Bayesian Prediction Analysis for the Exponential Model Under the Censored Sample with Incomplete Information

J ournal of K or ean

D a ta & I nf orma t ion S cience S ocie ty 2 002 , Vol. 13, N o.1 p p . 139～ 145

B ay e s i an P re dic t io n A n aly s i s

f o r t h e E x p o n e n t i a l M o d e l U n d e r t h e C e n s o re d S am p le w it h In c o m p le t e In f o rm at io n

Y e un g - H oon Kim , Je on g - H w an K o ¹⁾

A b s t ra c t

T his paper deals w ith the problem of obt aining the Bayesian predictiv e density function and the prediction int erv als for a future observ ation an d the p - th or der st atistics of n ^' fut ure observ ations for the exponential m odel under the censor ed sampling w it h incomplet e inform ation .

K ey W or d s a nd P hra s es : Bay esian Approach , Prediction , Incomplet e Censored Sample, Exponent ial distribut ion

1. In trodu c tion

T h e pr oblem of pr edict in g a fut u r e ob ser v at ion h a s r eceiv ed m u ch at t ent ion an d h a s b een dealt m ain ly in t w o appr oach es . On e is th e u su al cla s sical appr oach an d t h e ot h er is B ay esian appr oach . B a sed on t h e Bay esian appr oach , Ch hik ar a an d Gut t m an (1982), S inh a (1989 ), Up adhy ay an d P an dey (1989 ), an d Nigm an d A L - W ah ab (1996 ) su g g e st ed t h e B ay esian infer en ce ab out pr edict ion for Gau s sian , log n or m al, ex pon en tial an d Bur r dist r ib ut ion , r esp ect iv ely .

T h e ex pon en t ial dist r ibu t ion play s an im p or t ant r ole in m an y pr act ical r eliabilit y an aly s es . It is th e fir st m odel for w hich st at istical m eth od s w er e ex t en siv ely dev eloped an d is w idely u sed a s a r eliability m odel. E lper in an d Gert sb akh (1988 ) in v est ig at ed th e Bay e s in t er v al e st im at ion for ex pon ent ial p ar am et er in a r an dom cen sor ed sam plin g w ith in com plet e infor m at ion w h ich in clu des , a s part icular ca ses , b ot h t h e r an dom cen s or in g m odel an d t h e qu ant al - r e spon s e m odel. Calabr ia an d P u lcini (1990) pr opos ed t h e Bay esian pr ocedu r e for estim at in g t h e ex pon en tial m ean lifetim e an d t h e r eliability fun ct ion in a t im e cen sor in g m odel w it h in com plet e infor m at ion u sin g t h e s qu ar ed er r or 1. Department of St atistics, Andong National University.

E - m ail : jhko@anu .andong .ac.kr

los s . Kim (1995 ) pr op os ed th e B ay esian pr ocedu r e for est im at in g th e R ay leigh r eliabilit y fu n ction un der t h e cen sor ed s am ple w it h in com plet e in for m at ion .

In th is paper , w e w ill st u dy t h e B ay esian pr edict iv e den sity fu n ct ion an d t h e pr edict ion int er v als for a fu tu r e ob s er v at ion an d th e p - th or der st at is tic s of n ^' fu tu r e ob s erv ation s , at a specified m is s ion tim e t , b a s ed on a tim e c en s or ed s am ple w it h in com plet e infor m at ion ob s er v ed fr om th e ex p on ent ial m odel.

In s ect ion 2, w e deriv e t h e pr ediction an a ly sis of a fut ur e ob s er v a tion fr om t h e ob s er v at ion . In s ect ion 3, w e pr opos e th e pr e dict ion a n aly sis of th e p - th or der s t at ist ic s y ( p ) .

