Statistical properties of maximum likelihood estimates for accelerated lifetime data under the Weibull model

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Louisiana Tech Digital Commons

Doctoral Dissertations Graduate School

Spring 2001

Statistical properties of maximum likelihood

estimates for accelerated lifetime data under the

Weibull model

Mahmoud A. Yousef

Louisiana Tech University

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Recommended Citation

Yousef, Mahmoud A., "" (2001). Dissertation. 714.

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LIKELIHOOD ESTIMATES FOR ACCELERATED

LIFE-TIME DATA UNDER THE WEIBULL MODEL

by

Mahmoud A. Yousef, MA

A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree

Doctor of Philosophy

COLLEGE OF ENGINEERING AND SCIENCE LOUISIANA TECH UNIVERSITY

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THE GRADUATE SCHOOL

April 2 5 ,2 0 0 1

We hereby recommend that the dissertation prepared under our supervision by Mahmoud A. Yousef entided Statistical Properties o f Maximum Likelihood Estimates for Accelerated Life-Time Data Under the Weibull Model be accepted in partial fulfillment o f the requirements for the Degree o f Ph.D. in Computational Analysis and Modeling.

upervisor o f Dissertation Research

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Department Recommendation concurred in:

Approved:

Director

Advisory Committee

Approved:

Dean of the Graduate School

GS Form 13

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P ip e re h a b ilita tio n liners are often in stalled in h o st pipes th a t lie below th e w a te r table. As such, th e y are su b jected to e x te rn a l h y d ro sta tic pressure. T h e e x te rn a l pressure leads to early deform ation in the liners, w hich could ultim ately lead to its failing or buckling before its expected service lifetim e is achieved. E xperim ents in volving long te rm buckling behavior of liners are ty p ica lly accelerated lifetime te stin g procedures. I n a n accelerated testing procedure a liner is su b jec te d to a co n stan t ex te rn a l h y d ro static pressure a n d observed u n til it fails o r for a certain tim e, t w hichever occurs first. Liners t h a t do n o t fail at tim e t are deem ed censored observations. W hile a co n stan t pressu re is convenient to use in ex p erim en tal situ atio n s, in reality pressure fluctuates u n d er soil conditions over tim e dep en d in g o n th e w ater table.

In this study, co n sta n t an d variable pressures using th e W eibull m odel for tim e till buckling u n d er different sample sizes a n d different levels of censoring were investigated. D a ta w ere g enerated th ro u g h co m p u te r sim ulation and estim ates of p aram eters in th e W eibull m odel were o b tain ed using th e M axim um Likelihood a n d N ew ton-R aphson m ethods.

It was concluded th a t th e m axim um likelihood estim ate s under fixed or variable pressure, an d for different sam ple sizes w ith different levels of censoring, are unbiased. However, the estim ates for sam ple sizes as large as 100 are n o t norm ally d istrib u ted , especially w hen th e p a ra m e te r value being e stim a te d is sm all. It was seen th a t the

iii

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covariance m atrix a n d th e th eo retical variance-covariance m atrix . T hese results cast d o u b t on th e use of n o rm al th eo ry for inference concerning certain param eters.

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Finally, the author of this Dissertation reserves the right to publish freely, in the literature, at any time, any or all portions o f this Dissertation.

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TO THE M E M O R Y OF M Y FATHER A T IE H T O M Y M O TH E R GAMILEH

TO M Y WIFE GAMILEH A N D

M Y CH ILD R EN AHM ED, AB DALA, A N D JA N N A W IT H LOVE

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A B S T R A C T ... iii D E D I C A T I O N ... v L I S T O F T A B L E S ...viii L I S T O F F I G U R E S ...xx A C K N O W L E D G M E N T S ...xxii C H A P T E R 1 I N T R O D U C T I O N A N D R E S E A R C H O B J E C T I V E S . . . 1 1.1 In tro d u c tio n ...1 1.2 O b je c tiv e s... 4

1.3 O rg an izatio n of th e D is se rta tio n ... 4

C H A P T E R 2 R E L A T E D R E S E A R C H ... 5

2.1 R e lia b ility ...5

2.2 Some Im p o rta n t Lifetime M o d e ls ...7

2.2.1 T h e E xponential D is trib u tio n ...8

2.2.2 T h e W eibull D is tr ib u tio n ... 9

2.2.3 E x trem e Value D is trib u tio n ... 10

2.2.4 T h e G am m a D is trib u tio n ...11

2.2.5 T h e Lognorm al D is trib u tio n ...12

2.3 A ccelerated Life T e stin g ... 13

2.3.1 C onstant-S tress A ccelerated T e s t ...13

2.3.2 S tep-S tress A ccelerated T e s t ... 15

2.3.3 C um ulative E xposure M odel (O E M )... 17

2.3.4 T h e Kham is-Higgins M odel (K H M )... 19

C H A P T E R 3 M E T H O D O L O G Y ... 22

3.1 T h e M axim um Likelihood M e th o d ... 23

3.2 N ew ton-R aphson M e th o d ...25

3.3 S im u la tio n ... 28

vi

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P R E S S U R E ...30

4.1 T h e o r y ... 30

4.2 Choices for 7 . A, b, a n d pressure, x ... 36

4.3 R esu lts F ro m S im u la tio n ... 38 C H A P T E R 5 A C C E L E R A T E D L I F E T I M E U N D E R V A R I A B L E P R E S S U R E ... 73 5.1 T h e o ry ...73 5.2 R esu lts F ro m S im u la tio n ... 82 C H A P T E R 6 C O N C L U S I O N A N D F U T U R E R E S E A R C H ...116 6.1 C o n c lu sio n ... 116 6.2 F u tu re W o r k ... 118 A P P E N D I X A F O R T R A N P R O G R A M F O R T H E N E W T O N - R A P H S O N M E T H O D U N D E R C O N S T A N T P R E S S U R E ... 119 A P P E N D I X B F O R T R A N P R O G R A M F O R T H E N E W T O N - R A P H S O N M E T H O D U N D E R V A R I A B L E P R E S S U R E ... 128 A P P E N D I X C F O R T R A N P R O G R A M F O R F I N D I N G T H E I N V E R S E O F A M A T R I X ... 143 R E F E R E N C E S ...147

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T A B L E D E S C R IP T IO N PA G E 4.1 S tatistical p ro p erties o f th e M L e stim a te for 7 = 1.5 over

1000 replications. P ressu re= 2 7 .8 p si. censoring=10% ,

sam ple s iz e = 2 5 ... 41 4.2 S tatistical p ro p erties of th e M L e s tim a te for 7 = 1.5 over

1000 replications. P ressu re= 2 7 .8 psi, censoring=10% ,

sam ple s iz e = 5 0 ... 41 4.3 S tatistical p ro p erties o f th e ML e s tim a te for 7 = 1.5 over

sam ple s iz e = 1 0 0 ...42 4.4 S tatistical p ro p erties of th e ML e s tim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 2 5 ... 42 4.5 S tatistical p ro p erties o f th e ML e s tim a te for A = 5.7 x 10~ 5 over

1000 replications. P ressu re= 2 7 .8 psi. c e n so rin g =1 0%,

sam ple siz e = 5 0 ... 43 4.6 S tatistical p ro p erties o f the M L e s tim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 1 0 0 ... 43 4.7 S tatistical p ro p erties of th e ML e s tim a te for b=-0.25 over

sam ple s iz e = 2 5 ...44 4.8 S tatistical p ro p erties of the ML e s tim a te for b= -0.25 over

1000 replications. P ressu re= 2 7 .8 psi, censoring=10% .

sam ple size = 5 0 ... 44 4.9 S tatistical p ro p erties of th e ML e s tim a te for b=-0.25 over

v m

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4.10 S ta tis tic a l p roperties of th e M L e s tim a te for 7 = 1.5 over

