Monograph ΚΑΤΑΝΟΜΕΣ ΖΩΗΣ

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Εκθετική, Weibull, Pareto, Λογαριθμοκανονική, Γάμμα και Γεωμετρική

Transcript of Monograph ΚΑΤΑΝΟΜΕΣ ΖΩΗΣ

, (Survival Analysis). , (censored data), , , , . , ( / life or failure or survival time), ., : , Weibull, Pareto, , . : .

- . : ( ), (, , , ), .. survival, failure life time. , . , . . , .

. , , ., , () . , . . , 100 , . , , , .

(), , (). , 60 . . . (life distributions) (, Weibull, , ..). , , .

, (life distributions) :

Weibull Pareto

, : (, , , ...) , . , , . (reliability theory).

. , , 0. ( .. 1, 2 ,, ) ( (0,)). , , . , :

: .. , .

: .. (, , ), . : .. , , , , . U (censoring time), ( ).

(censored failure time random variable), = min{,U}. , .

. , .

Hald (1949). .1.

2.

3. (failure time) 4.

(exact data). (staggered entry). (censored failure data). 3, (censoring time), 4, / . (survival data)=

T =

(right censored data) (left censored data) (interval censored data)

(Right Censored Data) . , . , Ui ( ). .. Xi = min{Ti,Ui} (failure indicator) :i =

. , (censoring indicator) :ci =

(Left Censored Data) , . , Ui ( ). .. Xi = max{Ti, Ui} (failure indicator) :i =

.

: ( ). Stanford ( ). , . (left-censored observations). ( ). , , (right-censored observations).

(Interval Censored Data)

. , . , . Li, Ri (Li < Ri), Ti (Li, Ri).

1: . 2: ( ) . 3 .

(Truncated Data)

, ( ), . : : ( , . : , . , , . : ( - failure) , . : , , . .

n :(T1, U1), (T2, U2), , (Tn, Un) Ui ( ) Ti i . 3 () :

(Type I Censoring)

(Type II Censoring)

(Random Censoring)

T (Type I Censoring)

Ui . .. , 2 (Ui=2). , U=Ui ( i). ( ), / . , , U ( ). : ( - ). 100 (U = 10). ( > U =10 ). .

, , ( ) Ui. .

: ( ??? - ).

( D.J. Bartholomew) : ( ). 31 . 31 . :/ // i Ui

111-13-281

221---71

322-12-5170

42---60

521-23-3341

631-27-2731

731-14-1431

81-25-2430

92-6-429

1010---21

1 () U1=81 , ).

(31-) 81 . 2 U2=71 ( ). .. Xi = min{Ti,Ui} (failure indicator) :i =

. , 10 , :

X1 = min{T1,U1} = min{2,81} = 2 1=1X2 = min{T2,U2} = min{-,71} = 71 1=0X3 = min{T3,U3} = min{51,70} = 51 1=1X4 = min{T4,U4} = min{-,60} = 60 1=0X5 = min{T5,U5} = min{33,41}= 33 1=1X6 = min{T6,U6} = min{27,31}= 27 1=1X7 = min{T7,U7} = min{14,31}= 14 1=1X8 = min{T8,U8} = min{24,30}= 24 1=1X9 = min{T9,U9} = min{4,29} = 4 1=1X10 = min{T10,U10}= min{-,21} = 21 1=0 :

2, 71+, 51, 60+, 33, 27, 14, 24, 4, 21+ I (Type I Censoring)

Ui = T(r), r . .. 4/6 . . . , rN, r0, (t=T2) , (t=T2+). , 3, t>0, (t=T3), (t=T3+).

, . , . & :

( ) Ui, Ti. 1 : Ui ( 2 ), Ti ( ).

2 : Ui / , Ui Ti .

, . . , , .. , Ui Ti. . U U : U , , U . & (informative censoring) : ( ) Ui Ti.

& :

, , , . , , . Weibull, , ..

( )

0

( )

T

T

T

T

..

2+

3+

1

t=0

..

:

Ui Ti

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