極值分配中母數的加權最小平方估計

ໂࣃ̶੨̚ϓᇴ۞ΐᝋ౵̈π͞Ҥࢍ

ڒߋੑ ౙঔڒ

လݑᘽநࡊԫ̂ጯྤੈგநր

ၡ! ࢋ

ώቔ၆ໂࣃ̶੨೩ֻ˘࣎ܕҬ۞ΐᝋ౵̈π͞ڱ(WLS)ֽҤࢍໂࣃ̶੨

̚۞͎ޘϓᇴ(scale parameter)ĂϤٺໂࣃ̶੨Ξᖼೱј Weibull ̶੨Ă҃

Bergman [1]ޙᛉϡ̙Т۞ΐᝋ౵̈π͞ٺ Weibull ϓᇴ˯ઇਫ਼ᕩ̶ژ۞ࡁտĂ Тॡ΁ࣇ۞͞ڱϺ఼ϡٺໂࣃ̶੨˯ĄдΞҖ̶ّژ̚ͽሀᑢ͞ڱֽෞҤ͞ڱ

۞ᐹКĂ׎ඕڍ Bergman ۞͞ڱྵ՟ѣΐᝋ۞͞ڱ(OLS)ٙՐ଀ҤࢍࣃྵָĂ

൒҃ώ͛ٙ೩͞ڱځព۞˵ྵ Bergman ۞ඕڍࠎָĂТॡԧࣇ۞͞ڱՀߏྵࠎ ᖎಏ̈́टٽᒢྋĄ

ᙯᔣෟĈໂࣃ̶੨ăΐᝋ౵̈π͞ҤࢍณăѨԔ௚ࢍณă͎ޘϓᇴĄ

AN ALTERNATIVE WEIGHTED LEAST-SQUARES ESTIMATION OF THE PARAMETER OF THE EXTREME-VALUE DISTRIBUTION

Chun-Tsai Lin Hai-Lin Lu

Department of Management Information Science Chia-Nan University of Pharmacy and Science

Tainan, Taiwan 717, R.O.C.

Key Words: extreme-value distribution, weighted least-squares estimator, order statistics, scale parameter.

ABSTRACT

This paper proposes a weighted least square (WLS) method to estimate the scale parameter of the Extreme-value distribution. It is well known that Extreme-value distribution can be converted into Weibull distribution. Bergman(1986) has suggested a WLS approach that gave better performance than the ordinary least square (OLS) methods, on the estimation of Weibull parameters. It will be shown, through simulation results, that the WLS method proposed by this paper is better than that of Bergman on the parameter estimation. Furthermore, the proposed method is much more concise and easier to perceive.

˘ă݈! ֏

ໂࣃ̶੨۞௢᎕̶οבᇴࠎ

)]

exp(

exp[

1 )

( β

−α

−

−

= x

x

F (1)

׎̚x>0Ă−∞<α<∞ࠎҜཉϓᇴ(location parameter)Ă

>0

β ࠎ͎ޘϓᇴ(scale parameter)Ą

ໂࣃ̶੨Ξͽޝᆵᇃ۞ᑕϡд̍඀˯۞Ξያޘ̶ژ (Kececioglu[2]) Т ॡ ˵ Ξ ͽ ᄃ Weibull ̶ ੨ ઇ ̢ ࠹ ᖼ ೱ (Lawless[3])Ąѩ̶੨˵జણ҂јࠎ Gumbel ̶੨(Gumbel [4])Ăώ͛۞ϫ۞ࠎ੅ኢ׎ણᇴ۞ҤࢍĄ(1)ёޝटٽᖼೱ

ј˭ࢬԛёĂГ҃Ӏϡ౵̈π̶ֽ͞ژĂͽҤზ α ̈́ βĄ

β α β ⁻

=

−

− F X 1 X )) ( 1 ln(

ln( (2)

ன ઄ న X₁,X₂,L,X_n ࠎ ֽ ҋ ё (1) ̝ ᐌ ፟ ᇹ ώ ҃

) ( ) 2 ( ) 1

( ,X , ,X _n

X L ࠎ׎၆ᑕ۞ѨԔ௚ࢍณĂ݋ё(2)Ξᆷј 1 ,

)) ( 1 ln(

ln( _{() ()}

)

( β

α

β ⁻

=

−

−

= _{i i}

i F X X

Y i=1,L,n (3)

׎̚ࠎX₍₁₎<X₍₂₎<L<X_(n)Ą ၆F(x_(i))҃֏Ξϡ ˆ( )

)

x(i

F ֽҤზĂ҃ ˆ( )

)

x(i

F ۞Ҥณ

ΞనࠎĈ

πӮ৩(mean rank)Ҥࢍณ

) 1 ˆ(

)

( = +

n x i F _i

׶̚Ҝ৩(median rank)Ҥࢍณ

4 . 0

3 . ) 0 ˆ(

)

( +

= − n x i F _i

ְ၁˯πӮ৩ă̚Ҝ৩׌࣎Ҥࢍณ࠰̶ҾΞϤ˭ёጱ΍

(D’Agostino ᄃ Stephens[5])