1000 rep licatio n s. P ressnre= 26.8 p si, censoring= 20% ,

sam ple s iz e = 2 5 ... 45 4.11 S ta tistic a l properties of th e ML e stim a te for 7 = 1.5 over

1000 rep licatio n s. P ressure= 26.8 p si, censoring= 20% ,

sam ple s iz e = 5 0 ... 46 4.12 S ta tis tic a l properties of th e ML e stim a te for 7 = 1.5 over

1000 rep licatio n s. P ressure= 26.8 p si, censoring=20% ,

sam ple s iz e = 1 0 0 ... 46 4.13 S ta tis tic a l properties of th e M L e stim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 2 5 ... 47 4.14 S ta tis tic a l properties of th e ML e stim a te for A = 5.7 x 10- 5 over

1000 rep licatio n s. P ressure= 26.8 p si, censoring=2 0%,

sam ple s iz e = 5 0 ... 47 4.15 S ta tis tic a l properties of th e M L e stim a te for A = 5.7 x 10“ ° over

sam ple s iz e = 1 0 0 ... 48 4.16 S ta tis tic a l properties of th e M L e s tim a te for b=-0.25 over

1000 rep licatio n s. P ressure= 26.8 p si, censoring=20% .

sam ple s iz e = 2 5 ... 48 4.17 S ta tis tic a l properties of th e M L e s tim a te for b=-0.25 over

1000 rep licatio n s. P ressure= 26.8 p si, cen so rin g =2 0% ,

sam ple s iz e = 5 0 ... 49 4.18 S ta tis tic a l p roperties of th e M L e s tim a te for b=-0.25 over

sam ple s iz e = 1 0 0 ... 49 4.19 S ta tistic a l p roperties of th e ML e stim a te for 7 = 1.5 over

1000 rep licatio n s. Pressuxe=26.0 p si, censoring=30% ,

sam ple s iz e = 2 5 ... 50 4.20 S ta tistic a l p roperties of th e ML e stim a te for 7 = 1.5 over

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4.21 S ta tistic a l p ro p erties of th e M L e s tim a te for 7 = 1.5 over

1000 replications. P ressure= 26.0 p si, censoring=30% ,

sam ple size= 1 0 0 ...

4.22 S ta tistic a l pro p erties of th e M L e s tim a te for A = 5.7 x 10- 5 over

1000 replications. P ressure= 26.0 p si, censoring=30% .

sam ple s iz e = 2 5 ...- ... 4.23 S ta tistic a l p ro p erties of th e M L e s tim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 5 0 ...- ... 4.24 S ta tistic a l pro p erties of th e M L e s tim a te for A = 5.7 x 10- 5 over

1000 replications. P ressu re= 2 6 .0 psi,. censoring=30% ,

sam ple size= 1 0 0...

4.25 S ta tistic a l p ro p erties of th e M L e s tim a te for b=-0.25 over 1000 replications. P ressure= 26.0 psi„ censoring=30% ,

sam ple s iz e = 2 5 ... 4.26 S ta tistic a l pro p erties of th e M L e s tim a te for b=-0.25 over

1000 replications. P ressu re= 2 6 .0 psi_ censoring=30% ,

sam ple s iz e = 5 0 ... 4.27 S ta tistic a l pro p erties of th e ML e s tim a te for b=-0.25 over

1000 replications. P ressure= 26.0 psi_ censoring=30% ,

4.28 S ta tistic a l p ro p erties of the M L e s tim a te for 7 = 1.5 over

1000 replications. P ressure= 25.3 psi_ censoring=40% ,

sam ple s iz e = 2 5 ... 4.29 S ta tistic a l p ro p erties of the M L e s tim a te for 7 = 1.5 over

1000 replications. P ressure= 25.3 psi,. cen.soring=40%,

sam ple size= 5 0 ... 4.30 S ta tistic a l p ro p erties of the M L e s tim a te for 7 = 1.5 over

1000 replications. P ressure= 25.3 psi* censoring=40% ,

4.31 S ta tistic a l pro p erties of the ML e s tim a te for A = 5.7 x 10- 5 over

1000 replications. P ressure= 25.3 psi„ censoring=40% ,

x 51 51 52 52 53 53 54 54 55 55

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4.32 S ta tistic a l p ro p e rtie s of th e ML e s tim a te for A = 5.7 x 10~ 5 over

1000 replications- P ressu re= 2 5 .3 psi, censoring= 40% ,

sam ple s iz e = 5 0 ... 56 4.33 S ta tistic a l p ro p e rtie s of th e ML e s tim a te for A = 5.7 x 10- 5 over

1000 rep lic atio n s. P ressu re= 2 5 .3 psi, censoring= 40% ,

sam ple s iz e = 1 0 0 ... 57 4.34 S tatistic a l p ro p e rtie s of th e ML e stim a te for b= -0 .2 5 over

1000 rep licatio n s. P re ssu re =25.3 psi, c en so rin g = 4 0 %,

sam ple s iz e = 2 5 ... 57 4.35 S ta tistic a l p ro p e rtie s of th e ML e s tim a te for b= -0 .2 5 over

1000 rep licatio n s. P re ssu re =25.3 psi, censoring= 40% ,

sam ple s iz e = 5 0 ... 58 4.36 S ta tistic a l p ro p e rtie s of th e ML e stim a te for b= -0 .2 5 over

1000 rep licatio n s. P ressu re= 2 5 .3 psi, censoring= 40% ,

sam ple s iz e = 1 0 0 ... 58 4.37 T he E m p irical variance-covariance m a trix o f th e M L estim ates over

sam ple s iz e = 2 o ...59 4.38 T he E m p irical variance-covariance m a trix o f th e M L estim ates over

sam ple s iz e = 2 5 ...59 4.39 T he E m p irical variance-covariance m a trix o f th e M L estim ates over

sam ple size= 2 5 ...60 4.41 T he E m p irical variance-covariance m a trix o f th e M L estim ates over

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4.43 T h e E m pirical variance-covariance m atrix o f th e ML estim ates over 1000 replications. P ressu re= 2 6 .0 psi. censoring= 30% ,

sam ple s iz e = 5 0 ...61 4.44 T h e E m pirical variance-covariance m atrix of th e ML estim ates over

1000 replications. P ressu re= 2 5 .3 psi. censoring=40% ,

sam ple s iz e = 5 0 ...61 4.45 T h e E m pirical variance-covariance m atrix of th e ML estim ates over

sam ple size= 1 0 0 ...61

4.46 T h e E m pirical variance-covariance m atrix o f th e ML estim ates over 1000 replications. P ressu re= 2 6 .8 psi, censoring=20% ,

sam ple size= 1 0 0 ...62

4.47 T h e E m pirical variance-covariance m atrix of th e ML estim ates over 1000 replications. P ressu re= 2 6 .0 psi, censoring=30% ,

sam ple size= 1 0 0...62

4.48 T h e E m pirical variance-covariance m atrix of th e ML estim ates over 1000 replications. P ressu re= 2 5 .3 psi, censoring=40% ,

sam ple size= 1 0 0...62

4.49 T h e Fisher variance-covariance m atrix of th e M L estim ates over 1000 replications. P ressu re= 2 7 .8 psi, censoring=10% ,

sam ple s iz e = 2 5 ...63 4.50 T h e Fisher variance-covariance m atrix of th e M L estim ates over

sam ple s iz e = 2 5 ...63 4.51 T h e F isher variance-covariance m atrix of th e M L estim ates over

sam ple s iz e = 2 5 ...63 4.52 T h e Fisher variance-covariance m atrix of th e M L estim ates over

sam ple s iz e = 2 5 ...64 4.53 T h e F isher variance-covariance m atrix of th e M L estim ates over

