Estimation of the parameter in an Exponential Distribution using a LINEX Loss

2 002 , V ol. 13, N o.2 p p . 1～10

E s tim ation o f th e p aram e t e r in an E x pon e n ti al D i s tribu tion u s in g a LIN E X L o s s

Jun g s o o W o o ^{1 )} , H w ajun g L e e , K ab s o ok E u n ^{2 )}

A b s tra c t

A Bay e s estim at or of th e s cale par am et er in an ex pon en tial dist ribu t ion w ill b e con sider ed by a LIN EX err or , t h en t h e risk of t h e B ay es estim at or u sin g a LINE X los s w ill b e com pared w it h th at of a B ay es est im at or u sin g a squ ar e error .

K e y w o rd s : LINEX err or , S qu ar e error , P rior pdf, B ay es risk .

1 . A LIN E X lo s s an d a s t an d ard e x po n e n ti al prio r

F or cert ain t y pes of los s fun ction s , ex pr es sion s for t h e Bay es est im at or can b e det erm in ed . A s u su al, sy m m et ric los s fun ct ion of s qu ar e err or los s an d, ab s olut e err or los s ar e u s ed in B ay es an aly sis , s o th eir risk s can b e com p ar ed each oth er . Bu t LINE X a sy m m et ric los s fu ction h a s b een u sefu l w h en a g iv en positiv e e st im at ion err or is r eg ar ded a s m or e seriou s t h an a n eg at iv e err or . S o w e can ch oose a sy m m etric LIN EX los s fu n ct on t h at h a s b een u su ally applied t o econ om et rics . LINEX los s fun ction w a s ex t en siv ely discu s s ed by Zelln er (1986 ).

M any au t h or s h av e cosider ed a sy m m et ric lin ear los s fun ct ion s , specially Zelln er (1986 ) stu died a v ery u s efu l a sy m m etric LIN EX los s fu n ction . it rise s appr ox im at edly ex pon en t ially on on e side of zer o an d appr ox im at ely lin early on t h e ot h er side.

S o far , w e don ' t h av e com parison b et w een t w o Bay es risk s of p ar am et er in an ex p on ent ial dist ribu tion by LIN EX los s an d s qu ar e los s , s o w e n eed t o com par e

1. Professor, Department of Math. & Stat., Yeungnam University, Kyongsan, 712-749, Korea

2. Graduate Student, Department of Math. & Stat., Yeungnam University, Kyongsan, 712-749, Korea

t h eir Bay es risk s .

W e sh all con sider t h e follow in g ex p on ent ial pdf (Roh at g i (1976 )):

f ( x ; ) = e

, x > 0, w h er e > 0. (1.1) T h e B ay es est im at or of th e p ar am et er w ill b e con sidered by a LIN EX err or los s an d a st an dard ex pon en tial prior :

( ) = e

, > 0. (1.2)

W e sh all com pare t h e Bay es risk of a Bay e s est im at or of th e par am et er in an ex p on ent ial distribut ion u sin g a LIN EX error los s an d a st an dar d ex pon en tial prior w it h t h at of a B ay es est im at or of t h e par am et er in an ex p on ent ial dist ribu tion u sin g a s qu ar e err or los s an d a st an dar d ex p on ent ial prior w h ich w e h av e w ell k n ow n .

1 - 1 . L IN E X e rror l o s s

In t his section a st an dar d ex p on ent ial prior den sit y w ill b e u s ed a s (1.2), an d a LINE X err or los s w ill b e u s ed a s follow :

L (d , ( ) ) = e

- ( d - ( ) ) - 1 (s ee S adoogh i- A lv an di (1990)) A s su m e X

X

. . . , X

b e a sim ple r an dom s am ple fr om t h e den sit y (1.1).

T h en t h e j oint pdf of X

, X

, . . . , X

is f ( x

, x

, . . . , x

) =

e

n i = 1xi

. F or t h e st an dar d ex pon en tial prior , t h e j oin t pdf of X

, X

, . . . , X

an d is

f ( x

x

. . . , x

, ) =

e

- ( 1 +

n i = 1xi)

,

an d h en ce fr om t h e form u la 3.381(4 ) of Gra dsht ey n & Ry zhik (1965 ), t h e m ar g in al j oin t p df of X

, X

, . . . , X

is

f ( x

, x

, . . . , x

) = n ! ( 1 +

n

i = 1

x

)

.

