Simultaneously estimating the three Weibull parameters from progressively censored samples

Microelectron.Reliab.,Vol. 33, No. 15, pp. 2217-2224, 1993.

0026-2714/9356.00+.00 © 1993 PergamonPress Ltd

Printed in Great Britain.

SIMULTANEOUSLY ESTIMATING THE THREE WEIBULL PARAMETERS FROM PROGRESSIVELY CENSORED SAMPLES JSUN Y. WONG The University of New Brunswick, P. O. Box 5050, Saint John, New Brunswick, E2L 4L5, Canada

(Receivedfor publication 11 May 1992)

Abstract To e s t i m a t e t h e t h r e e W e i b u l l this

parameters simultaneously,

p a p e r u s e s t h e two common, a l t e r n a t i v e

maximum l i k e l i h o o d

estimation:

maximizing the log

likelihood

solving

approaches t o

one approach o f d i r e c t l y function

and a n o t h e r a p p r o a c h o f

t h e system o f e q u a t i o n s o b t a i n e d from d i f f e r e n t i a t i o n .

Specifically, population,

with one o f

a p r o g r e s s i v e l y c e n s o r e d sample f r o m a W e i b u l l t h e t h r e e e q u a t i o n s from d i f f e r e n t i a t i o n

a two-unknown e x p r e s s i o n f o r expression into

t h e o t h e r unknown.

the log likelihood

gives

Putting this

function helps solve this

e s t i m a t i o n problem. Computer p r o g r a m s and t h e i r e x a m p l e s f r o m an a r t i c l e results

shown i n

outputs for

are presented here.

the numerical The c o m p u t a t i o n a l

t h e s e o u t p u t s a r e compared w i t h

those from the

article. 1. I n t h e method o f maximize t h e

likelihood

approaches.

that

function.

one a p p r o a c h i s

Alternatively,

log

[I,

to

the

e s t i m a t e s can be f o u n d by s o l v i n g t h e

e q u a t i o n s o b t a i n e d by e q u a t i n g t o z e r o a f t e r

differentiation

following

maximum l i k e l i h o o d ,

log likelihood

maximum l i k e l i h o o d

Introduction

p.

51].

T h i s paper uses both o f

Specifically,

this

likelihood

function

each

t h e s e two

p a p e r a t t e m p t s t o maximize t h e w i t h t h e h e l p o f ~he c o n d i t i o n

v - Q o / n , which comes 4tom d i f f e r e n t i a t i o n

lnL=lnC+nlnW+(w- l)~_,/ln(Xl - u ) - n l n v

[2,

p.

98]:

-(l/o)Qo

Address correspondence to: 1418 15th Avenue, San Francisco, CA 94122, U.S.A. 2217

2218

J . Y . Wono where

Qa = E / (x/ - u) wJn + . ~ , r l ( T i - u) w-h

For example, t h i s of

e q u a t i o n v=Qo/n i s

t h e f o l l o w i n g computer program.

used i n

the next l i n e ,

likelihood It

line

Wong [ 3 ]

approach i s

then

538, which c o n t a i n s t h e l o g

have been g r e a t l y guided by t h e ÷ o l l o w i n g

number o f

and Lieberman [ 4 ,

t o a p p l y an a l g o r t i t h m i c

s t o p when i t

finds a local

best of

t i m e s from a v a r i e t y o f

these local

maxima i s

2. Example

1.

This i s

spans.

t h e f o l l o w i n g Type I

"A common

search p r o c e d u r e t h a t w i l l

initial

local

trial

it

t h e n chosen f o r

a

s o l u t i o n s in

maxima as p o s s i b l e .

Wingo's

[2,

p.

The

implementation."

99]

second example.

and t h u s p r o v i d e d c o m p l e t e l y

The r e m a i n i n g 17 i t e m s were s u b j e c t e d t o

c e n s o r i n g scheme:

T4=121 , T5=150;

p.99].