1 ) 2

ˆ(

)

( − +

= − c n

c x i

F _i Ă0≤ c≤1

˵ಶߏ૟ c = 0 ׶ c = 0.3 ̶Ҿ΃ˢ҃଀זĄ

Bergman ૻአдё(3)̚၆Տ˘̙࣎ТҜཉ۞X_(i)ֹϡ

࠹Т۞ΐᝋߏ̙Ъந۞Ă΁ᄮࠎ̙ТҜཉ۞X_(i)ᑕྍѣ̙

Т۞ΐᝋЯ̄ (weighting factor)Ă΁ޙᛉΐᝋЯ̄ߏ

2 ) ( )

( ))ln(1 ˆ( ))]

ˆ( 1

[( _{i i}

i F x F x

W = − − (4)

Faucher ᄃ Tyson [6]˵ࡁտё(3)ֹ֭ϡాᜈѡቢ҃ᒔ଀႙ ܕΐᝋЯ̄ (asymptotic weighting factor)Ă׎ܕҬΐᝋЯ̄

ܑϯт˭

] )) ˆ( 1 ( 1 [ 5 . 27 ) ˆ( 3 .

3 _{(i) (i)} ^0.025

i F x F x

W = − − − (5)

ۡז౵ܕ Drapella ᄃ Kosznik [7]ᄮࠎࢋҤࢍё(3)۞Y_(i)Ă

҂ᇋӀϡѡቢ۞࿀ܕڱՐ଀Ĉ

1 ) 1 ln(

5774 . 0

)!

1 (

! )!

1 ) (

1 )! ( ( )!

1 (

ˆ ! ¹

) 0 (

+ +

−

+ +

−

−

−

∑ ⋅

−

−

− −

−

= − ⁻

=

v i n

v i n

v i v

i i

n i

y n ⁱ

v v i

ઇࠎ ( )Y ۞ҤࢍณĂ֭Ր΍ણᇴ۞௚ࢍณâਠ׎Ҥࢍณi

۞តளൾ̂ٺ OLS ͞ڱ۞Ҥࢍณ (Drapella ᄃ Kosznik)Ą ώ͛ԧࣇ೩΍˘࣎າ۞ࡁտ͞ڱΝҤࢍໂࣃ̶੨̚

۞͎ޘϓᇴĂௐ˟༼݋၆׎͞ڱΐͽྎ௟۞ᄲځĂௐˬ༼

Ϥٺ Bergman ᄃ F&T ͞ڱඕڍ࠹ҬĂ߇ώ͛̚Ϊ૟ԧࣇ

ٙ೩΍۞͞ڱ׶ଳϡ Bergman ׶ OLS ׌჌͞ڱ۞ሀᑢඕ ڍઇͧྵĂ౵ޢઇ˘࣎ᖎൺ۞੅ኢĄ

˟ă͞ ڱ

ໂࣃ̶੨ࠎ޽ᇴ̶੨۞ؼҩ̶੨ĂЯѩё(2)۞νᙝ̚

1 )]

exp[(

)) ( 1

ln( β

α β ⁻

=

−

− F X X

ࠎ ᇾ ໤ ۞ ޽ ᇴ ̶ ੨ ׎ ഇ ୕ ࣃ ࠎ 1 Ą న )]

( 1 ln[ ₍₎

)

(i F Xi

Z =− − Ă݋ё(3)ΞԼᆷј 1 ,

ln _{() ()} β α

β ⁻

= _i

i X

Z i=1,L,n (6)

ன઄నX₁,X₂,L,X_nࠎໂࣃ̶੨̝ᐌ፟ᇹώ׶X₍₁₎

) ( )

2

( X n

X < <

< L ࠎ ׎ ࠹ ၆ ᑕ ۞ Ѩ Ԕ ௚ ࢍ ณ Ă ݋Z₍₁₎<

) ( )

2

( Zn

Z < L< ࠎ࠹၆ᑕ۞ᇾ໤޽ᇴ̶੨ѨԔ௚ࢍณĂ݋

׎πӮᇴ׶តளᇴ̶Ҿࠎ

∑ − +

= = i i j

j Z n

E

) 1

( 1

) 1

( Ă ∑

+

= −

= i i j

j n Z

Var

1 2

)

( ( 1)

) 1

( (7)

(Balakrishnan ᄃ Cohen[8])Ą҃lnZ_(i)۞តளᇴ(Bickel ᄃ Doksum[9])݋ࠎ

2 ) (

) ( )

( [ ( )]