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4.54 T h e F ish e r variance-covariance m a trix o f th e ML estim ates over 1000 replications. P ressu re= 2 6 .8 psi, censoring= 20%,

sam ple s iz e = 5 0 ... 4.55 T h e F ish e r variance-covariance m a trix o f the ML estim ates over

sam ple s iz e = 5 0 ... 4.56 T h e F ish er variance-covariance m a trix of the ML estim ates over

sam ple s iz e = 5 0 ... 4.57 T h e F ish e r variance-covariance m a trix of the ML estim ates over

4.58 T h e F ish er variance-covariance m a trix of the ML estim ates over 1000 replications. P ressu re= 2 6 .8 psi, censoring=20% ,

4.59 T h e F ish e r variance-covariance m atrix of the ML estim ates over 1000 replications. P ressu re= 2 6 .0 psi, censoring=30% ,

4.60 T he F ish e r variance-covariance m atrix o f the ML estim ates over 1000 replications. P ressu re= 2 5 .3 psi, censoring=40% ,

5.1 S ta tistic a l properties of th e ML e stim a te for 7 = 1.5 over

1000 replications. V ariable pressure, censoring=10% ,

sam ple s iz e = 2 5 ... 5.2 S ta tistica l properties of th e M L e stim a te for 7 = 1.5 over

sam ple s iz e = 5 0 ... 5.3 S ta tistic a l properties of th e ML e stim a te for 7 = 1.5 over

5.4 S ta tistic a l properties of th e M L estim ate for A = 5.7 x 10- 5 over

1000 replications. Variable pressure, censoring=10% ,

64 65 65 65 66 66 66 84 84 85

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5.5 S ta tistic a l p ro p ertie s of th e M L e stim a te for A = 5.7 x 10- 5 over

1000 rep licatio n s. V ariable pressu re. censoring=10% ,

sam ple s iz e = 5 0 ... 5.6 S ta tistic a l p ro p e rtie s of th e M L e stim a te for A = 5.7 x 10- 5 over

1000 rep licatio n s. Variable pressu re. censoring=10% ,

5.7 S ta tistic a l p ro p e rtie s of th e M L e stim a te for b= -0.25 over 1000 replications- Variable pressure, censoring=10% .

sam ple s iz e = 2 5 ... 5.8 S ta tistic a l p ro p e rtie s of th e M L e stim a te for b= -0.25 over

1000 rep licatio n s. Variable pressu re, cen so rin g = 10%,

sam ple s iz e = 5 0 ... 5.9 S ta tistic a l p ro p e rtie s of th e M L estim a te for b= -0.25 over

1000 rep licatio n s. V ariable pressure, censoring=10% ,

5.10 S ta tistic a l p ro p erties of th e ML estim ate for 7 = 1.5 over

1000 rep licatio n s. Variable pressure, censoring=20% ,

sam ple s iz e = 2 5 ... 5.11 S ta tistic a l p ro p erties of th e M L estim ate for 7 = 1.5 over

sam ple size= 5 0 ... 5.12 S ta tistic a l p ro p e rtie s of th e M L estim ate for 7 = 1.5 over

5.13 S ta tistic a l p ro p erties of th e M L estim ate for A = 5.7 x 10- 5 over

sam ple s iz e = 2 5 ... 5.14 S ta tistic a l p ro p erties of th e ML estim ate for A = 5.7 x 10- 5 over

sam ple s iz e = 5 0 ... 5.15 S ta tistic a l p ro p erties of th e M L estim ate for A = 5.7 x 10~ 5 over

x iv 86 86 87 87 88 88 89 89 90 90

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5.16 S ta tistic a l p ro p erties of th e ML e stim a te for b = -0.25 over 1000 replications. Vaxiable pressure, censoring= 20% ,

sam ple s iz e = 2 5 ... 5.17 S ta tistic a l p ro p erties of th e ML e stim a te for b= -0.25 over

1000 replications. V ariable pressure, cen so rin g =20% ,

sam ple s iz e = 5 0 ... 5.18 S ta tistic a l p ro p erties of th e ML e stim a te for b= -0.25 over

5.19 S ta tistic a l p ro p erties of th e ML e stim a te for 7 = 1.5 over

sam ple s iz e = 2 5 ... 5.20 S ta tistic a l p ro p e rtie s of th e ML e stim a te for 7 = 1.5 over

1000 replications. V ariable pressure, censoring= 30% ,

sam ple size= 5 0 ... 5.21 S ta tistic a l p ro p erties of th e ML e stim a te for 7 = 1.5 over

5.22 S ta tistic a l p ro p erties of th e ML e stim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 2 5 ... 5.23 S ta tistic a l p ro p erties of th e ML e stim a te for A = 5.7 x 10- 5 over

sam ple s iz e = 5 0 ... 5.24 S ta tistic a l p ro p erties of th e ML e stim a te for A = 5.7 x 10- 5 over

5.25 S tatistic a l p ro p erties of th e ML e stim a te for b= -0.25 over 1000 replications. V ariable pressure, cen so rin g =30% ,

sam ple s iz e = 2 5 ... 5.26 S tatistic a l p ro p erties of th e ML e stim a te for b= -0.2o over

91 92 92 93 93 94 94 95 95 96

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5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34 5.35 5.36 5.37

S ta tistic a l p ro p erties of the ML estim ate for b= -0 .2 5 over 1000 rep licatio n s. V ariable pressure, censoring=30% ,

sam ple s iz e = 1 0 0 ... 97 S ta tistic a l p ro p erties of the ML estim ate for 7 = 1.5 over

sam ple s iz e = 2 5 ... 97 S ta tistic a l p ro p erties of the ML estim ate for 7 = 1.5 over

sam ple s iz e = 5 0 ...98 S ta tistic a l p ro p erties of the ML estim ate for 7 = 1.5 over

1000 rep licatio n s. V ariable pressure, cen so rin g = 4 0 % ,

sam ple s iz e = 1 0 0 ... 98 S ta tistic a l p ro p erties of the ML estim ate for A = 5.7 x 10- 5 over

sam ple s iz e = 2 5 ... 99 S ta tistic a l p ro p erties of the ML estim ate for A = 5.7 x 10- 5 over

sam ple s iz e = 5 0 ...99 S ta tistic a l p ro p erties of the ML estim ate for A = 5.7 x 10- 5 over

sam ple size= 1 0 0 ...1 0 0

S ta tistic a l p ro p erties of the ML estim ate for b= -0.25 over 1000 rep licatio n s. V ariable pressure, censoring=40% ,

sam ple s iz e = 2 5 ... 100 S ta tistic a l p ro p erties of the ML estim ate for b= -0.25 over

sam ple s iz e = 5 0 ...101 S ta tistic a l p ro p erties of the ML estim ate for b= -0.25 over

sam ple size= 1 0 0 ...1 0 1

T he E m p irical variance-covariance m a trix of th e ML estim ates over 1000 rep licatio n s. V ariable pressure, censoring=10% ,

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5.38 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over 1000 rep lic atio n s. V ariable pressure, c e n so rin g =2 0% ,

sam ple s iz e = 2 5 ...1 0 2

5.39 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a tes over 1000 rep lic atio n s. V ariable pressure, censoring= 30% .

sam p le s iz e = 2 5 ... 102 5.40 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over

1000 rep lic atio n s. V ariable pressure, cen so rin g =40% ,

sam ple s iz e = 2 5 ...103 5.41 T h e E m p iric a l variance-covariance m a trix o f th e ML estim ates over

1000 rep lic atio n s. V ariable p ressure. censoring= 10% ,

sam ple s iz e = 5 0 ... 103 5.42 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over