560]:

The Examples from an A r t i c l e

" T h i r t y - t h r e e items f a i l e d determined l i f e

p.

maximum and then t o r e s t a r t

o r d e r t o ~ind as many d i s t i n c t

37 95 117

v is

477

should be p o i n t e d o u t a g a i n t h a t t h e computer programs

q u o t a t i o n from H i l l i e r

failed

This v a l u e f o r

line

function.

h e r e and i n

=99,

expressed i n

k=5~

T1=70, T2=80 , T 3

r1=4 , r2=5 , r3=4 , r4=3 , r 5 = I , "

The hours t o f a i l u r e

of

the t h i r t y - t h r e e

Wingo [ 2 ,

items t h a t

are p r e s e n t e d b e l o w .

55 97 120

64 98 120

72 100 120

74 I01 122

87 102 124

88 102 126

The computer program l i s t e d

89 105 130

91 105 135

92 107 138

94 113 182

below has been o b t a i n e d by using

t h e d a t a p r e s e n t e d above and by using t h e computer program i n Wong [ 3 ] .

Unlike the corresponding l i n e s

[3],

305,

lines

remark l i n e s , reason f o r

307,

308,

as shown i n

309,

310 o f

in

the e a r l i e r

t h e p r e s e n t paper a r e j u s t

t h e computer program l i s t e d

the incapacitation is

paper

t h a t t h e problem o f

below.

The

i n t e r e s t has

Weibull parameters

been r e d u c e d f r o m t h r e e by u s i n g X(1)

to

value. the

v=Qo/n. keep If

value

its

both of

PI

unknowns t o

Therefore, current X(1) of

value,

and

line

for

X(2)

2219

two unknowns,

e x a m p l e , when l i n e X(2)

were to

538 c o u l d

not

should

not

keep t h e i r be

279 IF RND(X)<=.1667 THEN 280 ELSE 283 280 X(K)=(1/IO)*FIX(IO~i(L(K)+RND(X)*U(K))+.5) 281GOTO 300 283 IF RND(X)<=.2 THEN 284 ELSE 287 284 X(K)=(I/IOO)*FIX(IOO*(L(K)+RND(X)*U(K))+.5) 285 SOTO 300 287 IF RND(X)<=.25 THEN 288 ELSE 290 288 X(K)=(1/IOOO)*FIX(IOOO*(L(K)+RND(X)*U(K))+.5) 289 SOTO 300 290 IF RND(X)<=.3333 THEN 292 ELSE 294 292 X(K)=(1/IOOOO)*FIX(IOOOO*(L(K)+RND(X)*U(K))+.5) 293 SOTO 300 294 IF RND(X)<=.5 THEN 296 ELSE 298 296 X(K)=(1/IOOOOO!)*FIX(IOOOOO!*(L(K)+RND(X)*U(K))+.5) 297 GOTO 300 298 X(K)=(L(K)+RND(X)*U(K)) 300 NEXT K 304 X(JJ)-A(aa) 3 0 5 REM K L P S = F I X ( I + K K L L * R N D ( X ) )

371

REM FOR KLAt=I TO KLPS REM KLA2=FIX(I+2*RND(X)) REM X(KLA2)=A(KLA2) REM NEXT KLA1 80TO 371 IF RND(X)<=.5 THEN 353 ELSE 355 x(Ja)=B(da) SOTO 371 X(Jg)=N(aa) REM

381Y(1)=37:Y(2)=551Y(3)=64:Y(4)=72:Y(5)=74 382 Y(6)=87:Y(7)=88:Y(8)=89=Y(9)=91:Y(10)=92 383 Y(11)=94:Y(12)=95=Y(13)=97:Y(14)=98:Y(15)=100 384 Y(Ib)=101:Y(17)=lO2:Y(18)-IO2:Y(19)=105:Y(20)=105 385 Y(21>=107:Y(22)=113:Y(23)=I17:Y(24)=120:Y(25)=120 386 Y(2&)=120:Y(27>=122:Y(28)=I24:Y(29)=126:Y(30)=130 387 Y(31)-135:Y(32)=138:Y(33)=182 430 PSUMI-O 431 FOR I I = l TO 33 433 IF ( Y ( I I ) - X ( 1 ) ) <.0000001 THEN 670 ELSE 434 434 PIDUM=LOG(Y(II)-X(1)) 435 PSUMI=PSUMI+PIDUM 437 NEXT I I 440 PSUM2=O

and

X(2),

instructs

its

current

current

values,

improved.