) ( ) (ln

i i

i E Z

Z Var Z

Var ≅

ٺ(6)ё̚΄

σ µ

σ ⁻

=

= _{() ()}

) (

ln _i 1 _i

i Z X

Y

д˘ਠਫ਼ᕩ̶ژ̚Ӯ઄నՏ˘࣎ᕇX_(i) (ٕՏ˘࣎Ҝ ཉ) ѣ࠹Т۞ࢦณٕΐᝋ(weight)ĂҭߏVar(Y_(i))ݒ̙ߏ˘

࣎૱ᇴĂ߇ԧࣇᄮࠎՏ˘࣎ᕇӮᑕΐ˯˘࣎ΐᝋЯ̄Ăͽ ႕֖ਫ਼ᕩ̶ژ̝̚઄నĂЯѩώ͛ԧࣇ૟ޙᛉֹϡ۞ΐᝋ Я̄ࠎVar(lnZ_(i))۞ࣆᇴ

) ( )]

(

[ _(i) ² _(i)

i E Z Var Z

W = (8)

׎ΐᝋπ͞׶΄ࠎ

∑ −

= = n

i WiYi Yi X i

Q

1

2 ) ( )

( ( )]

[

∑ − +

= = n

i WiYi aX i b

1

2 ) ( )

( ]

[

౵̼̈ Q ֽҤࢍໂࣃ̶੨۞ϓᇴα̈́βĂ׎̚Y_(i)=lnZ_(i)׶ b

aX X

Y_i( _(i))= _(i)− Ăa=1/β, b=α/β, i=1,L,n,ĂТॡ ଳ ϡ ё (7) ̚E(Z_(i))פ ΃Z_(i) Ă Ξ ͽ ଀ זY_(i) =lnZ_(i)=

) (

lnE Z_(i) ̝ࣃĂЯ҃ଂ Q ࣃ۞ໂ̼̈࿅඀̚

∑ − + − =

∂ =

∂

= n

i WiYi aXi b X i

a Q

1 [ () () ]( ()) 0

2

∑ − + =

∂ =

∂

= n

i Wi Yi aXi b

b Q

1 [ () () ] 0

2

ĂΞᒔ଀ໂࣃ̶੨۞ϓᇴα̈́β۞ΐᝋ౵̈π͞ڱҤࢍ

ณĂΞ଀

∑ ∑ −∑ ∑

∑ ∑ −∑ ∑

=

) ( ) ( 2

) (

) ( ) ( ) ( 2

) ( )

ˆ (

i i i i i i i

i i i i i i i i i

x w x w x w w

y x w x w x w y a w

∑ ∑ −∑ ∑

∑ ∑ −∑ ∑

=

) ( ) ( 2

) (

) ( ) ( ) ( )

ˆ (

i i i i i i i

i i i i i i i i

x w x w x w w

x w y w y x w b w

߇β۞Ҥࢍณࠎβˆ =1aˆĂα۞Ҥࢍณࠎα^ˆ=b^ˆβ^ˆĄ

ˬăሀᑢ͞ڱ

Ϥٺ F&TĂDrapella ׶ Kosznik ٙ೩͞ڱ̶Ҿд׎ኢ

͛ᄃ Bergman ઇ࿅ͧྵĂٺሀᑢ͞ڱ˭ٙ଀̝ඕڍӮᄃ Bergman ۞͞ڱ࠹ҬĂ၆ٺώ͛ٙ೩͞ڱΪᄃ Bergman ׶ OLS ͞ڱϡሀᑢ۞͞ё೩ֻ˘࣎ΞҖّ۞̶ژĄࠎ྿זͽ

˯۞ϫᇾĂԧࣇଳϡᄋгΙᘲ (Monte Carlo) ሀᑢ͞ڱĂ ੫၆ໂࣃ̶੨̚ણᇴ̶Ҿߏα=10Ăβ=0.5Ă1Ă5 (дѩ

ෛࠎৌࣃ(true value))Ă֭၆̙Т۞ᇹώ̂̈Ᏼפଂ 6 ז 100

̶ҾயϠ 10000 ௡ᇹώĂՏ௡ᇹώ၆̙Т۞͞ڱ̶Ҿࢍზ α^ˆĂβˆ Ą

ࢋෞҤң჌͞ڱٙޙᛉ۞௚ࢍณ౵ָĂӈѩ͞ڱࠎ׎

̚౵ָ۞͞ڱâ࣎Ъዋ۞ෞҤ͞ڱߏޝࢦࢋ۞ĄෞҤᇾ

໤ म ߏ ˘ ࣎ Ъ آ ͞ ڱ Ă Я ι ࠹ ᙯ ٺ Ҥ ࢍ ࣃ ۞ ໤ ቁ ّ (precision)ĂҭߏӮ͞ᄱ (mean square error : MSE) ඕЪ˞

ീ ณ Ҥ ࢍ ࣃ ۞ ត ள ӈ ᇾ ໤ म (໤ቁ ّ)ᄃ Ҥ ࢍࣃ۞ ઐम (bias) ӈჟቁؖ (accuracy)Ą˘࣎Ҥࢍณѣр۞ MSE ّኳ

˵ಶߏྵ̈۞តளᄃઐमĂЯѩԧࣇ҂ᇋ˘࣎р۞Ҥࢍณ

˵ಶߏѩҤࢍณѣྵ̈۞ MSEĄ၆ٙѣ۞͞ڱĂд 10000 Ѩ ሀ ᑢ ྏ រ ٙ ᒔ ଀ β ۞ Ҥ ࢍ ณ ࠎ βˆ⁽¹⁾, βˆ⁽²⁾, L ,

) 10000

ˆ(

β ĂԧࣇࢍზE(βˆ)ĂSˆ²(βˆ)ĂMSˆE(βˆ)Ă׎̚

= ∑

= 10000

1 )