1000 rep lic a tio n s. V ariable p ressure, censoring= 20% ,

sam ple s iz e = 5 0 ... 103 5.43 T h e E m p iric a l variance-covariance m a trix o f th e ML estim ates over

sam ple s iz e = 5 0 ... 104 5.44 T h e E m p iric a l variance-covariance m a trix o f th e ML e stim ates over

sam ple s iz e = 5 0 ...104 5.45 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over

1000 rep lic atio n s. Variable pressure, censoring= 10% ,

sam p le s iz e = 1 0 0 ...104 5.46 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over

1000 rep licatio n s. V ariable pressure, censoring= 20% ,

sam ple s iz e = 1 0 0 ... 105 5.47 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a tes over

1000 rep lic atio n s. V ariable pressure, censoring= 30% ,

sam p le s iz e = 1 0 0 ...105 5.48 T h e E m p iric a l variance-covariance m a trix o f th e ML estim a te s over

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5.49 T h e F isher variance-covariance m a trix o f th e ML estim ates over 1000 replications. V ariable pressure, censoring= 10% ,

sam ple s iz e = 2 5 ... 106 5.50 T h e F isher variance-covariance m a trix o f th e ML estim ates over

sam ple s iz e = 2 5 ... 106 5.51 T h e F ish er variance-covariance m a trix o f th e ML estim ates over

sam ple s iz e = 2 5 ... 106 5.52 T h e Fisher variance-covariance m a trix of th e ML estim ates over

sam ple s iz e = 2 5 ... 107 5.53 T h e Fisher variance-covariance m a trix of th e ML estim ates over

sam ple s iz e = 5 0 ... 107 5.54 T h e F isher variance-covariance m a trix of th e ML estim ates over

1000 replications. V ariable pressure, censoring=20%,

sam ple s iz e = 5 0 ... 107 5.55 T h e F isher variance-covariance m a trix of th e ML estim ates over

sam ple s iz e = 5 0 ... 108 5.56 T h e Fisher variance-covariance m a trix o f th e ML estim ates over

sam ple size = 5 0 ... 108 5.57 T h e Fisher variance-covariance m a trix o f th e ML estim ates over

sam ple size= 1 0 0... 108

5.58 T h e Fisher variance-covariance m a trix o f th e ML estim ates over 1000 replications. V ariable pressure, censoring= 20% ,

sam ple siz e = 1 0 0 ... 109 5.59 T h e Fisher variance-covariance m a trix of th e ML estim ates over

sam ple siz e = 1 0 0 ... 109 xviii

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1000 replications. P ressu re= 2 5 .3 psi. censoring=40% ,

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F IG U R E D E S C R IP T IO N PA G E 4.1 Relative frequency h isto g ram of th e ML estim ate for g a m m a = 1.5,

pressure= 27.8 psi, censoring=10% , sam ple s iz e = 2 5 ...

I

4.2 Relative frequency h isto g ram of th e ML estim ate for g a m m a = 1.5, pressure= 27.8 psi, censoring=10% , sam ple size = 1 0 0 ... 4.3 Relative frequency h isto g ram of th e ML estim ate for lam b d a= 0 .0 0 0 0 5 7 ,

pressure= 27.8 psi, censoring=10% , sam ple s iz e = 2 5 ... - ... 4.4 Relative frequency h isto g ram of th e ML estim ate for b = —0 .2 5 ,

pressure= 27.8 psi, censoring=10% , sam ple s iz e = 2 5 ... - ... 4.5 Relative frequency h isto g ram of th e ML estim ate for la m b d a = 0 .000057, pressure= 27.8 psi, censoring=10% , sam ple size = 1 0 0 ... 4.6 Relative frequency h isto g ram of th e ML estim ate for b = —0 .2 5 .

pressure=27.8 psi, censoring=10% , sam ple size= 1 0 0 ... 4.7 Relative frequency h isto g ram of th e ML estim ate for g a m m a = 1.5,

pressure=25.3 psi, censoring=40% , sam ple s iz e = 2 5 ... - ... 4.8 Relative frequency h isto g ram of th e ML estim ate for g a m m a = 1.5.

pressure=25.3 psi, censoring=40% . sam ple size = 1 0 0 ... 4.9 Relative frequency h isto g ram of th e ML estim ate for la m b d a = 0 .000057,

pressure=25.3 psi, censoring=40% , sam ple s iz e = 2 5 ... - ... 4.10 Relative frequency h isto g ram of th e ML estim ate for b = —0 .2 5 .

pressure=25.3 psi, censoring=40% , sam ple siz e = 2 5 ... 4.11 Relative frequency h isto g ram of th e ML estim ate for lam b d a= 0 .0 0 0 0 5 7 ,

pressure=25.3 psi, censoring=40% , sam ple s iz e = 1 0 0 ... 4.12 Relative frequency h isto g ram of th e ML estim ate for b — —0 .2 5 .

pressure=25.3 psi, censoring=40% , sam ple s iz e = 1 0 0 ... ...

x x 67 67 68 68 69 69 70 70 71 71 72 72

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variable p ressu re, censoring=10% , sam ple s iz e = 2 5 ...110 5.2 R elative frequency histo g ram of th e M L estim ate for lam bda= 0.000057,

variable p ressu re, cen so rin g =1 0%, sam p le s iz e = 2 5 ...1 1 0

5.3 R elative frequency h istogram of th e M L estim ate for b — —0.25,

variable p ressu re, cen so rin g =1 0%, sam ple s iz e = 2 5 ... I l l

5.4 R elative frequency h isto g ram of th e M L estim ate for g am m a= 1 .5 ,

variable p ressu re, cen so rin g =1 0%, sam ple size= 1 0 0... I l l

5.5 R elative freq u en cy histo g ram of th e M L estim ate for lam bda= 0.000057, variable p ressu re, cen so rin g =1 0%, sam ple size= 1 0 0...1 1 2

5.6 R elative frequency h istogram of th e M L estim ate for b = —0.25,

variable p ressu re, cen so rin g =1 0%, sam ple size= 1 0 0...1 1 2

5.7 R elative frequency histogram of th e M L estim ate for gam m a = 1 .5 ,

variable p ressu re, censoring=40% , sam ple s iz e = 2 5 ... 113 5.8 R elative frequency histogram of th e M L estim ate for la m b d a = 0 .000057,

variable p ressu re, censoring=40% , sam ple s iz e = 2 5 ... 113 5.9 R elative frequency h istogram of th e M L estim ate for b = —0.25,

variable p ressu re, censoring=40% , sam ple s iz e = 2 5 ... 114 5.10 R elative frequency h istogram of th e ML estim ate for gam m a = 1 .5 ,

variable p ressu re, censoring=40% , sam ple siz e = 1 0 0 ...114 5.11 R elative frequency h istogram of th e M L estim ate for la m b d a = 0 .000057,

variable pressu re, censoring=40% , sam ple siz e = 1 0 0 ...115 5.12 R elative frequency histogram of th e ML estim ate for b = —0.25,

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In tim es o f acknow ledgm ent, all too often we forget th e O ne behind it all - GOD. M y deepest g ra titu d e a n d ap p reciatio n go to A lm ighty A llah. It is He who has guided m e th ro u g h every success th a t I have h ad th ro u g h o u t m y life. W ith o u t His help an d su p p o rt, n eith er th is w ork n o r th e a u th o r him self w ould have seen the light.

M y acknow ledgm ent can never be com plete, however, w ith o u t expressing my sincere ap p reciatio n to m y advisor Dr. R aja N assar for his direction of this disserta tion. His suggestions, criticism s, an d encouragem ent were invaluable.

A ppreciation is also ex ten d ed to Dr. A lb ert E dw in A lexander, Dr. W eizhong Dai, an d Dr. R ay m o n d Leslie Sterling for their help, su p p o rt, encouragem ent, a n d service on my advisory com m ittee.