2&5 I F A ( K ) + ( N ( K ) - B ( K ) ) / H ( K ) A j )' N ( K ) THEN 2 6 6 ELSE 2 6 8 266 U(K)=N(K)-L(K) 2 6 7 GOTO 2 7 2 268 U(K)=A(K)+(N(K)-B(K))/H(K)^a-L(K) 2 7 2 I F YN<5 THEN 2 9 8 ELSE 2 7 4 2 7 4 I F R N D ( X ) < = . 5 THEN 2 9 8 ELSE 2 7 5 275 IF RND(X)<=.1429 THEN 2 7 7 ELSE 2 7 9 277 X(K)=(1/1)*FIX(I*(L(K)+RND(X)*U(K))+.5) 278 GOTO 3 0 0

304

keep

I DEFDBL A,X,P,M,L,U,Z,T,B,N,H,Y 2 DEFINT 1,3,K 5 DIM B ( 1 9 ) , N ( 1 9 ) , A ( 1 9 ) , H ( I ? ) , L ( 1 9 ) , U ( 1 9 ) , X ( I 1 1 1 ) , Y ( 1 1 1 ) 10 X=I 11 FOR 3JaJ=1 TO 9999 14 Z=RND(X) 16 M=-1.701411834&O4692D+38 18 YN=RND(X) 25 B(1)=O:B(2)=O 37 N(1)t36.9999?:N(2)=IO 50 A(1)=O+RND(X)*36.99999:A(2)=O+RND(X)*10 81H(1)=3:H(2)=3 88 FOR aoa=1 TO 1 100 FOR J-1 TO 10 101KKLL=FIX(I+2*RND(X)) 102 FOR gO=O TO 3 228 FOR I=1 TO 10 229 FOR K=I TO 2 230 IF A ( K ) - ( N ( K ) - B ( K ) ) / H ( K ) ^ 3 < B ( K ) THEN 250 ELSE 260 250 L(K)=B(K) 255 80TO 265 260 L ( K ) ' A ( K ) - ( N ( K ) - B ( K ) ) / H ( K ) ~ 3

307 308 309 310 342 352 353 354 355

X(1)

J . Y . WONG

2220

441 FOR IJ-1 TO 33 444 P2DUM=(Y(IO)-X(1))"X(2) 445 PSUM2=PSUM2+P2DUM 447 NEXT la 455 PSUM3=4*(70-(X1))"X(2)+5*(80-X(1))~'X(2)+4*(99-X(1))"X(2)+3*(121-X(1))^X(2)+1 *(150-X(1))^X(2) 477 X(3)=(PSUM2+PSUM3)/(33) 538 P1=O+33*LOG(X(2))+(X(2)-I)*(PSUM1)-33*LOG(X(3))-(I/X(3))*(PSUM2+PSUM3) 544 P=PI 551 IF P-9999999! THEN 702 ELSE 1001 702 NEXT JJ 706 NEXT a 711LPRINT J~JJ,33J,M,PPI,A(1),A(2),A(3) 712 NEXT JJJ 1001 NEXT JJ3J

Running t h i s for

the

earlier

computer

paper

[3]

p r o g r a m on t h e c o m p u t e r s y s t e m used yields

the

following

t h e computer p r o g r a m ' s complete o u t p u t

for

output,

JJJJ=l

which

is

through

JJJJ=15.

1 I 2.050291705114057D-06 2 1 3.524454027284455D-05 3 1 1.329298946806308 4 1 9.570125288197329D-O& 5 I 3.732632541410004 6 1 1.01628340748579&D-05 7 I 1.425898070526219D-05 8 1 5.567388641003852D-06 9 I 1.159815406191492D-05 10 1 3.74121802859597D-07 11 I 2.776736145906438D-05 12 1 5.841272661209895 13 1 3.&17579022680477D-06 14 I 2.314818117~192&&D-05 15 1 1.9667925187775120-05