T h er efor e, t h e post erior pdf of can b e obt ain ed : h ( x

, . . . , x

) = ( 1 +

n

i = 1

x

)

e

- ( 1 +

n i = 1xi)

/ ( n + 1) . (1.3) w h ich follow s a g am m a dist ribu t ion w it h p ar am et er s n +1 an d 1

1 +

n i = 1

x

.

By th e post erior pdf (1.3 ) an d th e r esu lt in S adooghi - A lv an di (1990), un der t h e

LIN EX err or los s , w e can ob t ain a B ay es estim at or of :

*

l

- ln E (e

X

, . . . , X

)

= ( n + 1) ln 2 +

n

i = 1

X

1 +

n i = 1

X

. (1.4 )

T h e Bay es risk of

un der a LINEX err or los s can b e obt ain ed a s follow in g s : R

( ,

) E [ E (e

* l-

- (

- ) - 1) ]

= E ( - 1) + E [ e

E ( ( 2 +

n i = 1

X

1 +

n i = 1

X

)

) ] - ( n + 1) E [ E ( ln 2 +

n i = 1

X

1 +

n i = 1

X

) ] (1.5 )

By th e st an dar d ex p on ent ial prior , th e Ist t erm in 2n d equ alit y (1.5 ) is zer o, b y th e form ula s 3.381(4 ) an d 3.914 (4 ) of Gra dsht ey n & Ry zh ik (1965 ), th e ex p ect ation of th e 2n d t erm in 2n d equ alit y (1.5 ) is 1,

b y form ula 3.381(4) of Gr adsht ey n & Ry zhik (1965), t h e ex pect at ion of th e 3r d t erm in 2n d equ alit y (1.5 ) can b e obt ain ed a s :

n

y

( 1 + y )

ln 2 + y 1 + y dy . T h er efor e, t h e Bay es risk is 1- n (n +1)

0

y

( 1 + y )

ln 2 + y

1 + y dy . (1.6 ) w h ich th e in t eg r al in (1.6) conv er g es by t h e adv an ced calculation .

T h e int egr al in (1.6 ) =

0

y

( 1 + y )

ln (2 + y ) dy -

0

y

( 1 + y )

ln (1 + y ) dy

=

y

( 1 + y )

ln (2 + y ) dy - B ( n , 1) ( ( n + 1) - ( 1) ) by 4.293 (14 ) of Gr ad sh t ey n & Ry zhik (1965 ),

=

1

0

( 1 - x )

ln ( x + 1

x ) dx - B ( n , 1) ( ( n + 1) - (1) ) by x = 1

y + 1 an d B (x ,y ) is a Bet a fun ct ion ,

=

n - 1 k = 0

n - 1

( k ) ^{( - 1)}

^[

x

ln (1 + x ) dx -

1

0

x

ln x dx ] - n

( ( n + 1) - ( 1) )

by t h e Bin om ial ex pan sion an d L ' H ospit al rule,

=

n - 1 k = 0

n - 1

( k ) ^{( - 1)}

^[ k + 1 ¹ ln 2 +

1 0

1

k + 1 x

dx

- 1 k + 1

1 0

x

1 + x dx ] - n

( ( n + 1) - ( 1) ) , by t h e part ial int egr at ion .

=

n - 1 k = 0

n - 1

( k ) ^{( - 1)}

^[ k + 1 ¹ ln 2 + 1 ( k + 1)

,

- 1

k + 1

2 1

( y - 1)

y dy ] - n

( ( n + 1) - ( 1) ) , (1.7) by a t ran sform at ion 1+x =y ,

w h er e B ( a , b) = ( a) ( b)

( a + b) an d ( x ) is t h e p si - fu n ct ion . T h e int egr al in th e la st equ ality (1.7 ),

2 1

( y - 1)

y dy =

k + 1 j = 0

k + 1

( j ) ^{( - 1)}

y

dy

= ( - 1)

ln 2 +

k + 1 j = 1

k + 1

( j ) ^{( - 1)}

^{( 2}

j ^{- 1)} (1.8 ) F r om th e r esu lt s (1.6 ), (1.7 ) an d (1.8 ),

t h e By es risk of

un der a LINE X err or los s can b e obt ain ed : R

( ,

) =1- n (n +1){

n - 1 k = 0

n - 1

( k ) ^{( - 1)}

^[ k + 1 ¹ ln 2 + 1 ( k + 1)