2 . P re dic t i on of a F u t u re Ob s e rv at io n

T h e pr ob abilit y den sit y fun ct ion of th e ex p on ent ial dist r ib ut ion , den ot ed by E ( ) , is g iv en by

f ( x | ) = ^{- 1} e ^{- x /} , >0 , 0 < x < . (2.1) T h en t h e r eliab ilit y at a s pecified tim e t > 0 is giv en by

P ( X > t ) = e ^{- t/} . (2.2)

L et t b e t h e fix ed tim e t o in spe ction an d let x _i b e t h e ex pon en tial lifetim e of it em i ( i = 1, 2 , , n ) . T h en th r ee pos sibilitie s ca n b e occu r r e d in t e st in g it em i : F ir s t , t h e it em fails at th e in st ant x _i ( x _i < t ) an d th e fa ilur e is n ot sign alled . T h e it em is foun d failed on t h e in sp ect ion t im e t . In th is ca s e , th e failu r e t im e is b efor e t h e in spe ction t im e . S econ d , t h e it em fails at th e in s t an t x _i ( x _i < t ) an d th e failu r e is im m edia t ely sig n a lled. In t his ca s e , t h e failur e t im e is ex a ctly k n ow n . T hir d , t h e it em is foun d un fa iled at tim e t . In t his ca s e , th e failur e tim e w ou ld b e b ey on d t h e in s pect ion t im e .

W h en n it em s a r e t e s t ed , th e cor r e s pon din g lik elih ood fun ct ion is giv en by

L ( | x ) ^{- n}

e ^{- u /} ( 1 - e ^{- t/} ) ⁿ

e ^{- tn}

^/ , (2.3)

w h er e n ₁ is t h e n um b er of elem ent s in t h e s et of n on c en s or ed a n d n on sign alled ob s erv ation s , n 2 is t h e n um b er of elem en t s in t h e s et of n on cen s or ed an d s ig n alle d ob s er v at ion s , n ₃ is t h e n um b er of elem ent s in t h e s et of cen s or e d ob s erv ation s , an d u is t h e su m of squ ar es of th e failu r e t im e of it em s in t h e set of n on cen sor ed an d sign alled ob s er v at ion s . T h en n = n ₁ + n ₂ + n ₃ an d th e v alu e of n ₁ an d n ₂ depen d on th e v alu e of t h e pr ob abilit y of failu r e - t o - sig n al p . On t h e av er ag e n 2 / ( n 1 + n 2 ) = p .

Un for t un at ely , sin ce t h e lik elih ood fu n ction is n ot ex pon en tial, a n at u r al

conju g at e pr ior can n ot b e fou n d . T hu s a pr ior den sit y g en er ally inv olv e s

n um er ical int egr at ion s . But u sin g t h e bin om ial for m u la , th e lik elih ood fun ct ion of can b e w rit t en a s

L ( | x ) ^{- n}

n

j = 0 c _j e ^{- {u + ( n}

^{+ j ) t }/} , (2.4)

w h er e c

= n

( ) j ^{( - 1)}

, j = 0 , 1, , n

. H en ce th e com m on ly u sed in for m ativ e pr ior for y ield s som e e st im at or s of w hich ar e in an an aly t ical for m .

N ow w e con sider t h e J effr ey s (1961) ' n oninform at iv e pr ior for . W h en t h e t im e t is sp ecified, t h e n on in for m ativ e pr ior for is

( ) ^{- 1} , >0 (2.5)

F r om (2.4 ) an d (2.5 ), t h e post er ior den sity of is g iv en by

( | x ) =

- ( n

+ 1) n

j = 0 c j e ^{- {u + ( n}

^{+ j ) t }/}

( n ₂ )

n

j = 0 c _j {u + ( n ₃ + j ) t } ^{- n}

, >0 , (2.6)

w h er e ( n ) is th e g am m a fun ction defin ed by ( n ) = ₀ z ^{n - 1} e ^{- z} dz . T h e dist r ib ut ion of a fut ur e ob ser v ation y g iv en is

f ( y | ) = ^{- 1} e ^{- y / ,} >0 , 0 <y < . (2.7) T h en t h e pr edict iv e den sit y fu n ct ion for y can b e obt ain ed an d is g iv en in t h e follow in g t h eor em .

T h e o re m 2 .1 . W it h a n on in for m ativ e pr ior for , th e pr edict iv e den sit y fun ct ion of y g iv en X = x is

(y | x ) = n ₂

n

j = 0 c _j ( w _j + y ) ^{- ( n}

^{+ 1)}

n

j = 0 c _j w _j ^{- n}

, (2.8)

w h er e w _j = u + ( n ₃ + j ) t.