The b e s t JJJa=lO,

-165.1113595270728 4.351258997796127 -165.1113598067854 4.351242308686784 -165.1239763528754 4.300374561340318 -1&5.1113595899435 4.351229112054059 -165.1544200736304 4.206818337759002 -165.111359595418 4.351219319220474 -165.1113596670459 4.351092147778061 -165.1113595883346 4.351372523349119 -165.11135960696 4.351240151826597 -165.1113595123808 4.351250080665094 -165.1113597437211 4.35124656919086 -165.190260947837 4.12884774518443 -165.111~595396108 4.351230841640509 -165.111~597045414 4.351234641916035 -165.1113596751269 4.351237109587112

solution

p r e s e n t e d above i s

(3.74121802859597D-7

1174&OSO7&.1738&7) w i t h

the

the solution

at

4.351250080665094

log

likelihood

This

log

o~ - 1 6 5 . 4 6 2 0 9 0 8 2 9 4 5 0 8 o b t a i n e d f r o m t h e

solution

presented

3.7596935 p.99] to

our

value

to

Table

3 of

41896335.59999053).

obtain

X(2)

in

likelihood

value of

-165.1113595123808. likelihood

log

-165.1113595270728 1174655579.355544 -165.111~5980&7854 1174559562.420165 -165.1239763528754 877168827.1895478 -165.1113595899435 1174485784.982288 -165.1544200736304 514~0172.0099406 -165.111359595418 1174430229.985157 -165.1113596670459 1173709170.649766 -165.1113595883346 1175299469.800135 -165.11135960696 1174548305.883455 -165.1113595123808 1174605076.173867 -165.1113597437211 1174584032.393056 -165.190260947837 328966910.1432168 -165.111~595396108 1174495838.355542 -165.1113597045414 1174516583.553323 -165.111~596751269 1174530720.83755

and

this X(3),

Wingo

value is

[2,

p.99],

bigger

than the

(14.451684

We used eW=v f r o m e ~ U i / w [ 2 ,

4 1 8 9 6 3 3 5 . 5 9 9 9 9 0 5 3 , where w and v c o r r e s p o n d respectively.

The b i g g e r

log

likelihood

2221

Weibull ~mme~rs v a l u e means t h a t produced [5, In

p.

the 99]

addition,

former the

A shorter this

paper

shorter

is

99], 642].

which

2.

15

6,5 92 124

This

The o t h e r

20 68 95 126

shown

in

to

the

values bigger the output

candidate

only

following are

the

other

table

that

following

"the

for

than

above. the

problem of

includes

example of

from Hatter

the

three

Wingo

[2,

and Moore [ 6 ,

p. p.

Type 1 c e n s o r i n g s were k=3;

T1=30 ,

T2=110,

T3

r2=3 ,

r3=2,"

Wingo

27 68 I00 127

42

42

43

44

48

64

6,5

71 102 134

74 102 149

75 112 152

75 113 153

76 116 161

77 117 168

78 124 205

the

e x a m p l e and i t s

p.99].

preceding program,

output

for

JJJJ=l

a computer program f o r

through

JJJJ=15 a r e

presented below. 1DEFDBL A,X,P,M,L,U,Z~T,B,N,H,Y 2 DEFINT I~J,K 5 DIM B(19),N(19),A(19)~H(19),L(19),U(19),X(1111),Y(111) 10 X-1 11 FOR JJJJ=l TO 9999 14 Z=RND(X) 1& M=-1.701411834&O4692D+38 18 YN=RND(X) 25 B(1)=O:B(2)=O 37 N(1)=14.99999:N(2)=10 50 A(1)=O+RND(X)*14.99999:A(2)=O+RND(X)*10 81H(1)-3:H(2)=3 88 FOR JJJ=! TO I 100 FOR J=l TO 10 101 KKLL=FIX(I+2*RND(X)) 102 FOR JO=O TO 3 228 FOR I=1 TO 10 229 FOR K=I TO 2 230 IF A(K)-(N(K}-B(K))/H(K}~'J < B ( K ) THEN 250 ELSE 260 250 L(K)=B(K) ~ 5 GOTO 265 260 L(K)=A(K)-(N(K}-B(K))/H(K)~J 265 IF A(K)+(N(K)-B(K))/H(K)^J > N(K) THEN 266 ELSE 268 2b& UtK)=N(K)-LtK) 267 GOTO 272

268 272 274 275

have

solutions.

scheme: [2,

to

solution.

t h e a p p e n d i x , which

three

is

data

Modeled a f t e r this

in

first

uses t h e

r 1=2~

latter

likelihood

also

presented

made a c c o r d i n g =165!