I would like to e x te n d m y appreciation to D r. R ich ard Greechie for providing me w ith th e o p p o rtu n ity to receive a n assistan tsh ip a n d stu d y at Louisiana Tech University. Also, I w ould like to th a n k Dr. B ernd Siegfried Schroeder for the help he provided w ith L A T E X a n d his su p p o rt an d encouragem ent th roughout my graduate studies. M any th a n k s to C y n th ia V anlandingham for th e help she provided w ith the com puter program m ing.

I would like also to express m y deep a p p reciatio n to m y m other, brothers, an d sisters for th e ir endless love, a n d unlim ited su p p o rt th ro u g h o u t m y g rad u ate studies.

Last, b u t n o t least, I w ould like also to express m y deep appreciation to my xxii

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a n d understanding, b u t m ore im p o rtan t for th e ir love a n d perseverance th a t enabled th e m to provide me w ith an ap p ro p riate home environm ent to com plete this research.

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I N T R O D U C T I O N A N D R E S E A R C H O B J E C T IV E S

1.1 Introduction

It is know n th a t th e u n d erg ro u n d in frastru ctu re sy ste m in th e U nited S tates is in urgent need for re p a ir or u p g rad e. It has b een th e p ractice, w henever there is a pro b lem w ith a n u n d e rg ro u n d pipeline, to use th e o p e n -tre n c h m e th o d which includes digging th e ground, rem oval of th e deterio rated p ip e(s), a n d replacem ent w ith new one(s). It is clear th a t th is m e th o d is not desirable b ecau se of th e am ount of work req u ired for th e jo b , th e cost associated w ith it, th e tim e p e rio d to finish the work, a n d th e inconvenience to th e businesses an d th e g en eral public.

Recently, w ith developm ent of trenchless techniques, it has becom e possible to re p a ir underground p ip e(s) w ith o u t excavating th e g ro u n d . W ith th e new trenchless m ethods in effect today, th e problem s associated w ith th e o p en -tren ch m eth o d are slowly disappearing.

A relatively recen t a p p ro a ch for pipeline re h a b ilita tio n w hich provides signif icant economical, social a n d environm ental benefits involves pipeline repair by in se rtio n of a fining m a te ria l th ro u g h existing m anholes. C u red -In -P lace P lastic P ipe (C IP P ) an d F old -an d -F o rm ed P ip e (F F P ) are p e rh a p s th e m ost well known of the refining m ethods (G uice et al., 1994).

1

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in trenchless p ip e lin e rehabilitation. It involves th e in stallatio n o f p lastic liners inside the dam ag ed p ip e lin e th ro u g h existing m anholes o r o th er e n try p o in ts. T h e process inverts a re sin -im p reg n a te d fabric tu b e in to th e dam aged pip e u sin g a hydraulic head or w inching it in place. W hen circulating h o t w ater inside th e pipe, th e resin will cure an d h a rd e n in to a continuous, sn u g -fittin g tu b e inside th e original host pipe. T he C IP P tech n iq u e n o t only seals the jo in ts a n d restores th e pipeline integrity, but also increases th e s tre n g th of the existing p ip e a n d provides im proved corrosion re sistance for th e in n e r surface of the pipe. T h is m eth o d was in tro d u ce d by Insituform in E urope in 1971 a n d th e n was b rought to th e U nited S ta te s in 1977 (Li. 1994).

T h e ex a c t definition of C IP P according to th e A m erican S ociety for T estin g and M aterials (A S T M ) is "a hollow cylinder co n tain in g a non-woven m a te ria l surrounded by the cured th e rm o se ttin g resin. P la stic coatings may be included. T h is pipe is form ed w ith in a n ex istin g pipe. T herefore, it takes the sh ap e o f a n d fits tig h tly to the existing p ip e ” (A S T M F1216, 1993).

In th e F o ld a n d Form ed P ipe system s, th e cross-sectional a re a of th e new pipe is te m p o ra rily re d u ce d before in stallation. A fte r installation, it is th e n expanded to its original size a n d sh ap e to provide a close fit w ith th e ex istin g pipe. T h is fitting is accom plished b y folding the lining p ip e into a LT-shape, a fte r w hich it is inserted inside th e old p ip e an d reverted by h eat a n d pressure (Li, 1994).

Liners a re o ften installed in host pipes th a t lie below th e w a te r table, and as such th e y a re su b jec te d to ex tern al h y d ro sta tic pressure. T h e e x te rn a l pressure leads to early d efo rm atio n in the liners w hich could u ltim ately le a d to its failing or

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buckling before th e ir expected service life is achieved. Insufficient u n d erstan d in g of th is buckling phenom enon is a lim iting facto r in th e C IP P liner industry. To design a dependable liner, one needs to have a good knowledge about th e long-term buck ling behavior o f a p ip e under ex tern al p ressu re a n d models to pred ict such behavior. M any studies exist on predicting sh o rt-te rm buckling behavior of a free o r confined pipe (Tim oshenko a n d Gere, 1961; A ggarval a n d Cooper, 1984: Glock, 1977: Guice a n d Li, 1994: O m ara et al., 1996: F alter, 1996: W elch, 1989, Boot a n d Welch, 1996: Boot, 1998: B oot a n d Javadi, 1998; an d H all a n d Zhu, 2000).

O n th e o th e r hand, th e long-term buckling behavior of a pipe liner u nder pres sure has been u n d e r stu d y for a relatively sh o rt perio d of tim e. O nly few m odels exist for p red ictin g th e long-tim e behavior (W elsh, 1989; Guice et al.. 1994; S trau g h n

et al., 1995; B oot a n d Welch, 1996; M oore, 1998; a n d Zhao, 1999). T h ere is a need

for m odels to p re d ic t th e tim e until failure of a p ip e liner system u nder a h ydrostatic pressure load.

E x p erim en ts involving long-term buckling behavior of liners are typically ac celerated l i f e t i m e te stin g procedures. In a n accelerated testing p rocedure a liner is su b jected to a co n sta n t ex tern a l h y d ro static p ressu re an d observed u n til it fails, o r for a certain t i m e , t w hichever occurs first. Liners th a t do not fail a t tim e t are deem ed

censored observations. W hile a constant pressure is convenient to use in experim ental situations, in re a lity pressure fluctuates u n d er soil conditions over tim e depending on the w ater table. It is desirable then to have accelerated lifetime m odels to predict tim e u n til buckling un d er constant as well as variable external h y d ro static pressure.

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1.2 Objectives

T h e objectives o f th is s tu d y are

1. to exam ine a c c e le rate d lifetim e m odels for c o n s ta n t a n d variable pressure for

the W eibull lifetim e d istrib u tio n an d show how to o b ta in m axim um likelihood estim ates (M LE) o f m o d el p aram eters.

2. to stu d y th ro u g h sim u la tio n th e sta tistic a l p ro p e rtie s of th e M LE as a function

of sam ple size a n d p e rc e n t censoring a n d co m p are re su lts for th e co n stan t a n d variable pressure situ a tio n s.

1.3 Organization of the D issertation

T his d isserta tio n is org an ized as follows:

In ch ap ter 2. a review o f lite ra tu re is presented. In c h a p te r 3, th e m axim um likelihood m e th o d an d th e N ew to n -R ap h so n technique are discussed. In c h a p te r 4, th eo ry a n d sim ulation results for acce le ra ted lifetime te stin g u n d e r c o n sta n t pressure are p re sented. In c h a p ter 5. th e o ry a n d results from sim u la tio n for th e accelerated lifetim e te stin g under variable p re ssu re are discussed. In c h a p te r 6. a conclusion an d fu tu re

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R E L A T E D R E S E A R C H

2.1 R eliability

T h e S ta tistic a l analysis of lifetim e d a ta is of in terest to sta tistic ia n s, engineers, physicians, an d researchers in biological sciences.