ˆ(

10000 ) 1 (ˆ

i

E β β i

∑ −

= =

10000

1

2 )

(

2 (ˆ (ˆ))

9999 ) 1 (ˆ ˆ

i

i E

S β β β

∑ −

= =

10000

1

2 )

( )

(ˆ 10000 ) 1 (ˆ ˆ

i

E i

S

M β β β ,βࠎৌࣃ

ࠎ˞͞ܮԧࣇ૟ BergmanăOLS ̈́ώ͛ٙ೩͞ڱͽܑ

˘ֽᄲځĈ

ԧࣇ۞ሀᑢඕڍពϯдܑ˟Ҍܑα̚Ăܑ˟ҌܑαӮ ࠎԆБྤफ़Ă׎̚ৌࣃ۞ଳϡ̙ε˘ਠ̼Ăͽβ =0.5Ă1Ă 5 ֽ఍நĄ൑ኢтңϤܑ˟זܑαΞځព۞࠻΍n≥14ॡ

͞ڱ 7 ѣ౵̈۞Sˆ(βˆ)Ăn>20ॡ͞ڱ 7 ۞MSˆE(βˆ)Ϻࠎ౵

̈Ąϡ OLS ͞ڱٙ଀۞ඕڍ׎Sˆ(βˆ)д̙Тᇹώ˭Ӯྵ

Bergman ᄃώ͛ٙ೩າ͞ڱࢋ̂Ă҃дΐᝋ۞ Bergman ͞ ڱ̚ͽפ c = 0.5 ॡ׎Sˆ(βˆ)౵̈Ą

ܑ˘! Ч჌͞ڱ̝௑ܑཱིϯ

͞ڱ פ c ࣃ ଳϡ͞ڱ

1 0.0 OLS 2 0.3 OLS 3 0.5 OLS 4 0.0 Bergman 5 0.3 Bergman 6 0.5 Bergman

7 ů ώ͛͞ڱ

αă၁ּᛚᛖ

̫ͽ˘࣎၁ּֽᄲځԧࣇٙ೩͞ڱᄃ׎΁͞ڱٙࢍ

ზ̝ඕڍĄώּଳҋ McCool[10]׎ܑ̣ࠎࣧؕྤफ़Ĉ ၆ٺܑ̣۞εୀॡมפҋ൒၆ᇴॡ௑Ъໂࣃ̶੨Ăд

̙Т۞ᑅ˧˭̶Ҿϡ݈˘༼ٙࢗ۞͞ڱࢍზણᇴ۞Ҥࢍࣃ

׎ඕڍЕٺܑ̱̚ĄࢵА၆ٺ఺ֱᇹώߏӎ௑Ъٺֽҋໂ ࣃ̶੨ĂԧࣇᅮАઇዋЪޘ۞ᑭؠ(goodness-of-fit tests)Ą

׎Վូт˭Ĉ

઄నT₍₁₎<T₍₂₎<L<T_(n)ࠎѨԔᐌ፟ᇹώĂ׎ֽҋᇹ ώ̂̈ࠎ n ۞Ϗۢ۞ϓវ̶੨F(T)Ă΄F(T)ࠎপؠ۞௢

᎕̶੨בᇴĂԧࣇ۞ϫ۞ߏࢋᑭؠĈ ) ( ) (

0:F T F T

H = _E

၆ϲ ၆ٙѣ T Ă

) ( ) (

1:F T F T

H ≠ _E

҂ᇋ Cramer-Von Mises ௚ࢍณ (Kendall ᄃ Stuart[11]ć Stephens[12])Ĉ

n n

t i F

W ⁿ

i E i

n 12

] 1 5 . ) 0 (

[ ²

1 ()

2=∑ − − +

=

၆ٺα჌̙Тᑅ˧ٺ˛჌Ҥࢍ͞ڱٙ̚଀̝Ҥࢍณ̶ҾՐ

଀W Еٺܑ˛̚Ą _n²

ͽ ˯W_n² Ӯ ̈ ٺW_c²=0.461 ( ༊ ព ඾ ௚ ࢍ ͪ ໤ 05

.