log

more l i k e l y

computer program t a i l o r - m a d e

program's

E>:ample

is

data than the

other

-1&5.4620908294508 are

solution

U(K)=A(K}+(N(K)-B(K))/H(K)^J-L(K) IF YN<5 THEN 298 ELSE 274 IF RND(X)<-.5 THEN 298 ELSE 275 IF RND(X)<=.1429 THEN 277 ELSE 279 277 X(K)=(1/1)*FIX(I*(L(K)+RNDtX)*U(K))+.5) 278 GOTO 300 279 IF RND(X)<-.I&b7 THEN 280 ELSE 283 280 X(K}=(1/IO)*FIX(IO*(L(K)+RNDtX)*U(K))+.5) 281GOTO 300 283 IF RND(X)<=.2 THEN 284 ELSE 287

2222

J.Y. Woso

284 X ( K ) = ( 1 / I O O ) * F I X ( I O O * ( L ( K ) + R N D ( X ) * U ( K ) ) + . 5 ) 285 80T0 300 287 IF RND(X)<=.25 THEN 288 ELSE 290 288 X ( K ) = ( 1 / I O O O g * F I X ( I O O O * ( L ( K ) + R N D ( X ) * U ( K ) ) + . 5 } 289 GOTO 300 290 IF RND(X)<=.3333 THEN 292 ELSE 294 292 X ( K ) = ( 1 / I O O O O ) * F I X ( I O O O O * ( L ( K ) + R N D ( X ) * U ( K ) ) + . 5 ) 293 SOTO 300 294 IF RND(X)<=.5 THEN 296 ELSE 298 296 X(K)=(1/IOOOOO!)*FIX(IOOOOO!*(L(K)+RND(X)*U(K))+.5) 297 80T0 300 298 X ( K ) = ( L ( K ) + R N D ( X ) * U ( K ) ) 300 NEXT K 304 X(JO)=A(JJ) 305 REM KLPS=FIX(I+KKLL*RND(X)) 307 REM FOR KLAI=I TO KLPS 308 REM KLA2=FIX(I+2*RND(X)) 309 REM X(KLA2)=A(KLA2) 310 REM NEXT KLA1 342 GOTO 371 352 IF RND(X)<=.5 THEN 353 ELSE 355 353 X(JJ)=B(J3) 354 SOTO 371 355 X ( J J ) = N ( J J ) 371 REM 381 Y ( 1 ) = 1 5 | Y ( 2 ) = 2 0 : Y ( 3 ) = 2 7 : Y ( 4 ) = 4 2 z Y ( 5 ) = 4 2 382 Y ( 6 ) = 4 3 z Y ( 7 ) = 4 4 : Y ( 8 ) = 4 6 : Y ( 9 ) = 6 4 : Y ( 1 0 ) = 6 5 383 Y ( 1 1 ) = 6 5 : Y ( 1 2 ) = b 8 : Y ( 1 3 ) = 6 8 : Y ( 1 4 ) = 7 1 : Y ( 1 5 ) = 7 4 384 Y ( 1 6 ) = 7 5 : Y ( 1 7 ) = 7 5 : Y ( 1 8 ) = 7 6 : Y ( 1 9 ) = 7 7 : Y ( 2 0 ) = 7 8 385 Y ( 2 1 ) = 9 2 : Y ( 2 2 ) = ? 5 : Y ( 2 3 ) = l O O : Y ( 2 4 ) = 1 0 2 : Y ( 2 5 ) = 1 0 2 386 Y ( 2 6 ) = 1 1 2 : Y ( 2 7 ) = 1 1 3 : Y ( 2 8 ) = 1 1 6 : Y ( 2 9 ) - 1 1 7 : Y ( 3 0 ) = 1 2 4 387 Y ( 3 1 ) = 1 2 4 : Y ( 3 2 ) = 1 2 6 : Y ( 3 3 ) = I 2 7 : Y ( 3 4 ) = 1 3 4 z Y ( 3 5 ) = 1 4 9 388 Y ( 3 6 ) = 1 5 2 : Y ( 3 7 ) = 1 5 3 : Y ( 3 8 ) = 1 6 1 z Y ( 3 9 ) = I 6 8 : Y ( 4 0 ) = 2 0 5 430 PSUMI=O 431 FOR I I = l TO 40 433 IF ( Y ( I I ) - X ( 1 ) ) <.0000001 THEN 670 ELSE 434 43~ P I D U M = L O G ( Y ( I I ) - X ( 1 ) ) 435 PSUHI=PSUMI+PIDUM 437 NEXT I I 440 PSUM2=O 441 FOR I J = l TO 40 444 P2DUM=(Y(IJ)-X(1))'~X(2) 4 4 5 PSUM2=PSUM2+P2DUM 447 NEXT I J 455 P S U M 3 = 2 * ( 3 0 - ( X 1 ) ) ~ X ( 2 ) + 3 * ( I 1 0 - X ( 1 ) ) ~ X ( 2 ) + 2 * ( 1 6 5 - X ( 1 ) ) " X ( 2 ) 477 X(3)=(PSUM2+PSUM3)/(40) 538 P I = O + 4 0 * L O G ( X ( 2 ) ) + ( X ( 2 ) - i ) * ( P S U M I ) - 4 0 * L O G ( X ( 3 ) ) - ( 1 / X ( 3 ) ) * ( P S U M 2 + P S U M 3 ) 544 P=PI 551 IF P-9999999! THEN 702 ELSE 1001 702 NEXT JJ 706 NEXT J 711LPRINT J J J J , J J J , M , P P 1 , A ( 1 ) , A ( 2 ) , A ( 3 ) 712 NEXT J J J 1001 NEXT J J J J