A pplications of lifetim e d istrib u tio n s range from in v estig atio n s into the en d u ran ce o f item s to research involving h u m an diseases. L ifetim e d istrib u tio n m ethod ology h as its m ost frequent a p p lic a tio n in engineering, a n d biom edical sciences.

L et T be a nonnegative ra n d o m variable re p resen tin g th e failure tim e of an in d iv id u al from some p o p u latio n . Let /(£ ) denote th e p ro b a b ility density function (p .d .f) o f T an d let th e d istrib u tio n function be

T h e survivor function is defined as th e p ro b ab ility th a t T is a t least as great as £; th a t is,

In cases involving lifetim es of m an u factu red item s, S ( t ) is referred to as th e (2.1)

(2.2)

5

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reliability fu n ctio n . S(t) is a m onotone d ecreasing continuous fu n ctio n with. 5 (0 ) = 1 a n d 5(o o ) = limt_).oo S{t) = 0.

F rom Eqs. (2.1) a n d (2.2), we see t h a t th e survivor function o r th e reliability function is given by

S ( t ) = R ( t ) = l - F ( t ) . (2.3)

A n o th e r im p o rta n t concept dealing w ith a lifetim e d istrib u tio n is th e hazard function h(t), defined as

= Hm Pr<f < 1 < t + A t \ T > t )

v A t —>-0 A t

= m

s(ty

(2.4)

T h e h a z a rd function specifies the in sta n ta n e o u s ra te of failure a t tim e T = t. given th a t th e individual will survive u n til tim e T = t .

Now, since f ( t ) = — S'(t), it is seen from Eq. (2.4) th a t

h{t) = —^ ]os S ( t ) . (2.5)

Hence.

lo g S (x )|o — h(x)dx, ( 2 6)

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S(t) = e x p (—

J

h{x)dx) (2.7)

T h e cum ulative h azard fu n ctio n is defined as

H(t) =

f

h ( x ) d x .

o (2.8)

H ence, from Eq. (2.6). we have

S(t) = exp[— H{t)\. (2.9)

Now. since S(oo) = 0. th e n H(oo) = lim ^o o H(t) - oo. T herefore, th e h azard fu n ctio n h(t) for a continuous lifetim e d istrib u tio n has the following properties:

T h ro u g h o u t the literatu re on failure tim e d a ta , num erous p ara m etric m odels are used to analyze problem s related to aging or a failure process. A m ong these m odels, few a re frequently used because of th e ir d em o n strated usefulness in a wide range o f situ a tio n s. T h e exponential an d W eibull m odels, for exam ple, are often em ployed. T hese distrib u tio n s a dm it closed-form expressions for ta il area probabilities a n d hence sim ple form ulas for survivor a n d h a zard functions. Also, th e lognorm al an d g am m a d istrib u tio n s are frequently used despite th e fact th a t th ey are gener ally less convenient com putationally. A n o th e r im p o rta n t d istrib u tio n is the extrem e

1. h(t) > 0

2. J£° h( t)dt = oo

2.2 Some Im portant Lifetime M odels

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value distribution, which, describes certain ex trem e p h en o m en a like electrical s tre n g th of m aterials a n d c e rta in ty p es o f lifetime d a ta (K alb feisah a n d Prentice. 1980: Law less. 1982: Nelson, 1990: a n d C o llett 1999).

2.2.1 The E xponential D istribution

Suppose t h a t th e h a z a rd function is c o n s ta n t. T h en , the hazaxd fu n ctio n c a n be w ritten as

h{t) = A for 0 < t < oo. £ 9

T h e p a ra m e te r A is a positive co n stan t e s tim a te d by fittin g the m odel to ob served d a ta . From E q. (2.7). we have th a t

S{t) = exp(—y Xdx) = e -At. ^ n )

Hence, th e p ro b a b ility density function (p .d .f) o f survival times is given by

f ( t ) = Xe~xt for 0 < t < oo. ^

E q u a tio n (2.1 2) rep resen ts the p ro b ab ility d e n sity function of a ran d o m vari

able T th a t has a n e x p o n e n tia l d istribution. It c an b e easily verified th a t th e m ean is ^ a n d the variance is p- (Ross, 1997).

A very im p o rta n t ch aracteristic of th e e x p o n e n tia l d istrib u tio n is its lack of m em ory. T his lack of m em o ry p ro p erty can b e seen as follows:

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For £2 > t i > 0,

P [ T > t 2 \ T > t l ] = ^

= e _A(t2_£l). (2-13)

Hence, th e survival p robability depends o n th e interval (£2 — £1) an d is in

dependent of w h a t h a p p e n e d before tim e £1.

2.2.2 The W eibull D istribution

T h e a ssu m p tio n o f a constant h azard fu n ctio n is ra th e r restrictive. A m ore general form of a h a z a rd function is such th a t

h(t) = Ayt7 - 1 for £ > 0. (2-14)

Here, th e h a z a rd function depends on two p a ra m e te rs, A an d 7, bo th g reater

th a n zero. In th e sp ecial case w hen 7 = 1, th e h a z a rd function takes the co n stan t

value A, an d hence th e survival tim es have th e ex p o n en tial d istribution. If 7 7^ 1, th e

h az a rd function increases or decreases m onotonically. T h e p aram eter 7 is know n as

th e shape p ara m e ter. Hence, the shape of th e h a z a rd function depends on 7 . T h e

p ara m e ter A is know n as th e scale p aram eter. F rom Eq. (2.7), it is seen th a t th e survivor function is given by

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S ( t ) = e x p (— f Ayx7 l dx)

J 0

= e x p (—A£7). (2.15)

Hence, th e probability density function is given by

/(£ ) - Ay£7 r e x p (—A£7) for t > 0. _(2.16)

T h e fu n ctio n in Eq. (2.16) is th e d en sity of a random variable th a t has the W eibull d istrib u tio n w ith shape p a ra m e te r 7 an d scale p a ra m ete r A.

2.2.3 E xtrem e Value Distribution

T h is d istrib u tio n is also know n as th e G um bel d istrib u tio n (G um bel. 1958). T h e p .d .f for th e extrem e value d istrib u tio n is given by

w here b > 0 and — 0 0 < u < 0 0 are param eters. It can b e seen th a t if T

is a ra n d o m variable w ith a W eibull d istrib u tio n , then X = lo g T has a n extrem e value d istrib u tio n w ith b = A a n d u = — log A7. Also, the survivor function of the extrem e value distrib u tio n is given by

X — XL

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2.2.4 T he Gam m a D istribution

A n o th er d istrib u tio n for survival d a ta is th e gam m a d istrib u tio n w hich is defined for a ra n d o m variable th a t takes positive values. T h e h a z a rd function for th e gam m a d istrib u tio n is given by

^k-^k—

lg—

At

h{t)

= r(fc)(i-rA

t(&)):

(2'19)

w here F(k) is a g a m m a function a n d Fxt(k) is th e incom plete g am m a function a n d is given by

FAt(/c) = f (fc) lo XdX' (2'20)

T h e gam m a d istrib u tio n has a p ro b ab ility d en sity function of th e form

. A(A£)*- 1e - At r .

f ( t ) = ' for t > 0, (2.2 1)

w here A > 0 is called th e scale p a ra m e te r, a n d k > 0 is th e index o r shape p aram eter.

For k = 1, th e g am m a d istrib u tio n reduces to th e ex p o n en tial d istrib u tio n .

In te g ratin g Eq. (2.21). one o b tain s th e su rv iv al fu n ctio n for the g am m a as

S(t) = l - r xt(k), (2.2 2)

w here F \ t (k) is given in Eq. (2.20).

