Zastosowanie stochastycznego modelu Cairnsa-Blake'a-Dowda do prognozowania oczekiwanej długości trwania życia

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(37) . =DVWRVRZDQLHVWRFKDVW\F]QHJRPRGHOX«. 0RGHO/HH&DUWHUD $XWRU]\]DSURSRQRZDOLSURVW\PRGHORSLVXMćF\]PLDQ\XPLHUDOQRŋFLZF]DVLH :PHWRG]LHWHMORJDU\WPQDWĕŧHQLD]JRQyZMHVWRSLVDQ\SU]H]VXPĕGZyFKVNãDG QLNyZ]NWyU\FKMHGHQQLH]DOHŧ\RGF]DVXDGUXJLMHVWLORF]\QHPSDUDPHWUXRNUH ŋODMćFHJRRJyOQ\SR]LRPXPLHUDOQRŋFLRUD]SDUDPHWUXZVND]XMćFHJRMDNV]\ENR ]PLHQLDVLĕXPLHUDOQRŋþRVyEZGDQ\PZLHNXZ]DOHŧQRŋFLRG]PLDQRJyOQHM XPLHUDOQRŋFL3DUDPHWU\WHJRPRGHOXVćHVW\PRZDQH]GDQ\FKKLVWRU\F]Q\FK 0RGHO/HH&DUWHUDPDSRVWDþ log mx ( t ) = β(x1) + β(x2 )κ (t 2 ) ,. .

(38). JG]LH (1) β(x1)²RV]DFRZDQHZDUWRŋFLVćUyZQHŋUHGQLHM]ORJP[ W

(39) SRF]DVLHWGODWHJReβ x UHSUH]HQWXMHRJyOQ\NV]WDãW]PLDQSR]LRPXXPLHUDOQRŋFLF]\OLRNUHŋODXŋUHGQLRQć ZF]DVLHWNU]\ZćVWDU]HQLDVLĕRVyE β(x2 )²SDUDPHWUSU]HGVWDZLDMćF\]PLDQ\XPLHUDOQRŋFLRVyEZZLHNX[RNUH ŋODMćF\ZUDŧOLZRŋþORJDU\WPXQDWĕŧHQLDXPLHUDOQRŋFLRVyEZZLHNX[QD]PLDQ\ ZF]DVLHLQGHNVXκ (t 2 )RUD]Z]JOĕGQćV]\ENRŋþ]PLDQXPLHUDOQRŋFLRVyEZZLHNX[ ZF]DVLHW κ (t 2 )²SDUDPHWURSLVXMćF\RJyOQćWHQGHQFMĕ]PLDQXPLHUDOQRŋFLZF]DVLHW :PRGHOX]DNãDGDVLĕSRQDGWRŧH tn. . ∑ κ (t 2) = 0RUD]∑x β(x2) = 1GODNDŧGHJRW. WW«WQL[ [[«[P

(40). t =t1. JG]LH Q²OLF]EDREVHUZRZDQ\FKODW NDOHQGDU]RZ\FK

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(43) . log mx ( t ) = β(x1) + β(x2 )κ (t 2 ) + β(x3) γ (t 3−)x .

(44).

(45) . 0RQLND3DSLHŧ. :PRGHOX]DNãDGDVLĕSRQDGWRŧH tn. (2). ∑ κt. (2). ∑ βx. = 0,. t =t1. . tn. = 1,. x. ∑ γ (t 3−)x = 0, ∑ β(x3) = 1.. t =t1.

(46). x. 0RGHO$JH3HULRG&RKRUW 0RGHO$JH3HULRG&RKRUWMHVWRNUHŋORQ\UyZQDQLHP log mx ( t ) = β(x1) + κ (t 2 ) + γ (t 3−)x. .

(47). :PRGHOX]DNãDGDVLĕSRQDGWRŧH tn. (2). ∑ κt. t =t1. . tn. = 0,. ∑ γ (t 3−)x = 0..

(48). t =t1. 0RGHO&DLUQVD%ODNH·D'RZGD 0RGHO&DLUQVD%ODNH·D'RZGD &%'

(49) MHVWRSLVDQ\Z]RUHP logit qx ( t ) = β(x1)κ (t1) + β(x2 )κ (t 2 ). .

(50). :FHOXXSURV]F]HQLDPRGHOXDXWRU]\]DSURSRQRZDOLŧHPRŧQD]DãRŧ\þLŧ 1 β(x1) = 1RUD]β(x2 ) = ( x − x )JG]LHx = ∑ i xiMHVWŋUHGQLć]SUyE\>&DLUQV%ODNH n L'RZG@:yZF]DVPRGHO&%'SU]\MPXMHSRVWDþ logit qx ( t ) = κ (t1) + κ (t 2 ) ( x − x ). .

(51). 5R]V]HU]RQ\PRGHO&DLUQVD%ODNH·D'RZGD 5R]V]HU]RQ\PRGHO&DLUQVD%ODNH·D'RZGDRNUHŋODZ]yU logit qx ( t ) = β(x1)κ (t1) + β(x2 )κ (t 2 ) + β(x3) γ (t 3−)x. .

(52). :FHOXXSURV]F]HQLDPRGHOXDXWRU]\]DSURSRQRZDOLŧHPRŧQD]DãRŧ\þLŧ β x = 1, β(x2 ) = ( x − x ) , β(x3) = 1>&DLUQV%ODNHL'RZG@:yZF]DVUR]V]HU]RQ\ PRGHO&%'SU]\ELHUDSRVWDþ (1). . logit qx ( t ) = κ (t1) + κ (t 2 ) ( x − x ) + γ (t 3−)x.

(53).

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(62) . a). –3,4 –3,6. Wartości ln mx. –3,8 –4,0 –4,2 –4,4 –4,6 –4,8 –5,0 –5,2 –5,4 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005. a). –3,4 –3,6. Wartości ln mx. –3,8 –4,0 –4,2 –4,4 –4,6 –4,8 –5,0 –5,2 –5,4 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005. 5\V:DUWRŋFLORJDU\WPXFHQWUDOQHJRZVSyãF]\QQLND]JRQyZNRELHW D

(63) LPĕŧF]\]Q E

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