=0

α ॡW ̝ᓜࠧࣃ² W_c²=0.461) (Kececioglu)Ą߇α

჌ᑅ˧۞ྤफ़פҋ൒၆ᇴޢӮ௑Ъໂࣃ̶੨Ą

Ϥٺ(x_i−α)/β ׍ѣᇾ໤޽ᇴ̶੨Ă׎ഇ୕ࣃࠎ 1Ą

̫ࢍზЧ჌͞ڱдα჌ᑅ˧ٙ଀Ҥࢍࣃαˆ Ăβˆ ̶Ҿࢍზ β

α)/

(x_i− ̝׶۞πӮЕٺܑˣĄ

̣ăඕ ኢ

Bergman(1986)͞ڱϡдໂࣃ̶੨˯ĂពϯӀϡΐᝋ౵

̈π͞ڱ̙ͧϡΐᝋ۞౵̈π͞ڱٙՐ଀ҤࢍࣃົՀჟቁ ᄃৌࣃमளՀ̈Ă҃ٙՐ΍۞Ҥࢍࣃдn≥20ॡӮ͞ᄱ MSE ͽາ͞ڱࠎ౵ָĄдᇹώn<20ॡ Bergman ۞͞ڱᄃ າ͞ڱ֭՟ѣព඾۞मள (πӮ۞Ӯ͞ᄱ MSE ࠹मд 5%

ͽ̰)Ă൒҃ Bergman ۞͞ڱ̪ᅮҿᕝ c ࣃ۞פ଺Ăԧࣇ۞

ܑ˟ дໂࣃ̶੨̚ϓᇴࠎα=10,β =0.5۞ሀᑢҤࢍඕڍ(πӮ ± ᇾ໤मE(βˆ)±sˆ(βˆ)ă(Ӯ͞म) (MSˆE(βˆ)))

͞ ڱ

OLS Bergman ώ͛͞ڱ

n 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 0.6791±0.2830 0.5880±0.2431 0.5215±0.2140 0.6891±0.2627 0.6120±0.2350 0.5610±0.2203 0.6386±0.2441 6 (0.3349) (0.2586) (0.2151) (0.3237) (0.2603) (0.2286) (0.2806)

0.6538±0.2382 0.5779±0.2085 0.5217±0.1865 0.6534±0.2111 0.5929±0.1945 0.5535±0.1868 0.6149±0.1983 8 (0.2836) (0.2225) (0.1877) (0.2610) (0.2156) (0.1943) (0.2292)

0.6363±0.2086 0.5705±0.1849 0.5213±0.1673 0.6301±0.1788 0.5807±0.1683 0.5491±0.1637 0.5993±0.1684 10 (0.2492) (0.1979) (0.1686) (0.2211) (0.1866) (0.1709) (0.1955)

0.6222±0.1877 0.5639±0.1681 0.5200±0.1534 0.6112±0.1588 0.5698±0.1517 0.5434±0.1488 0.5861±0.1492 12 (0.2240) (0.1798) (0.1547) (0.1939) (0.1670) (0.1549) (0.1722)

0.6109±0.1723 0.5582±0.1556 0.5183±0.1429 0.5961±0.1451 0.5603±0.1401 0.5377±0.1381 0.5755±0.1353 14 (0.2049) (0.1661) (0.1441) (0.1740) (0.1526) (0.1431) (0.1549)

0.6042±0.1594 0.5558±0.1448 0.5190±0.1336 0.5862±0.1319 0.5550±0.1282 0.5352±0.1266 0.5689±0.1228 16 (0.1904) (0.1551) (0.1350) (0.1576) (0.1395) (0.1314) (0.1408)

0.5963±0.1491 0.5515±0.1362 0.5175±0.1263 0.5768±0.1233 0.5490±0.1205 0.5315±0.1193 0.5617±0.1141 18 (0.1175) (0.1456) (0.1275) (0.1452) (0.1301) (0.1234) (0.1297)

0.5897±0.1399 0.5481±0.1283 0.5163±0.1195 0.5693±0.1151 0.5444±0.1129 0.5286±0.1120 0.5562±0.1062 20 (0.1162) (0.1371) (0.1206) (0.1344) (0.1213) (0.1156) (0.1201)

0.5688±0.1111 0.5371±0.1036 0.5127±0.0978 0.5457±0.0921 0.5293±0.0913 0.5190±0.0908 0.5381±0.0835 30 (0.1307) (0.1101) (0.0986) (0.1028) (0.0959) (0.0928) (0.0918)

0.5572±0.0957 0.53112±0.0901 0.5110±0.0857 0.5345±0.0786 0.5224±0.0781 0.5147±0.0778 0.5292±0.0709 40 (0.1115) (0.0953) (0.0864) (0.0858) (0.0812) (0.0792) (0.0767)

0.5495±0.0838 0.5271±0.0794 0.5097±0.0759 0.5277±0.0693 0.5183±0.0690 0.5121±0.0688 0.5238±0.0623 50 (0.0973) (0.0839) (0.0766) (0.0747) (0.0714) (0.0699) (0.0667)

0.5316±0.0573 0.5177±0.0553 0.5068±0.0536 0.5144±0.0482 0.5097±0.0481 0.5066±0.0480 0.5125±0.0431 100 (0.0655) (0.0580) (0.0541) (0.0503) (0.0491) (0.0485) (0.0449)

