to compute standard error for and approximate confidence intervals for F(t)

Show how to extend nonparametric estimation to allow for multiply right-censored data

Illustrate the Kaplan-Meier nonparametric estimator for data with observations reported as exact failures

ˆ ( )F t

for data with observations reported as exact failures Describe and illustrate a generalization that provides a

nonparametric estimator of F(t) with arbitrary censoring


2

Data for Plant 1 of the Heat Exchanger Data for Plant 1 of the Heat Exchanger Tube Crack DataTube Crack Data


A Nonparametric Estimator of A Nonparametric Estimator of F(tF(ti i )) Based on Based on Binomial Theory for Binomial Theory for Interval SinglyInterval Singly--Censored DataCensored Data


3

Plant 1 Estimate of CDFPlant 1 Estimate of CDF


Comments on the Nonparametric Estimate Comments on the Nonparametric Estimate of of F(tF(ti i ))


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Confidence IntervalsConfidence Intervals


Some Characteristic Features of Some Characteristic Features of Confidence IntervalsConfidence Intervals

The level of confidence expresses one’s confidence (not probability) that a specific interval contains the (not probability) that a specific interval contains the quantity of interest

The actual coverage probability is the probability that the procedure will result in an interval containing the quantity of interest

A confidence interval is approximate if the specified level of confidence is not equal to the actual coverage probabilityp y

With censored data most confidence intervals are approximate. Better approximations require more computations


5

Pointwise BinomialPointwise Binomial--Based Based Confidence Interval for Confidence Interval for F(tF(ti i ))


Pointwise NormalPointwise Normal--Approximation Approximation Confidence Interval for Confidence Interval for F(tF(ti i ))


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Plant 1 Heat Exchanger Tube Crack Nonparametric Estimate Plant 1 Heat Exchanger Tube Crack Nonparametric Estimate with Conservative Pointwise 95% Confidence Intervals Based with Conservative Pointwise 95% Confidence Intervals Based on Binomial Theoryon Binomial Theory


Calculation of the Nonparametric Estimate of Calculation of the Nonparametric Estimate of F(F(ttii) ) for for Plant 1 from the Heat Exchanger Tube Crack DataPlant 1 from the Heat Exchanger Tube Crack Data


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Integrated Circuit (IC) Failure Times in Integrated Circuit (IC) Failure Times in Hours Data from Meeker (1987)Hours Data from Meeker (1987)


Lfp1370.ld

Nonparametric Estimator of F(t) Based on Binomial Nonparametric Estimator of F(t) Based on Binomial Theory for Theory for Exact Failures and Singly Right Censored DataExact Failures and Singly Right Censored Data


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JMP AnalysisJMP Analysis



0 008

0.010

0.012

0.014

0.016

0.018

0.020

Fa

ilin

g

0.000

0.002

0.004

0.006

0.008

0 100200 300 400 500600 700 800900 1100 1300

Hours


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Comments on the Nonparametric Comments on the Nonparametric Estimate of Estimate of F(t)F(t)


Delta Method and Delta Method and Derivative of the Logit of the CDFDerivative of the Logit of the CDF

2

Delta Method:

2ˆ ˆ ˆ( ) = '( ) ( )

Derivative of the Logit Function:

( ) log log log 11

Var f f Var

xf x x x

x

ˆˆlogit

ˆ ˆ1F

seF

F F


1 1 1'( )

1 (1 )f x

x x x x

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PointwisePointwise NormalNormal--Approximation Confidence Approximation Confidence Interval for Interval for F(t F(t ii )) Based on the Based on the LogitLogit TransformationTransformation


PointwisePointwise NormalNormal--Approximation Confidence Approximation Confidence Interval for Interval for F(t F(t ii )) Based on the Based on the LogitLogit TransformationTransformation


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Nonparametric Estimate for the IC Data with Normal Nonparametric Estimate for the IC Data with Normal Approximation Pointwise 95% ConfidenceApproximation Pointwise 95% Confidence

Interval Based on the Logit TransformationInterval Based on the Logit Transformation


Notation ExampleNotation Example

-1 -1

10 0

13 sample size

3 # of failures in the interval

2 # of right censored observation at

7 risk set at

3ˆ estimate of the probability of

7

thi

i i

i i

i i j jj j

i

n

d i

r t

n t n d r

p


7

failing in the intethi rval given that item

has survived to the begining of the interval

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A Nonparametric Estimate of A Nonparametric Estimate of F(tF(tii) ) BasedBased ononInterval Data and Interval Data and Multiple Right CensoringMultiple Right Censoring


Pooling of the Heat Exchanger Tube Pooling of the Heat Exchanger Tube Crack DataCrack Data


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Calculation of the Nonparametric Estimate of Calculation of the Nonparametric Estimate of F(F(ttii)) for the Heat Exchanger Tube Crack Datafor the Heat Exchanger Tube Crack Data

0.0133, 0.9867

0.0254, 0.9746


0.0206, 0.9794

Nonparametric Estimate for the Heat Nonparametric Estimate for the Heat Exchanger Tube Crack DataExchanger Tube Crack Data