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2.2.5 T he Lognormal D istribution

A n o th e r im p o rta n t d istrib u tio n for lifetim e d a ta is th e lognorm al d istrib u tion. T h is d istrib u tio n h as b een widely used in e n g in eerin g a n d biom edical science.

A ra n d o m v ariab le T is said to have a lo g n o rm al d istrib u tio n w ith param eters

11 a n d cr2 if y = lo g T has a n o rm al d istrib u tio n w ith m e a n (J, a n d variance cr2 . T h e

p ro b ab ility d en sity fu n ctio n o f Y is given by

f ( y ) = exP ~

^)21

for ~

00

< y < °°- (2.2.3)

From Eq. (2.23), it is seen th a t the p .d .f for T is given by

/ W = ^ 7 U exP i - 5 5 2 (lo£ ( - f ‘)2l; s > 0 - (2'24)

T h e survivor a n d h a z a rd functions for th e lo g n o rm al d istrib u tio n involve th e s ta n d a rd norm al d istrib u tio n fu n ctio n

<F(x) = f e~^~ du. (2-25)

J— oo v 2tt

T h e lognorm al survival fu n ctio n is given by

a n d the h a z a rd fu n ctio n is given by

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2.3 A ccelerated Life T esting

T h e lifetime of a high-reliability device is usually v ery long. Thus, it is pro hibitive tim e-w ise to te st such a device un d er no rm al conditions. A ccelerated lifetim e te stin g (ALT) is a m eth o d th a t exposes devices to higher stre ss levels th a n th ey expect to receive under norm al use to induce early failures, a n d o b ta in inform ation quickly o n th e ir lifetim e d istrib u tio n .

Schabe an d V iertl (1995) p resen ted an axio m atic ap p ro ach to accelerated life tim e testin g . Clark, G arganese, an d Swarz (1997) p resen ted an ap p ro ach to designing accelerated-lifetim e te stin g experim ents. T h e basis of th e ir ap p ro ach is a destructive evalu atio n perform ed on a sm all num ber of te st item s to m easure th e design lim its. As such, e n v ir o n m e n ta l stress levels can be tailo red to achieve th e objectives of th e accelerated lifetime test.

A ccelerated te st conditions involve-higher-than u su al load or stress (such as te m p e ra tu re , voltage, pressure, etc.. or some com b in atio n o f them ) on the device. A ccelerated lifetime te stin g is a com m on m eth o d for assessing th e reliability of a n ite m because, for p rac tica l reasons, lifetim e testin g is p erfo rm ed in a relatively sh o rt tim e interval. Two ty p es of accelerated testin g exist in th e lite ra tu re , constant-stress te stin g a n d step-stress testing.

2.3.1 Constajit-Stress A ccelerated Test

One w ay of ap plying stress to a test device is a co n stan t-stress. Each device is assigned only one stress level in a com pletely ra n d o m m anner. Regression m eth

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ods are used to estim ate lifetim e u n til failure a t a given design s tre s s. A functional relationship betw een constant-strress a n d lifetim e u n til failure is assu m ed . T h e test d a ta are th e n used to estim ate t l i e p aram eters of the d istrib u tio n o f tim e u n til fail ure. E stim ates of th e p a ra m e te rs in th e m odel can be o b ta in e d b y m axim izing the log-likelihood function using th e IN ew ton-R aphson Technique.

A very im p o rtan t problem i n co n stan t-stress te stin g is d e te rm in in g th e num ber of devices to be allocated to each stress. Inferential procedures for th e c o n sta n t stress te st have been given by Nelson ([1980). w hen lifetim e follows a W e ib u ll d istrib u tio n a n d b y Nelson a n d H ahn (1 9 7 2 ), w hen th e lifetim e of a n ite m follows a Lognorm al or a W eibull distrib u tio n . K ie lp ln sk i a n d Nelson (1975) a n d N elson a n d Kielpinski (1976) presented o p t im u m plans a n d th e th eo ry of o p tim u m p lan s in th e case of ALT for estim atin g a sim ple linear re la tio n s h ip betw een stress a n d th e lifetim e of a n item, w hich has a N orm al or Lognorma.1 d istrib u tio n , w hen th e d a ta a re to b e analyzed be fore all test devices fail. T h e ir m o d e l assum es th a t the n o rm a l d is trib u tio n location p a ra m e te r jj. (m ean) is a linear fu n c tio n of the stress a n d th a t th e scale p aram eter cr (sta n d a rd deviation) does n o t d ep en d on stress. Nelson (1975) p re se n te d simple least-squares m ethods for a n a ly z in g accelerated lifetim e te st d a t a w ith th e inverse pow er law model, w hen all test d ev ices are rim to failure. N elson a n d M eeker (1978) p resen ted the th eo ry of m ay im u m likelihood for large-sam ple o p tim u m ALT plans. T h e y showed how th e plans can T e used to estim ate a sim ple lin e a r relatio n sh ip be tw een stress an d p ro d u ct lifetim e in th e case of a W eibull or S m allest E x tre m e Value d istrib u tio n . T h e y assum ed t h a t th e sm allest extrem e-value lo c a tio n p a ra m e te r (i is a linear function of stress an d . th a t th e scale p a ra m e te r is c o n s ta n t. A itk in and

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C aly to n (1980) show ed how regression m odels c an b e fitte d to censored survival d a ta by th e use of th e ex p o n en tial, W eibull, a n d ex trem e value d istrib u tio n s in generalized linear in te ra c tiv e m o d elin g (GLIM ). B ugaighis (1990) p resented resu lts show ing th a t exchange o f censorship ty p es resulted in m in o r re d u c tio n in th e efficiency of various estim ato rs. M e e te r a n d M eeker (1994) e x te n d e d th e m axim um likelihood th eo ry for test p la n n i n g; to a no n co n stan t scale p a ra m e te r cr. T h e y also p re se n te d test plans for a large ran g e o f p ra c tic a l testing situ a tio n s. T h ia g a ra ja h (1995) considered tests o n tim e-censored d a ta for th e equality o f several ex p onential scale p a ra m e te rs in the presence o f unspecified location p ara m e te rs. He derived 3 s ta tistic s for testin g the hom ogeneity of M ( M > 2) exponential scale p a ra m e ters. He also com pared, th ro u g h a sim u latio n stu d y , th e size an d power for th e 3 developed sta tistic s.

2.3.2 S tep -S tress Accelerated Test

A n o th e r w ay of applying stress to a device is a step -stress schem e which allows th e stre ss se ttin g of a device to b e changed a t prespecified tim es or u pon th e occurrence of a fixed n um ber of failures.

S te p -stre ss te s tin g reduces tim e a n d assures th a t failures o ccu r v e ry quickly. A te s t device s ta r ts a t a specified low stress. If th e device does n o t fail in a specified tim e, th e stress o n it is raised and held a t th a t level for a specified tim e. If th e device does n o t fail a t th is stress, its stress is in creased a n d held, a n d th e p ro cess continues in the sam e fash io n u n til all devices fail.

T h e d esig n p ro b le m in step-stress te stin g is to determ ine th e tim e to change stresses, p ro v id e d t h a t a fixed num ber of stress levels has b een selected. T h e choice of

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th e se tim es will determ ine how m any devices fail a t each stress. C onstant-stress a n d step -stress have th e sam e o p tim a lity criterion: i.e., th e y choose the tim es to change stress th a t m inim ize th e variance of some e stim a to r of a p aram eter.

As is th e case w ith constant-stress test, one needs to estim ate th e p aram eters o f th e lifetim e m odel u n d e r step-stress. P a ra m e te r estim ates are th e n used to de term in e w ith in reason th e lifetim e of a n item a t a co n sta n t design stress. As such, one needs a m odel th a t relate s th e lifetim e d istrib u tio n u n d er constant-stress to th a t u n d e r step-stress.