ܑˬ дໂࣃ̶੨̚ϓᇴࠎα=10,β =1۞ሀᑢҤࢍඕڍ(πӮ±ᇾ໤म E(βˆ)±sˆ(βˆ)ă(Ӯ͞म) (MSˆE(βˆ)))

͞ ڱ

OLS Bergman ώ͛͞ڱ

n 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 1.3582±0.5660 1.1760±0.4862 1.0430±0.4281 1.783±05253 1.2240±0.4700 1.1219±0.4406 1.2771±0.4881 6 (0.6698) (0.5171) (0.4303) (0.6474) (0.5206) (0.4572) (0.5613)

1.3077±0.4764 1.1557±0.4169 1.0433±0.3729 1.3067±0.4223 1.1857±0.3891 1.1069±0.3736 1.2299±0.3967 8 (0.5671) (0.4451) (0.3754) (0.5219) (0.4311) (0.3886) (0.4585)

1.2726±0.4172 1.1410±0.3699 1.0427±0.3346 1.2601±0.3577 1.1615±0.3366 1.0982±0.3274 1.1985±0.3368 10 (0.4983) (0.3958) (0.3373) (0.4423) (0.3733) (0.3418) (0.3910)

1.2444±0.3754 1.1277±0.3363 1.0399±0.3067 1.2224±0.3176 1.1396±0.3050 1.0867±0.2975 1.1722±0.2983 12 (0.4480) (0.3597) (0.3093) (0.3877) (0.3340) (0.3099) (0.3445)

1.2218±0.3446 1.1164±0.3111 1.0367±0.2858 1.1921±0.2901 1..1207±0.2802 1.0754±0.2761 1.1509±0.2707 14 (0.4098) (0.3322) (0.2881) (0.3479) (0.3051) (0.2862) (0.3099)

1.2084±0.3188 1.1116±0.2895 1.0380±0.2673 1.1726±0.2639 1.1100±0.2565 1.0703±0.2533 1.1378±0.2457 16 (0.3808) (0.3103) (0.2700) (0.3153) (0.2790) (0.2629) (0.2817)

1.1925±0.2983 1.1031±0.2724 1.0350±0.2526 1.1536±0.2466 1.0981±0.2411 1.0630±0.2387 1.1234±0.2282 18 (0.3550) (0.2913) (0.2550) (0.2905) (0.2603) (0.2469) (0.2594)

1.1794±0.2798 1.0962±0.2567 1.0326±0.2390 1.1386±0.2302 1.0888±0.2259 1.0572±0.2240 1.1123±0.2124 20 (0.3324) (0.2741) (0.2412) (0.2687) (0.2427) (0.2312) (0.2403)

1.1376±0.2223 1.0742±0.2072 1.0254±0.1955 1.0914±0.1843 1.0587±0.1825 1.0379±0.1816 1.0762±0.1670 30 (0.2614) (0.2201) (0.1972) (0.2057) (0.1917) (0.1855) (0.1835)

1.1145±0.1914 1.0623±0.1801 1.0219±0.1713 1.0691±0.1572 1.0448±0.1561 1.0294±0.1556 1.0585±0.1419 40 (0.2230) (0.1906) (0.1727) (0.1717) (0.1625) (0.1583) (0.1535)

1.0991±0.1676 1.0542±0.1588 1.0194±0.1519 1.0556±0.1387 1.0365±0.1380 1.0241±0.1376 1.0476±0.1246 50 (0.1947) (0.1678) (0.1531) (0.1494) (0.1427) (0.1397) (0.1333)

1.0632±0.1147 1.0354±0.1105 1.0137±0.1072 1.0287±0.0964 1.0194±0.0962 1.0132±0.0961 1.0250±0.0863 100 (0.1309) (0.1160) (0.1081) (0.1006) (0.0981) (0.0970) (0.0898)

ܑα дໂࣃ̶੨̚ϓᇴࠎα=10,β =5۞ሀᑢҤࢍඕڍ(πӮ±ᇾ໤मE(βˆ)±sˆ(βˆ)ă(Ӯ͞म) (MSˆE(βˆ)))

͞ ڱ

OLS Bergman ώ͛͞ڱ

n 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 6.7911±2.8298 5.8801±2.4312 5.2150±2.1405 6.8914±2.6267 6.1202±2.3497 5.6097±2.2032 6.3855±2.4406 6 (3.3490) (2.5856) (2.1513) (3.2368) (2.6031) (2.2860) (2.8064)

6.5383±2.3820 5.7786±2.0847 5.2165±1.8647 6.5335±2.1115 5.9282±1.9456 5.5345±1.8679 6.1494±1.9835 8 (2.8356) (2.2254) (1.8772) (2.6096) (2.1557) (1.9428) (2.2924)

6.3628±2.0858 5.7049±1.8495 5.2133±1.6728 6.3006±1.7884 5.8075±1.6828 5.4909±1.6368 5.9925±1.6840 10 (2.4916) (1.9792) (1.6863) (2.2113) (1.8665) (1.7089) (1.9548)