N ow w e w an t t o con str u ct th e pr edict ion int er v als for a fu tu r e ob ser v at ion . If a n on in for m ativ e pr ior for is u sed , t h e 100 ( 1 - ) % equ al - t ail pr ediction int er v al ( C _{N L} , C _{N U} ) for a fut u r e ob serv ation y ar e t h e s olut ion s of th e equ at ion s

2 =

n

j = 0 c _j ( 1 - w _j

w _j + C _{N L} ) ⁿ

an d 2 =

n

j = 0 c _j ( w _j

w _j + C _{N U} ) ⁿ

. (2.9)

T h e 100 ( 1 - ) % m ost plau sible pr edict ion int er v al ( M _{N L} , M _{N U} ) for y can b e obt ain ed by solv in g sim u lt an eou sly t h e follow in g s :

n

j = 0 c _j ( 1 + M _{N L} w _j ) ^{- n}

-

n

j = 0 c _j ( 1 + M _{N U}

w _j ) ^{- n}

= 1 - (2.10)

an d

n

j = 0 c j ( w j + M N L ) ^{- ( n}

^{+ 1)} =

n

j = 0 c j ( w j + M N U ) ^{- ( n}

^{+ 1)} . (2.11)

A s a pr ior distr ibut ion for , w e con sider an inv ert ed g am m a pr ior dist ribu tion I G( , ) w it h t h e pr ob abilit y den sity fun ct ion

( | , ) = ( )

- ( + 1)

e ^{- /} , , >0 , >0 . (2.12) T h en on e can obt ain ea sily th e post er ior den sity of giv en X = x w hich is g iv en by

( | x ) =

- ( n

+ + 1) n

j = 0 c j e ^{- {u + ( n}

^{+ j ) t + }/}

( n ₂ + )

n

j = 0 c _j {u + ( n ₃ + j ) t + } ^{- n}

, , >0 , >0 . (2.13)

H en ce th e follow in g th eor em can b e obt ain ed .

T h e o re m 2 .2 . W it h an in v er t ed g am m a pr ior for , th e pr edict iv e den sit y fu n ct ion of y giv en X = x is

(y | x ) =

( n ₂ + )

n

j = 0 c _j ( v _j + y ) ^{- ( n}

^{+ + 1)}

n

j = 0 c j v j ^{- ( n}

^{+ )}

, (2.14)

w h er e v j = u + ( n 3 + j ) t + .

Un der t h e inv er t ed g am m a prior distr ibut ion for , t h e 100 ( 1 - ) % equ al - t ail pr ediction in t erv al ( C _{G L} , C _{G U} ) for y ar e th e solut ion s of th e equ at ion s

2 =

n

j = 0 c _j ( 1 - v j

v _j + C _{G L} ) ⁿ

⁺ an d 2 =

n

j = 0 c _j ( v j

v _j + C _{G U} ) ⁿ

⁺ . (2.15)

A lso t h e 100 ( 1 - ) % m ost plau sible pr ediction in t er v al ( M _{G L} , M _{G U} ) for y can

b e obt ain ed by solv in g sim u lt an eou sly t h e follow in g s :

n

j = 0 c j ( 1 + M _{G L}

v _j ) ^{- ( n}

^{+ )} -

n

j = 0 c j ( 1 + M _{G U}

v _j ) ^{- ( n}

^{+ )} = 1 - (2.16)

an d

n

j = 0 c j ( v j + M G L ) ^{- ( n}

^{+ + 1)} =

n

j = 0 c j ( v j + M G U ) ^{- ( n}

^{+ + 1)} . (2.17)

3 . P re di c tion of th e Orde re d Ob s erv ation

In th is sect ion w e con sider t h e pr edict iv e den sity fu n ct ion of th e p - th or der st at ist ic y _{( p )} of n ^' fut ur e ob s er v at ion s . T h e pr ob ability den sit y fun ction of

y ( p ) is a s follow s :

(y ( p ) | ) =

p - 1

i = 0 d _i e ^{- {( n}

'

- p + 1 + i)y

}/

( p , n ^' - p + 1) , y ( p ) 0 , 1 p n ^' , (3.1) w h er e ( a , b) is t h e g am m a fu n ct ion defin ed b y ( a , b) = ₀ ¹ z ^{a - 1} ( 1 - z ) ^{b - 1} dz an d d

= p - 1

( i ) ^{( - 1)}

^.