- 1

k + 1 ( - 1)

ln 2 - 1 k + 1

k + 1 j = 1

k + 1

( j ) ^{( - 1)}

^{( 2}

j ^{- 1)} ]

- n

[ ( n + 1) - ( 1) ] } (1.9 )

1 - 2 . S qu are e rro r lo s s

On t h e oth er h an d , by u sin g a s qu ar e err or los s an d a st an dar d ex p on ent ial prior , a B ay es est im at or of th e p ar am et er in an ex pon en tial dist rib ut ion w it h th e p df (1.1) an d it s B ay es risk ar e w ell kn ow n a s

t h e B ay es estim at or of an d it s Bay e s risk are

*

s

= n + 1 1 +

n i = 1

X

. (1.10)

an d R

( ,

) = 2

n + 2 , re spectiv ely . (1.11)

F urt h erm or e, fr om t h e post erior pdf (1.3), w e can obt ain t h e g en er alized

M LE (m ode ) of λ:

*

m

n

1 +

n i = 1

X

(1.12)

B a sed on th e m ode (1.12) an d th e form u la s 3.381(4 ) an d 3.914 (3 ) of Gr ad sh t ey n &

Ry zhik (1965), th en t h e B ay es risk of th e m ode

un der a squ ar e err or los s is :

R

( ,

) = 2

n + 1 (1.13)

F r om t h e B ay es risk s (1.11) an d (1.13 ), it ' s n o w on der t h at t h e B ay es est im at or h a s les s B ay es risk th an th e m ode un der th e s qu ar e err or los s . A n d fr om t h e r esult s (1.10) an d (1.12), th e m ode locat es at t h e left side of t h e B ay es e st im at or u n der t h e post erior den sity cu rv e.

On t h e ot h er h an d , by th e post erior pdf (1.3 ), 2 ( 1 +

n

i = 1

X

) ( x

, . . . , x

) follow s a ch i- squ ar e dist ribu tion w it h a df 2 (n +1), an d h en ce, w e can obt ain a Bay es con fiden ce int erv al :

(

2

2 ( n + 1) , / 2

2 ( 1 +

n i = 1

X

)

,

2

2 ( n + 1) , 1 - / 2

2 ( 1 +

n i = 1

X

)

) is a (1- ) 100% B ay es confiden ce in t erv al of .

2 . A LIN E X lo s s an d a g am m a prior

T h e B ay es est im at or of th e p ar am et er w ill b e con sidered by a LIN EX err or los s an d a g am m a prior :

( ) = ( )

- 1

e

, >0 (2.1)

W e sh all com pare t h e Bay es risk of a Bay e s est im at or of th e par am et er in an ex p on ent ial distribut ion u sin g a LIN EX error los s an d a g am m a prior w it h t h at of a B ay es estim at or of th e par am et er in an ex pon en t ial dist ribu tion u sin g a squ ar e err or los s an d a g am m al prior w hich w e h av e w ell kn ow n .

2 - 1 . L IN E X e rror l o s s

T h e Bay es estim at or of t h e p ar am et er w ill b e con sider ed by a LINEX err or

los s :

L (d , ( ) ) = e

- ( d - ( ) ) - 1 (see S adoogh i- A lv an di (1990)) and a gamma prior density (2.1).

.

A s su m e X

X

. . . , X

b e a sim ple r an dom sam ple fr om th e den sit y (1.1). T h en t h e j oint pdf of X

, X

, . . . , X

is f ( x

, x

, . . . , x

) =

e

- ⁿ

i = 1x_i

.

F or th e g am m a prior , th e j oint pdf of X

, X

, . . . , X

an d is f ( x

x

. . . , x

, ) =

( )

n + - 1

e

- (

n i = 1xi+ )

,

an d t h e m ar g in al j oin t p df of X

, X

, . . . , X

is f ( x

, x

, . . . , x

) = ( n + )

( )(

n

i = 1

x

+ )

.