6.2222±1.8772 5.6386±1.6813 5.1997±1.5336 6.1120±1.5880 5.6977±1.5173 5.4336±1.4875 5.8612±1.4917 12 (2.2400) (1.7985) (1.5466) (1.9386) (1.6700) (1.5494) (1.7224)

6.1089±1.7231 5.5819±1.5557 5.1834±1.4289 5.9604±1.4506 5.6036±1.4011 5.3770±1.3804 5.7546±1.3533 14 (2.0491) (1.6610) (1.4406) (1.7397) (1.5256) (1.4310) (1.5495)

6.0419±1.5938 5.5578±1.4477 5.1902±1.3364 5.8630±1.3193 5.5500±1.2823 5.3517±1.2265 5.6889±1.2284 16 (1.9041) (1.5515) (1.3498) (1.5764) (1.3953) (1.3144) (1.4083)

5.9627±1.4914 5.5155±1.3620 5.1748±1.2630 5.7679±1.2330 5.4905±1.2054 5.3149±1.1935 5.6170±1.1409 18 (1.7751) (1.4563) (1.2750) (1.4525) (1.3013) (1.2344) (1.2907)

5.8972±1.3989 5.4809±1.2835 5.1629±1.1948 5.6928±1.1511 5.4438±1.1293 5.2860±1.1199 5.5615±1.0621 20 (1.6619) (1.3706) (1.2058) (1.3435) (1.2134) (1.1558) (1.2014)

5.6880±1.1115 5.3711±1.0362 5.1269±0.9777 5.4573±0.9212 5.2938±0.9125 5.1897±0.9081 5.3807±0.8350 30 (1.3072) (1.1007) (0.9859) (1.0285) (0.9586) (0.9277) (0.9177)

5.5723±0.9568 5.3115±0.9007 5.1096±0.8567 5.3453±0.7859 5.2243±0.7807 5.1468±0.7778 5.2924±0.7094 40 (1.1149) (0.9530) (0.8637) (0.8584) (0.8123) (0.7915) (0.7673)

5.4953±0.8379 5.2712±0.7939 5.0972±0.7594 5.2781±0.6933 5.1823±0.6900 5.1206±0.6881 5.2378±0.6228 50 (0.9733) (0.8390) (0.7656) (0.7470) (0.7137) (0.6986) (0.6667)

5.3161±0.5733 5.1768±0.5526 5.0684±0.5362 5.1434±0.4821 5.0968±0.4810 5.0664±0.4803 5.1249±0.4315 100 (0.6546) (0.5802) (0.5405) (0.5030) (0.4907) (0.4849) (0.4492)

ܑ̣ ᐀ඏᇹώдα჌̙Тᑅ˧۞εୀॡม

ᑅ˧ εୀॡมѨԔ

0.87 1.67, 2.20, 2.51, 3.00, 3.90, 4.70, 7.53, 14.70, 27.8, 37.4 0.99 0.80, 1.00, 1.37, 2.25, 2.95, 3.70, 6.07, 6.65, 7.05, 7.37 1.09 0.012, 0.18, 0.20, 0.24, 0.26, 0.32, 0.32, 0.42, 0.44, 0.88 1.18 0.073, 0.098, 0.117, 0.135, 0.175, 0.262, 0.270, 0.350, 0.386, 0.456

ܑ̱ α჌̙Тᑅ˧ٺ˛჌͞ڱ̝Ҥࢍณ

͞ ڱ

OLS Bergman ώ͛͞ڱ

ᑅ˧ 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 0.87 αˆ

β^{ˆ 2.37095} 1.17675

2.34628 1.06681

2.33028 0.98523

2.25431 1.49115

2.20594 1.41125

2.17231 1.35865

2.17549 1.34600 0.99 αˆ

β^{ˆ 1.52044} 0.86091

1.50054 0.77693

1.48727 0.71457

1.56800 0.97105

1.55776 0.93289

1.55290 0.89680

1.44378 0.86699 1.09 αˆ

β^{ˆ -0.82534} 1.28494

-0.86774 1.13534

-0.89903 1.02260

-0.98312 0.86387

-1.01616 0.71703

-1.03959 0.61345

-1.07856 0.95801 1.18 αˆ

β^{ˆ -1.30989} 0.63973

-1.32471 0.57727

-1.33466 0.53079

-1.29962 0.73049

-1.31283 0.69336

-1.32085 0.67160

-1.37330 0.65560

ܑ˛ Ч჌͞ڱٙ଀Ҥࢍ̶੨̝ Cramer-Von Mises ௚ࢍณ

͞ ڱ

OLS Bergman ώ͛͞ڱ

ᑅ˧ 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 0.87 0.0916 0.0983 0.1102 0.1051 0.0980 0.0944 0.0925 0.99 0.0438 0.0434 0.0494 0.0558 0.0508 0.0473 0.0504 1.09 0.2030 0.1723 0.1478 0.1124 0.0780 0.0541 0.1579 1.18 0.0302 0.0325 0.0410 0.0394 0.0340 0.0318 0.0363