Nelson (1980) p resen ted sta tistic a l m odels a n d m eth o d s for analyzing accel e ra te d lifetime te st d a ta from step-stress tests. He used th e m axim um likelihood e stim a tio n technique to e stim a te the p aram eters of such models. He applied his m e th o d to th e W eibull d istrib u tio n an d the Inverse Pow er Law. M iller and Nelson (1983) o b tain ed o p tim u m p lan s for two stresses w here all devices are ru n to failure. T h e y o b tained a n o p tim al stress test th a t m inim izes th e asym ptotic variance of the m axim um likelihood e stim a te (M LE) of m ean lifetim e for a n exponential model w here th e m ean lifetim e is a log-linear function of stress. Bai, K im , an d Lee (1989) derived a n o p tim u m sim ple (two stresses) stress ALTs for th e case w here censoring was in volved. T h ey o b tain ed an o p tim u m test p lan th a t m inim izes th e asym ptotic variance o f th e M LE of th e m ean lifetim e a t a design stress w ith censored observations. Bai a n d C hun (1991) o b tain ed o p tim u m sim ple step -stress ALT w ith com peting causes of failure. Tyoskin an d K rivolapov (1996) developed a n o n p aram etric m odel for interval estim atio n , based on results from step-stress ALT. T h ey presented, through sim ula tio n , a num erical exam ple to verify their approach.

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O p timum sim ple (two stresses) step-stress ALT p la n s have some lim itations because th e y depend on th e assu m p tio n of a lin ear relatio n sh ip betw een stress a n d tim e-u n til failure. K ham is a n d Higgins (1996) p resen ted 3-step stress plans for ALT. T h e y derived an o p tim u m q u a d ra tic p la n an d ev alu ated a 3-step stress test p lan (th e com pound linear plan) in lieu of the optim um sim ple-stress p lan . Kham is (1997) o b tain ed o p timum M -step, step -stress designs w ith k-stress variables. Xiong an d Mil- liken (1999) stu d ied s ta tis tic a l m odels in step stress ALT w h en th e stress-change tim es are random . T h e y presen ted th e m arginal lifetime d istrib u tio n of a device under a step-stress test p la n w hen th e stress-change tim es are ran d o m variables. They also, p resented a n o ptim um ALT for sim ple step-stress (two stresses) w hen the lifetime under an y constant-stress follows th e exponential d istrib u tio n .

T h e m ain goal of using step-stress testing is to avoid censoring. One knows th a t th e device m ay have high reliability and not fail w ith in a reasonable tim e. By increasing the stress on th e device th e problem of censoring could be avoided.

2.3.3 Cumulative Exposure M odel (CEM)

To analyze d a ta from a step-stress scheme, one needs a m odel th a t relates the lifetim e d istrib u tio n of th e step-stress to th a t of the co n stan t-stress. One such m odel is th e C um ulative E xp o su re M odel (CEM ) b y N elson (1980).

Suppose th ere are n increasing levels of stresses aq < x<i < < • • • < xn. Let

Fj(f) denote the failure d istrib u tio n u nder xz w ith a co n sta n t-stress testing. Let tz be the tim e it takes to change a stress from Xi to xi+1; i = 1, 2, • • • ,n — 1. Then, the

CEM is given by

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F (£) = Fi(t) F2(t — ti 4- s x) Fz{t — to -F s 2) if 0 < £ < £x if t i < t < t2 if t 2 < t < £3 Fn[t tn —1 ~f~ s n—x) if tn—1 ^ £ <c 0 0. (2.28) w here s 1 is th e solution of F2(s x) = F x(£x). Here, F0(£) = F x(£), 0 < £ < £x. Hence, Fq = F 2[(£ — £x) -F s x] h < t < t2

Also, S2 is th e so lu tio n of F3(.s2) = F2 ( £ 2 ~ ti + s

i)-Hence, F0(£) = F 3[(£ — t 2) + s2], £2 < £ < £3

-If we continue in th is m anner, we see th a t .sx is th e solution of

Fi(Si_i) = 1 [tx—x — U- 2 + S i_2]

and, F0 = Fx[(£ — £*_x) + sx_ x], £i_x < £ < £*

This m odel assum es th e following:

1. T h e rem ain in g lifetim e of a device d ep en d s on

(a) th e c u rre n t cum ulative fraction failed, and (b) th e c u rre n t stress.

2. If held a t th e cu rren t stress, survivors fail according to th e cum ulative distrib utio n for th a t stress, but s ta rtin g a t th e previously acc u m u late d fractio n failed.

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on lifetime.

2.3.4 T he Khamis-Higgins M odel (KHM )

From Eq. (2.16). it is seen th a t th e cum ulative d istrib u tio n function for the W eibull d istrib u tio n is given by

F(w ) = 1 — exp (—Aw''); w > 0. (2.29)

U sing th e tra n sfo rm a tio n t = v f f, it is seen th a t

F{t) = 1 — exp (—At); t > 0. (2.30)

T h e above tra n sfo rm atio n from W to T transform s th e W eibull into a n ex p o n e n tia l d istrib u tio n a n d facilitates m any of th e inferential re su lts. However, such p ro p e rty does n o t carry over to step -stress te stin g for the W eib u ll CEM ; i.e., the above tra n sfo rm atio n does not resu lt in th e ex p o n en tial exposure m odel. To over com e th is difficulty, K ham is an d Higgins (1998) o b ta in e d a new m o d e l for step-stress testin g , th e K ham is-H iggins M odel(K H M ). T h e K H M is based on a tim e transform a tio n of th e exponential. T h e tim e tra n sfo rm a tio n enables th e user to know results for m ultiple-step, m ultiple-stress m odels developed for th e ex p o n en tial step -stress model. T h e K H M is given by

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F ( w ) = 1 - <

ex p (—Ajif/7)

ex p (—A2(tu7 — t j ) — A it j)

if 0 < w < ti

if £x < w < £2

e x p (—A„(u/ 7 - ( J . j ) ---A2(iJ - t j ) - AitJ) if i < w < oo,

(2.31) a n d F (£) = 1 -ex p (—Ax£) e x p (-A 2(£ - £x) - Axq ) if 0 < £ < £x if t \ < t <

e x p (-A n (£ - £ ;_ x) ---A2( q ~ t\) - Ax£*) if t*n_ v < £ < oo.

T h e KHM h azard function is riv en bv

(2.32) h{w) = AX7vf* 1 if 0 < w < £x A2 7u f/ _ 1 if £x < w < t-z An7iu7 - 1 if £n_ x < w < oo. T h e KHM assum es th e following:

1. W i j = T£j, w here Titj follows the ex p onential CEM .

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2. T h e Ai is re la te d to th e stress level xt- by

In (A*) = A0 -I- Ajx*,

(2.34)

w here A0,A 1; a n d 7 a re constants, in d ep en d en t o f tim e a n d stress, a n d are

d eterm in ed from th e te s t d a ta .

3. A ll th e n devices a re in itia lly placed on test a t s tre ss level x x an d ru n u n til tim e

Ti w h en the stress is changed to the stress level x 2. A t stress level x2, testin g

continues u n til tim e t2 w h en stress is changed to th e stress level x3, and so on,

u n til stress level xn . A t stress level xn , te stin g co ntinues until all rem aining devices fail or u n til tim e r c, whichever occurs first, w here t c is the censoring tim e a t stress level x*,.

T h e K H M has a very in te re stin g p ro portional h a z a rd p ro p erty . It is also as flexible as th e W eibull CEM for d a ta fitting, an d its m a th e m a tic a l form makes it easy to o b ta in p a ra m e te r e stim a te s a n d sta n d a rd deviations o f estim ates.