T h er efor e, th e post erior pdf of can b e obt ain ed :

h ( x

, . . . , x

) = (

n

i = 1

x

+ )

( n + )

n + - 1

e

- (

n i = 1xi+ )

. (2.2)

w hich follow s a g am m a dist ribu tion w it h par am et er s n + an d 1

n i = 1

x

+

.

By th e post erior pdf (2.2) an d t h e re sult in S adoogh i- A lv an di (1990), u n der t h e LIN EX err or los s w e can obt ain a Bay es e st im at or of :

*

l

- ln E (e

X

, . . . , X

)

= - ( n + ) ln

n i = 1

X

+

n

i = 1

X

+ + 1

(2.3)

fr om th e form ula 3.381(4) of Gr ad sht ey n & Ry zh ik (1965 ).

T h e Bay es risk of

u n der a LINEX err or los s can b e obt ain ed a s follow in g s :

R

( ,

) E [ E (e

* l-

- (

- ) - 1) ]

= ( n + ) ( n ) ( )

y

( y + )

dy - ( n + )( n + ) ( n ) ( )

0

y

( y + )

ln y + + 1

y + dy + - 1

fr om t h e form u la 3.381(4 ) of Gra dsht ey n & Ry zhik (1965 ).

= - n +

B ( n , )

1

0

( 1 - t)

t

ln (1 + t ) d t, by in t egr at ion (1.4 ) an d

y + t,

= - n +

B ( n , )

n - 1 k = 0

n - 1

( k ) ^{( - 1)}

k + ¹ [ ln (1 + 1

) - 1

( k + + 1) F ( 1; k + + 1; k + + 2; - 1

) ] , (2.4)

from t h e form u la 3.381(4 ) of Gr ad sh t ey n & Ry zhik (1965), w h er e B (x ,y ) an d F (a ;b ;c ;x ) are t h e b et a fu n ct ion an d th e h y per g eom et ric fun ction , r espect iv ely , in Gr adsht ey n & Ry zh ik (1965 ),

H er e, alth ou gh a st an dar d ex pon en tial prior is a special ca se of a g am m a prior , it is n ot ea sy for u s t o sh ow th at th e B ay es risk (2.4 ) can r epre sen t by t h e risk (1.9 ) w h en α=1 an d β=1 in t h e g am m a prior . T h er efor e, w e h av e con sider ed t w o s epar at ed sect ion s .

1 - 2 . S qu are e rro r lo s s

On t h e ot h er h an d , by u sin g a squ are err or los s an d a g am m a prior , a Bay es e st im at or of th e p ar am et er in an ex pon ent ial dist ribu tion w it h th e pdf (1.1) an d it s B ay es risk ar e w ell kn ow n a s

t h e B ay es estim at or of is

*

s

= n +

n i = 1

X

+

. (2.5 )

T o fin d it s B ay es risk , Let I ( n ; n + + 2 )

0

y

( y + )

dy = 1

+ 2

B ( + 2 , n ) , (2.6 ) fr om in t egr at ion by su b st itu tion , t

y + , w h er e B (x ,y ) is a b et a fu n ct ion . F r om th e in t egr at ion (2.6 ) an d th e form ula 3.381(4 ) of Gr ad sh t ey n &

Ry zhik (1965), w e can ob t ain t h e Bay e s risk of t h e Bay es e st im at or

un der a

s qu ar e err or los s :

R

( ,

) =

( )

+ 1

e

d - 2 ( n + )

( n ) ( )

y

y +

0

n + - 1

e

d dy + ( n + )

( n ) ( )

y

( y + )

0

n + - 1

e

dy d

= ( + 1)

2

- ( n + )

( n + )

( n ) ( ) I ( n ; n + + 2) , by in t egr al (1.5 ),

= ( + 1)

2

( n + + 1) . (2.7)

F r om t h e p ost erior p df (2.2), th e g en er alized M LE of is n + - 1

n i = 1

X

+

, w hich

locat e s at a left of t h e B ay es estim at e of u n der th e post erior pdf (2.2) .

F or a giv en 0< γ< 1,

a 0

1

( ) x

e

dx = , defin e a

( ; ) .