1 1 2.357913091620224 2 1 2.699096340758992 3 1 4.863192394855919 4 1 3;40884975818358 5 1 2.201240235859545 6 1 2.131121495518566

7 1 2.083208528841196 8 1 1.906848965665504 9 1 2.100733268033991 10 1 2.922066179195487 11 1 2.478940012239667 12 1 2.530416552527102

-214.4484374097897 2.144978800524872 -214.4490189087053 2.13516540975579 -214.466384802812 2.069063251649742 -214.4519351406668 2.11365492714603 -214.4483366890336 2.149552715383546 -214.448324321151 2.151607312200521 -214.4483273243422 2.152979499188906 -214.4484170375479 2.15831378327574 -214.4483251466083 2.152499582417855 -214.4496773132214 2.128331831663079 -214.4485858409108 2.141467603176874 -214.4486679332387 2.139929195276064

--214.4484374097897. 24652.9~718071963 -214.4490189087053 23367.74120548693 -214.466384802812 Ib348.b?726334849 -214.4519351406668 20797.38863779457 -214.4483366890336 25275.11574199767 -214.448324321151 25559.632554248&3 -214.4483273243422 25751.96555496548 -214.4484170375479 26508.88415460595 -214.4483251466083 25684.17153190432 -214.4496773132214 22519.08064038417 -214.4485858409108 24185.52?95624731 -214.4486679332387 23984.25590097

Weibull parameters 13 I 2.124376730918297 14 1 2.618310159879766 15 1 3.875097994348093

-214.4485241856193 2.151827520473557 -214.4488348680192 2.137372654828837 -214.455232387785 2.099355798578717

Among t h e c a n d i d a t e s o l u t i o n s solution

is

a t JJJJ=13,

2.151827520473557 value of

(4.102572 p.

solution the latter

value is

bigger

p.

99],

which i s

We used ~W=v f r o m

~ ~ 0I/w

18318.509765625~ where w and v

and X(3)~ r e s p e c t i v e l y .

The b i g g e r l o g

means t h a t

t o have p r o d u c e d [ 6 ,

the former

p.99] the data than

solution.

In a d d i t i o n ,

other log

likelihood

- 2 1 4 . 4 5 7 3 3 5 1 2 6 7 3 9 9 a r e a l s o shown i n 3.

the listed

v a l u e s b i g g e r than

t h e o u t p u t p r e s e n t e d above.

