분포추정
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Transcript of 분포추정
-
2001. 11.2001. 11.
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IE, POSTECH IE, POSTECH
- - -
- () - , , MTTF -
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IE, POSTECH IE, POSTECH
(cumulative distribution function) t X F(.) t
.
(Probability Density Function)
f(x) f(x) X .
)()( tXPtF =
).()(}{}{}{ aFbFaXPbXPbXaP == ,
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IE, POSTECH IE, POSTECH
: =2: =2 0,2)( 2 = tetf t tetF 21)( =
tetFtR 2)(1)( == == t
t
eet 2
22)(
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IE, POSTECH IE, POSTECH
0 ,)()( )(1 = tettf t
)(0
1)()( tt
edxxftF == )()(1)( tetFtR ==
1)(
)(1)()(
)(1)()(
===
te
ettFtft
t
t
:
(shape parameter):
(scale parameter)
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IE, POSTECH IE, POSTECH
=0.5
=1
=3
=1.5
0 ,)()( )(1 = tettf t
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IE, POSTECH IE, POSTECH
=0.5
=1
=3
=1.5
11
1)()( = tt >1: increasing=1: constant
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IE, POSTECH IE, POSTECH
MTTFMTTF
X
X
1 )(][
00 ==== dxexdxxxfXEMTTF x
== 11][XEMTTF
)1()1()(0
1 == dxex x
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IE, POSTECH IE, POSTECH
=2, =0.0001 . -
t=10,000 ()
t=10,000 ()
t=20,000 ()
27.88625000 )5.0()0002.0/1(][ ==== XEMTTF
3679.0)10000( 110000)0001.0( 2 === = eeR tt
0002.0)1(0002.0)10000( 12 ==
0004.0)2(0002.0)20000( 12 ==
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IE, POSTECH IE, POSTECH
:
2][;][ sXVarXXE ==
F(t)
t
F(t)
t
x Y
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IE, POSTECH IE, POSTECH
-2.3 0 2.3 4.6
)(11lnlntF
Y =
tX ln=
0.1 1 10 100
Y=063%
10%
0
-2
-2
-4
2
X=0
1
F(t)
t
X=1,Y=01
T
)( TF
m m
)/11(/10 mtm += { } 2/12/10 )/11()/21( mmt m ++=
mt /10=0/1)( ttmetF =
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IE, POSTECH IE, POSTECH
n 0
(type I censored data)
12n
0
t1t2
tn
12n
0
t1t2
tnT
(type II censored data)
n
T
r
12n
0
t1t2
tnyr
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IE, POSTECH IE, POSTECH
((Maximum Likelihood Estimation)Maximum Likelihood Estimation)
f(t; ), : parameter(s) Data:
(likelihood function)
:
),...,,( 21 nttt
);();();();( 21 ntftftfdataL L=
L()
*
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IE, POSTECH IE, POSTECH
(complete or uncensored data)
);();(1
=
=n
iitfdataL
(type I censored data)
[ ] rnri
i TRtfdataL
== );();();(
1
(type II censored data)
(r: T )
[ ] rnrr
ii yRtfdataL
=
= );();();(1
( : r )ry
-
IE, POSTECH IE, POSTECH
0 ,);( = tetf t
=
= ==
n
ii
it
nn
i
t eedataL 11
);(
=
=n
iitndataL
1
ln);(ln
0ln
1
==
=
n
iit
nL
tt
nn
ii
1
1
==
=
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IE, POSTECH IE, POSTECH
(())
(type I censored data)
( ) ==
== r
ii
it
rTrnr
i
trnT eeeedataL 1)(1
);(
=
+=r
iitrTrnL
1
ln)(ln
0)(ln
1
=+=
=
r
iit
rTrnL
=+
= ri
i Trnt
r
1
)(
(type II censored data)
=
+= ri
ri yrnt
r
1
)(
=/
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IE, POSTECH IE, POSTECH
=0.1, 0.2,
lnL (, )
0 ,)(),;( )(1 = tettf t
== =
=
=
n
ii
itn
ii
nnn
i
ti etetdataL 1
1
1
1
)(1)();,(
==
++= ni
i
n
ii ttnnL
11ln)1(lnlnln
0ln
0lnlnlnlnln
1
1
111
==
=++=
=
===n
ii
i
n
ii
n
ii
n
ii
tnL
ttttnnL
=
itn
=
= ni
it
n
1
-
IE, POSTECH IE, POSTECH
(())
(type I censored data)
))((
1
1 1);,( Trntr
ii
rr eetdataL
r
ii
= = =
TrnttrrL ri
i
r
ii )(ln)1(lnlnln
11++=
==
Trnt
rL r
ii
1
1
1 )(ln =
=
=
+= ri
i Trnt
r
1)(
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IE, POSTECH IE, POSTECH
MinitabMinitab
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IE, POSTECH IE, POSTECH
MinitabMinitab (())
Weibull, uncensored Case
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IE, POSTECH IE, POSTECH
Minitab Minitab WeibullWeibull, Uncensored Case, Uncensored Case
AD* 1.488Goodness of Fit
1000 10000
1
5
10
20
304050607080909599
Weibull Probability
P
e
r
c
e
n
t
0 5000 10000 15000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Survival Function
P
r
o
b
a
b
i
l
i
t
y
0 5000 10000 15000
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
Hazard Function
R
a
t
e
0 5000 10000 15000
0.00000
0.00005
0.00010
Probability Density Function
Overview Plot for F-timeML Estimates - Complete Data
ShapeScale
MTTF
FailureCensor
2.10317576.4
6710.3
80
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IE, POSTECH IE, POSTECH
Censoring OptionsCensoring Options
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IE, POSTECH IE, POSTECH
Minitab Minitab WeibullWeibull, type I censored, type I censored
AD* 14.81Goodness of Fit
1000 10000
1
5
10
20
304050607080909599
Weibull Probability
P
e
r
c
e
n
t
0 10000 20000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Survival Function
P
r
o
b
a
b
i
l
i
t
y
0 10000 20000
0.0000
0.0001
0.0002
0.0003
Hazard Function
R
a
t
e
0 10000 20000
0.00000
0.00005
0.00010
Probability Density Function
Overview Plot for F-timeML Estimates - Type 1 (Time) Censored at 10000
ShapeScale
MTTF
FailureCensor
1.58168614.4
7731.9
62
-
IE, POSTECH IE, POSTECH
Minitab Minitab WeibullWeibull, type II censored, type II censored
AD* 14.81Goodness of Fit
1000 10000
1
5
10
20
304050607080909599
Weibull Probability
P
e
r
c
e
n
t
0 10000 20000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Survival Function
P
r
o
b
a
b
i
l
i
t
y
0 10000 20000
0.0000
0.0001
0.0002
0.0003
Hazard Function
R
a
t
e
0 10000 20000
0.00000
0.00005
0.00010
Probability Density Function
Overview Plot for F-timeML Estimates - Type 2 (Failure) Censored at 7
ShapeScale
MTTF
FailureCensor
1.58168614.4
7731.9
62