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Approximate Variance of Estimated CDFApproximate Variance of Estimated CDF

ˆ ˆˆ ˆRecall, ( ) 1 ( ) the Var ( ) Var ( )

ˆ

i i i i

i i i

F t S t F t S t

1 1 1ˆ ˆ ˆAlso ( ) 1 and ( )

ˆThen a Taylor series first-order approximation of ( ) is

i i i

i j j i jj j j

i

S t p q S t q

S t

1

( )ˆ ˆ( ) ( )i i

i i j jjj

S tS t S t q q

q


1

( )ˆ( )

jj q

i ii j jj

j

q

S tS t q q

q

Approximate Variance of Estimated CDFApproximate Variance of Estimated CDF

2 2

Then it follows that

1 1

( ) ( )ˆ ˆVar ( ) ( )

ˆbecause the are approximately

uncorrelated binomial proportions.

ˆ(The values are asymtotically as unc

i i j ji ii jj j

j j j

j

j

q pS t S tS t Var q

q q n

q

q n

orrelated).


2 2

1 1ˆVar ( ) ( ) ( )

(1 )

i ij ji i ij j

j j j j

p pS t S t S t

n q n p

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Estimating the Standard Error of the Estimating the Standard Error of the Estimated CDFEstimated CDF


Standard Errors for the Estimated CDF Standard Errors for the Estimated CDF of of the the Heat Heat Exchanger Tube Crack DataExchanger Tube Crack Data

0.0133, 0.9867

0.0254, 0.9616,

0.0206, 0.9418


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Recall:Recall: PointwisePointwise NormalNormal--Approximation Confidence Approximation Confidence Interval for Interval for F(t F(t ii )) Based on the Based on the LogitLogit TransformationTransformation


NormalNormal--Approximation Pointwise Confidence Approximation Pointwise Confidence Intervals of the Heat Exchanger Tube Crack DataIntervals of the Heat Exchanger Tube Crack Data


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18


0 1 2 3



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Recall:


Shock Absorber Failure DataShock Absorber Failure Data

F il i i b f kil f f hi l

First reported in O’Connor (1985)

Failure times in number of kilometers of use, of vehicle shock absorbers

Two failure modes, denoted by M1 and M2 One might be interested in the distribution of time to

failure for mode M1, mode M2, or the overall failure-time distribution of the part

Data Table C.2 in the Appendix, page 630


Here we do not differentiate between mode M1 and M2.We will estimate the distribution of time to failure by either mode M1 or M2.

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Failure Pattern in the Shock Absorber Data: Failure Pattern in the Shock Absorber Data: Failure Mode IgnoredFailure Mode Ignored


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Nonparametric Estimates for the Shock Nonparametric Estimates for the Shock Absorber Data up to 12,220 kmAbsorber Data up to 12,220 km


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JMP JMP JMP JMP AnalysisAnalysis


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24



25


Theory of Simultaneous Confidence Theory of Simultaneous Confidence BandsBands


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27



28



29



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SPLIDA GRAPH:SPLIDA GRAPH:

0.8

- - -

-

Turbine Wheel Crack Initiation Data with Nonparametric Pointwise 95% Confidence Bands

0.2

0.4

0.6F

ract

ion

Fa

iling

-

- - -

- - -

-- -


0

10 20 30 40 50

Sat Aug 23 22:36:34 EDT 2003Hundreds of Hours

- - --

- --

1

Turbine Wheel Crack Initiation Data with Nonparametric Simultaneous 95% Confidence Bands

SPLIDA GRAPH:

0 2

0.4

0.6

0.8

Fra

ctio

n F

ailin

g

- - -

- - -

- - -

- - -

-


0

0.2

10 20 30 40 50

Sat Aug 23 22:31:59 EDT 2003Hundreds of Hours

- - - -- -

-

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m

10.0000

14 0000

Start Time

10.0000

14 0000

End Time

0.9302

0 9302

Survival

0.0698

0 0698

Failure

0.0337

0 0473

SurvStdEr

Combined

14.0000

18.0000

22.0000

26.0000

30.0000

34.0000

14.0000

18.0000

22.0000

26.0000

30.0000

34.0000

0.9302

0.9041

0.8333

0.7778

0.7778

0.5385

0.0698

0.0959

0.1667

0.2222

0.2222

0.4615

0.0473

0.0345

0.0680

0.0657

0.0650

0.1383


38.0000

42.0000

46.0000

38.0000

42.0000

46.0000

0.4190

0.4190

0.4165

0.5810

0.5810

0.5835

0.0865

0.0766

0.0822

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Omitted Topic in Chapter 3Omitted Topic in Chapter 3

Uncertain censoring time◦ Have assumed that censoring takes place at ◦ Have assumed that censoring takes place at

the end of the observation intervals◦ Can assume censoring happens in the

“middle” of the observation intervals◦ Leads to actuarial or life table nonparametric

estimate of cdf See Table 3 6 Page 64estimate of cdf. See Table 3.6 Page 64.


unit 3: nonparametric estimationweb.utk.edu/~leon/rel/fall04pdfs/567unit3handout.pdf · 9/3/2009...

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