ܑˣ α჌ᑅ˧˭Ч჌͞ڱٙ଀Ҥࢍࣃαˆ Ăβˆ ̶ҾՐზ ∑ −

= n i

xi

n 1 ˆ 1 ˆ

β α

͞ ڱ

OLS Bergman ώ͛͞ڱ

ᑅ˧ 1

c = 0.0

2 c = 0.3

3 c = 0.5

4 c = 0.0

5 c = 0.3

6 c = 0.5

7 WLS 0.87 0.92141 0.97761 1.03386 0.94700 0.99224 1.02860 1.02945 0.99 0.86970 0.90702 0.84292 0.81791 0.82809 0.83462 0.94923 1.09 0.76551 0.78402 0.80408 0.88960 0.96965 1.07413 0.96998 1.18 0.89305 0.93840 0.98313 0.86254 0.88457 0.89979 0.97921

͞ڱݒ̙֭ᅮࢋՙؠ c ࣃĄд၁ּ੅ኢ̚Ăͽໂࣃ̶੨

̚׍ѣ޽ᇴপّ۞ّኳֽ࠻Ăα௡ᑅ˧۞ྤफ़Ă̂ౌځ ព۞ܑϯώ͛ٙ೩۞͞ڱѣྵָ۞ඕڍĄТॡԧࣇ۞͞

ڱ̙ညϡ౵̂ໄҬҤࢍ͞ڱ(maximum likelihood estima- tion method; MLE)૱ืϡᝑ΃ڱՐࣃĂԧࣇ۞ٙ೩͞ڱ

ۡତྋĂͷ Bergman(1986)˵ϡ࠹Т۞͞ё૟΁۞͞ڱᄃ MLE ۞͞ڱͧĂ׎ѣड़ّ˵࠹༊Ă߇ώ͛۞າ͞ڱᑕߏ Ъآ۞Ą

௑ཱི৶͔

) (x

E x ̝πӮᇴ

) (x

F ໂࣃ̶੨̝௢᎕̶οבᇴ

) ˆ x(

F F(x)̝Ҥࢍณ E

S

M ˆ Ӯ͞ᄱ̝Ҥࢍณ

Sˆ ᇾ໤म̝Ҥࢍณ

) (x

Var x ̝តளᇴ

W i ΐᝋЯ̄

2

W Cramer-Von n Mises ௚ࢍณ Xn

X

X₁, ₂,L, ໂࣃ̶੨̝ᐌ፟ᇹώ

) ( ) 2 ( ) 1

( ,X , ,X _n

X L ࠎX₁,X₂,L,X_n࠹၆ᑕ۞ѨԔ௚ࢍณ

)

Y(i Z_(i)̝ҋ൒၆ᇴבᇴ

) ( ) 2 ( ) 1

( ,Z , ,Z_n

Z L ࠎX₍₁₎,X₍₂₎,L,X_(n)࠹၆ᑕ۞ᇾ໤޽ᇴ

̶੨ѨԔ௚ࢍณ

α Ҝཉϓᇴ

αˆ α̝Ҥࢍณ

β ͎ޘϓᇴ

βˆ β̝Ҥࢍณ

ણ҂͛ᚥ

1. Bergman, B., “Estimation of Weibull Parameters Using a Weight Function,” Journal of Materials Science Letters,

Vol. 5, pp. 611-614 (1986).

2. Kececioglu, D., Reliability & Life Testing Handbook, Prentice-Hall Inc., Vol. 1, New Jersey (1993).

3. Lawless, J. F., Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York (1982).

4. Gumbel, E. J., Statistics of Extremes, Columbia University Press, New York (1958).

5. D’Agostino, R. B., and Stephens, M. A., Goodness-of-Fit Techniques, Marcel Dekker Inc., New York (1986).

6. Faucher, B., and Tyson, W. R., “On the Determination of Weibull Parameters,” Journal of Materials Science Letters, Vol. 7, pp. 1199-1203 (1988).

7. Drapella, A., and Kosznik, S., “An Alternative Rule for Placement of Empirical Point on Weibull Probability Paper.,” Quality and Reliability Engineering Interna- tional ,Vol. 15, pp. 57-59 (1999).

8. Balakrishnan, N., and Cohen, A. C., Order Statistics and Inference, John Wiley & Sons., New York , pp. 35 (1991).

9. Bickel, P. J., and Doksum, K. A., Mathematical Statistics-Basic Ideas and Selected Topics, Prentice-Hall Inc., New Jersey, Vol. 1, pp. 113 (2001).

10. McCool, J. I., “Confidence Limits for Weibull Regression with Censored Data,” IEEE Transaction. Reliab, R29, pp.

145-150 (1980).

11. Kendall, M. G., and Stuart, A., The Advanced Theory of Statistics, Vol. 3, 2^nd ed, Landon (1986).

12. Stephens, M. A., “EDF Statistics for Goodness of Fit and Some Comparisons,” Journal American Statistics As- sociation, Vol. 69, pp. 730-737 (1974).

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