F r om t h e p ost erior p df (2.2), 2 (

n

1

X

+ ) follow s a g am m a dist ribu tion w ith a sh ape p ar am et er n +α an d a s cale p ar am et er 2, fr om defin it ion of

an d th e

p ost erior pdf (2.2) of , (

*

( ;

2 ) 2 (

n

i = 1

X

+ ) ,

*

( ; 1 - 2 ) 2 (

n

i = 1

X

+ )

) can b e obt ain ed a s an

(1- γ)100% B ay es confiden ce in t erv al for . (2.8 )

By t h e Bay es risk s (1.9 ) an d (1.11) an d T able s of t h e p si- fun ct ion in A br am ow it z & S t eg un (1970), an d th e Bay es risk s (2.4) an d (2.7 ) an d t h e r ecu r sion form u la of hy p er g eom etric fu n ct ion in A br am ow it z & S t egu n (1970), T able 1 sh ow s B ay es risk s of t w o B ay es estim at or s un der t w o differ en t err or los s es .

T able 1. B ay es risk s of t w o Bay e s e st im at or s

an d

b a s ed on a g am m a

prior an d a st an dar d prior

n = 1

2 , = 1 = 1 , = 1 = 2 , = 1

* s

* l

* s

* l

* s

* l

5 0.11538 0.047776 0.2857 1 0.11574 0.75000 0.2932 1

10 0.06522 0.028943 0.16667 0.07270 0.46154 0.19533

15 ^{0.04545 0.020824 0.11765 0.053 15 0.33333 0.14686} 20 0.03488 0.0 16276 0.0909 1 0.04 193 0.26087 0.11778 25 0.02830 0.0 13364 0.07407 0.03463 0.2 1429 0.09836 30 ^{0.0238 1 0.0 11338 0.06250 0.0295 1 0.18 182 0.08445}

(to continue)

n = 1

2 , = 2 = 2 , = 2 = 1

2 , = 1

2 = 2 , = 1

2

* s

* l

* s

* l

* s

* l

* s

* l

5 ^{0.028846 0.0 13029 0.18750 0.082007 0.46154 0.16528 3.00000 0.97633} 10 0.0 16304 0.007656 0.11538 0.052796 0.26087 0.10488 1.84615 0.68354 15 0.0 11364 0.005429 0.08333 0.038989 0.18 182 0.07724 1.33333 0.52843 20 ^{0.00872 1 0.004208 0.06522 0.030922 0.13953 0.06124 1.04348 0.43 15 1} 25 ^{0.007075 0.003435 0.05357 0.025626 0.11321 0.05077 0.857 14 0.36494} 30 ^{0.005952 0.002903 0.04545 0.02 188 1 0.09524 0.04337 0.72727 0.3 1632}

T hr ou gh ou t nu m erical ev alu at ion s in T able 1, for a g am m a prior an d a st an dar d prior , t h e Bay es risk s of a Bay e s est im at or

of th e s cale p ar am et er in an ex p on ent ial distribut ion u n der t h e LINE X err or los s ar e les s t h an t h ose of a B ay es e st im at or

of th e scale par am et er in an ex pon ent ial dist ribu t ion un der t h e s qu ar e err or los s w h en = 1

2 , = 1, = 2 an d = 1

2 , 1 , 2 , an d n =5 (5 )30.

R ef e re n c e s

1. A br am ow it z, M & S t egu n , I.A .(1970), H an db ook of M at h em atical F un ction s , Dov er P u blicat ion s , In c., N ew Y ork

2. Gr ad sh t ey n , I.S .& Ry zh ik , I.M .(1965 ), T ab le of Int eg r als , S erie s , an d P r odu ct s , A cadem ic P r es s , N ew Y ork

3. R oh at g i, V .K .(1976 ), A n Int r odu ction t o P r ob abilit y T h eory an d M at h em atical S t at istics , J oh n W iley & S on s , N ew Y ork

4. S adooghi - A lv an di, S .M .(1990), E st im at ion of t h e p ar am et er of a P ois son Dist ribu t ion u sin g a LIN EX Los s F un ct ion , A u st ral, J . S t atist ., 32 (3 ), 393 - 398

5. Zelln er , A .(1986 ). B ay esian est im ation an d pr edict ion u sin g a sy m m et ric los s fun ct ion . J . A m er . S t at ist . A s soc . 81, 446- 451.

[ 2002년 6월 접수, 2002년 9월 채택 ]