Conclusion

The c o m p u t a t i o n a l r e s u l t s that

likelihood

Wingo [ 2 ,

the former solution

more l i k e l y

the log likelihood

-214.4573351267399 o b t a i n e d from

Table 2 of

to obtain this

value of

is

(2.124376730918297

18318.509765625}.

c o r r e s p o n d t o o u r X(2) likelihood

p r e s e n t e d above, t h e b e s t

This log

value of

presented in 2.090069

99]

-214.4483241836193 25589.95824723053 -214.4488348680192 23652.50920500331 -214.455232387785 19251.05726851394

25589.95824723053) w i t h

than the log likelihood

[2,

which i s

-214.4483241836193.

the solution

2223

presented in

this

paper suggest

computer p r o g r a m s can be u s e f u l as models t o

e s t i m a t e f r o m a p r o g r e s s i v e l y c e n s o r e d sample t h e t h r e e W e i b u l l parameters simultaneously. likelihood

function

unknown l i k e l i h o o d function

is

The m a x i m i z a t i o n o f

facilitated

function

the log

by t r a n s f o r m i n g t h e t h r e e -

t o t h e two-unknown l i k e l i h o o d

through substitution.

References 1.

M. J.

C r o w d e r , A.

Analysis of 2.

Reliability

D. R. Wingo,

C. Kimber, and R. L.

Smith,

D a t a , Chapman and H a l l ,

Statistical

London ( 1 9 9 1 ) .

" S o l u t i o n o f t h e Three-Parameter Weibull

E q u a t i o n s by C o n s t r a i n e d M o d i f i e d Q u a s i l i n e a r i z a t i e n ( P r o g r e s s i v e l y Censored S a m p l e s ) " , 22,

No. 2 ,

3.

J.

Y.

pp.

F.

S.

Vol.

R-

9 6 - 1 0 2 (June 1973).

Wong, "A N o t e on S o l v i n g a System R e l i a b i l i t y

Microelectron~ Reliab. 4.

IEEE T r a n s . R e l i a b . ,

Hillier

(to appear).

and G. L.

L i e b e r m a n , In_fir e d u c t i o n t o

M a t h e m a t i c a l Proqramminq, M c 8 r a w - H i l l , New York

(1990).

Problem",

2224

J . Y . WONG

5.

B.

Basic

J.

Bickel

Ideas

and

and K .

A.

Selected

Doksum,

Topics,

Mathematical

Holden-Day,

Statistics:

San

Francisco,

(1977). b.

H.

o÷

the

L.

Hatter

parameters

and

from

643

(November

Appendix:

censored

and of

A.

H.

Moore,

gamma and

samples",

"Maximum l i k e l i h o o d

Weibull

populations

Technometrics,

Vol.

7~

estimation

from No,4~

complete pp.

639-

1965).

S h o r t e r I n p u t and I t s Output f o r

Example 1

1DEFDBL A~X,P~M~L~U~ZqT,B,N,H~Y 2 DEFINT I , J , K 5 DIM B ( 1 9 ) , N ( 1 9 ) ~ A ( 1 9 ) , H ( 1 9 ) ~ L ( 1 9 ) ~ U ( 1 9 ) ~ X ( I 1 1 1 ) ~ Y ( 1 1 1 ) 10 X=I 11 FOR J333=i TO 9999 16 M=-I.701411834604692D+38 25 B(1)=O:B(2)=O 37 N(1)=36.99999:N(2)=15 50 A(1)=O÷RND(X)*36.99999:A(2)=O+RND(X)*I5 81H(1)=3:H(2)=3 88 FOR JJO=1 TO 1 100 FOR J=1 TO 20 102 FOR JO=O TO 3 228 FOR I = I TO 10 229 FOR K=I TO 2 230 IF A ( K ) - ( N ( K ) - B ( K ) ) / H ( K ) ~ J < B(K) THEN 250 ELSE 260 250 L(K)=B(K) 255 GOTO 265 260 L ( K ) = A ( K ) - ( N ( K ) - B ( K ) ) / H ( K ) ~ J 265 IF A ( K ) + ( N ( K ) - B ( K ) ) / H ( K ) " J > N(K) THEN 266 ELSE 268 266 U(K)=N(K)-L(K) 267 GOTO 298 268 U ( K ) = A ( K ) + ( N ( K ) - B ( K ) ) / H ( K ) ' ~ J - L ( K ) 298 X(K)=(L(K)+RND(X)*U
1 1 4.351238068187414 2 1 4.308143269224753 3 1 4.35123801185718

-165.1113595089649 11745~6967.882&36 -165.1217723948011 917424281.653093b -165.111~595089594 1174536648.465566

6.S&~S58441989643D-lO 1.117051563114305 8.431490822631~94D-12