H en ce t h e pr edict iv e den sit y fu n ct ion of y ( p ) can b e obt ain ed an d is giv en a s follow s :

T h e o re m 3 .1 . U n der a n oninform at iv e pr ior for an d t h e pr ob ability den sit y fu n ct ion for y _{( p )} , th e pr edict iv e den sit y of p - th or der st atist ic of n ^' fu t ur e ob serv ation s is giv en b y

(y _{( p )} | x ) = n ₂

n

j = 0 c _j

p - 1

i = 0 d _i {w _j + ( n ^' - p + 1 + i)y _{( p )} } ^{- ( n}

^{+ 1)} ( p , n ^' - p + 1)

n

j = 0 c _j w _j ^{- n}

. (3.2)

Un der a n on in for m at iv e prior for an d t h e den sity fun ct ion of t h e p - th or der st atist ic, t h e low er an d upper 100 ( 1 - ) % pr edict ion b oun d s for y ( p ) ar e t h e s olut ion s of th e equ at ion s

2 = 1

( p , n ^' - p + 1)

n

j = 0 c _j

p - 1

i = 0 d _i 1 - {w _j / ( ( n ^' - p + 1 + i) C ^* _{N L} + w _j ) } ⁿ

n ^' - p + 1 + i (3.3)

an d

2 = 1

( p , n ^' - p + 1)

n

j = 0 c _j

p - 1

i = 0 d _i {w _j / ( ( n ^' - p + 1 + i) C ^* _{N U} + w _j ) } ⁿ

n ^' - p + 1 + i . (3.4)

A lso t h e 100 ( 1 - ) % m ost plau sible pr edict ion in t er v al ( M ^* _{N L} , M ^* _{N U} ) of y _{( p )} is t h e sim u lt an eou s solu tion s of

1 ( p , n ^' - p + 1)

n

j = 0 c _j

p - 1

i = 0 d _i {1 + ( n ^' - p + 1 + i) M ^* _{N L} / w _j ) } ^{- n}

n ^' - p + 1 + i

- 1

( p , n ^' - p + 1)

n

j = 0 c j p - 1 i = 0 d i

{1 + ( n ^' - p + 1 + i) M ^* N U / w j ) } ^{- n}

n ^' - p + 1 + i

= 1 - (3.5)

an d

n

j = 0 c _j

p - 1

i = 0 d _i {w _j + ( n ^' - p + 1 + i) M ^* _{N L} } ^{- ( n}

^{+ 1)}

=

n

j = 0 c _j

p - 1

i = 0 d _i {w _j + ( n ^' - p + 1 + i) M ^* _{N U} } ^{- ( n}

^{+ 1)} . (3.6)

In st ead of u sin g a n oninform at iv e pr ior of , on e can u se an in v er t ed g am m a pr i or dist r ibu tion w it h pa r a m et er s an d . T h en sim ilar t o t h e ca s e of t h e n on in for m at iv e prior , on e can obt ain t h e follow in g t h eor em .

T h e o r e m 3 .2 . W ith an inv er t ed g am m a pr ior dist r ibu t ion for an d t h e pr ob ab ilit y den s it y fun ction for y _{( p )} , t h e pr edict iv e den s it y of p - th or der st at is tic of n ^' fut ur e ob s er v at ion s is g iv en by

( y _{( p )} | x ) =

( n ₂ + )

n

j = 0

c _j

p - 1 i = 0

d _i { v _j + ( n ^' - p + 1 + i ) y _{( p )} } ^{- ( n}

^{+ + 1)}

( p , n ^' - p + 1)

n

j = 0

c _j v _j ^{- ( n}

^{+ )}

. (3.7)

U n der an inv er t ed g am m a pr ior dist r ibu t ion for is u s ed, t h e low er an d u pp er 100 ( 1 - ) % pr edict ion b ou n d s for y ( p) ar e th e s olut ion s of th e equ at ion s

2 = 1

( p , n ^' - p + 1)

n

j = 0

c _j

p - 1 i = 0

d _i 1 - { v _j / ( ( n ^' - p + 1 + i ) C ^* _{G L} + v _j ) } ⁿ

⁺

n ^' - p + 1 + i (3.8)

an d

2 = 1

( p , n ^' - p + 1)

n

j = 0

c _j

p - 1 i = 0

d _i { v _j / ( ( n ^' - p + 1 + i ) C ^* _{G U} + v _j ) } ⁿ

⁺

n ^' - p + 1 + i . (3.9)

A ls o th e 100 ( 1 - ) % m ost pla u sible pr edict ion in t er v a l ( M ^* _{G L} , M ^* _{G U} ) of y _{( p)} is

t h e s im u lt an e ou s s olu tion s of 1

( p , n ^' - p + 1)

n

j = 0

c _j

p - 1 i = 0

d _i {1 + ( n ^' - p + 1 + i ) M ^* _{G L} / v _j ) } ^{- ( n}

^{+ )} n ^' - p + 1 + i

- 1

( p , n ^' - p + 1)

n

j = 0 c _j

p - 1

i = 0 d _i {1 + ( n ^' - p + 1 + i ) M ^* _{G U} / v _j ) } ^{- ( n}

^{+ )} n ^' - p + 1 + i

= 1 - (3.10)

an d

n

j = 0

c _j

p - 1 i = 0

d _i { v _j + ( n ^' - p + 1 + i ) M ^* _{G L} } ^{- ( n}

^{+ + 1)}

=

n

j = 0

c _j

p - 1 i = 0

d _i { v _j + ( n ^' - p + 1 + i ) M ^* _{G U} } ^{- ( n}

^{+ + 1)} . (3.11)

R e f e re n c e s

1. Calab ria , R . an d P u lcin i, G .(1990). B ay e s e s tim at ion in ex pon en tial cen s or ed s a m ple s w it h in com plet e infor m at ion . Com m un ica t ions in S ta t is t ics, P art A - T h e ory an d M e thods , 19, 3037 - 3049.

2. Chhik ar a , R . S . an d Gu tt m an , I.(1982), P r edict ion lim it s for t h e inv er s e Gau s s ian dist r ibu tion , T echn om e tr ics, 24, 319 - 324.

3. E lp er in , T . an d Ger t sb akh , I.(1988). E s tim ation in a r a n dom cen s orin g m odel w ith in com plet e in for m ation : ex pon ent ial lifetim e dist r ibu tion . I E E E

T ransactions on R e liability , R - 37, 223 - 229.

4. J effr ey s , H .(1961). T h e ory of P robab ility , 3r d ed ., Ox for d U niv er sit y P r e s s , L on don .

5. K im , Y .- H .(1995). B a y e sian r eliabilit y e st im at ion for t h e R ay leigh m odel u n der th e cen s or ed s am ple w ith in c om plet e in for m a tion . J ournal of S ta tis tical T he ory & M e th ods , 6, 39 - 51.

6. N igm , A . M . an d A L - W ah a b , N . Y . A BD (1996 ), B ay e s ian pr edict ion w ith s am ple s ize for t h e Bu r r lifet im e dis tr ibut ion . Com m un ica t ions in S ta t is t ics P art A - T he ory and M e thods , 25, 1289 - 1303.

7. S in h a , S . K .(1989 ), B ay e sian infer en ce ab out th e pr edict ion cr edible int er v als an d r elia bilit y fu n ction for log n or m al dis tr ibut ion , J ournal of the I n d ian S ta tis tical A s s ocia tion , 27, 73 - 78.

8. U pa dhy a y , S . K . an d P an dey , M .(1989 ), P r ediction lim it s for an ex pon en t ial

dis tr ibut ion : A B ay e s pr edict iv e dist r ibu t ion appr oa ch , I E E E T ransactions

on R eliab ility , R - 38, 599 - 602.