Συγκριτική...

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  • e-mail: bozikas@unipi.gr

    , 4-7 2016

    . , ..

  • I ( , )

    I , 50 (1961-2010): 70.2 78 73.8 83.3

    I

    I

    . , ..

  • I ( , )

    I , 50 (1961-2010):

    70.2 78 73.8 83.3

    I

    I

    . , ..

  • I ( , )

    I , 50 (1961-2010): 70.2 78 73.8 83.3

    I

    I

    . , ..

  • I ( , )

    I , 50 (1961-2010): 70.2 78 73.8 83.3

    I

    I

    . , ..

  • I ( , )

    I , 50 (1961-2010): 70.2 78 73.8 83.3

    I

    I

    . , ..

  • I 3 :

    1 2

    . , ..

  • I 3 :1

    2

    . , ..

  • I 3 :1 2

    . , ..

  • 1

    2

    3

    . , ..

  • 1 2

    3

    . , ..

  • 1 2

    3

    . , ..

  • 1 2

    3

    . , ..

  • 1 2

    3

    . , ..

  • I

    : Human Mortality Database (2015) x : 60-94 t: 1981-2010 t x ( 5 ): 1887-1891 1946-1950

    I

    .. : Dx,t : Ex,t() : E 0x,t Ex,t + 1/2 : x,t = dx,t/Ex,t ( mx,t) : qx,t = dx,t/E

    0x,t

    . , ..

  • I

    : Human Mortality Database (2015) x : 60-94 t: 1981-2010 t x ( 5 ): 1887-1891 1946-1950

    I

    .. : Dx,t : Ex,t() : E 0x,t Ex,t + 1/2 : x,t = dx,t/Ex,t ( mx,t) : qx,t = dx,t/E

    0x,t

    . , ..

  • I Lee and Carter (1992): logx ,t = x + (1)x

    (1)t (M1)

    I Renshaw and Haberman (2006): logx ,t = x + (1)x

    (1)t + tx (M2)

    I Cairns et al. (2006): logit qx ,t = (1)t + (x x)

    (2)t (M3)

    , logit qx,t = logqx,t

    1 qx,tI :

    x : x

    (i)t : t

    (1)x : ,

    tx : t x

    . , ..

  • I Lee and Carter (1992): logx ,t = x + (1)x

    (1)t (M1)

    I Renshaw and Haberman (2006): logx ,t = x + (1)x

    (1)t + tx (M2)

    I Cairns et al. (2006): logit qx ,t = (1)t + (x x)

    (2)t (M3)

    , logit qx,t = logqx,t

    1 qx,tI :

    x : x

    (i)t : t

    (1)x : ,

    tx : t x

    . , ..

  • I Lee and Carter (1992): logx ,t = x + (1)x

    (1)t (M1)

    I Renshaw and Haberman (2006): logx ,t = x + (1)x

    (1)t + tx (M2)

    I Cairns et al. (2006): logit qx ,t = (1)t + (x x)

    (2)t (M3)

    , logit qx,t = logqx,t

    1 qx,t

    I :

    x : x

    (i)t : t

    (1)x : ,

    tx : t x

    . , ..

  • I Lee and Carter (1992): logx ,t = x + (1)x

    (1)t (M1)

    I Renshaw and Haberman (2006): logx ,t = x + (1)x

    (1)t + tx (M2)

    I Cairns et al. (2006): logit qx ,t = (1)t + (x x)

    (2)t (M3)

    , logit qx,t = logqx,t

    1 qx,tI :

    x : x

    (i)t : t

    (1)x : ,

    tx : t x . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)

    I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • I R

    I

    I M1 M2 :

    .. Dx ,t Poisson(Ex ,t x ,t)I M3:

    .. Dx ,t Binomial(E 0x ,t , qx ,t)

    . , ..

  • - M1

    60 65 70 75 80 85 90 95

    4.5

    3.5

    2.5

    1.5

    x vs. x

    age

    60 65 70 75 80 85 90 95

    0.00

    0.01

    0.02

    0.03

    0.04

    x(1)

    vs. x

    age

    1980 1990 2000 2010

    86

    42

    02

    4

    t(1)

    vs. t

    year

    60 65 70 75 80 85 90 95

    54

    32

    x vs. x

    age

    60 65 70 75 80 85 90 95

    0.01

    0.02

    0.03

    0.04

    x(1)

    vs. x

    age

    1980 1990 2000 2010

    10

    50

    5

    t(1)

    vs. t

    year

    . , ..

  • - M2

    60 65 70 75 80 85 90 95

    4.5

    3.5

    2.5

    1.5

    x vs. x

    age

    60 65 70 75 80 85 90 95

    0.015

    0.025

    0.035

    x(1) vs. x

    age

    1980 1990 2000 2010

    64

    20

    24

    t(1)

    vs. t

    year

    1890 1910 1930 19500

    .200

    .100.0

    00.1

    0

    tx vs. tx

    cohort

    . , ..

  • - M2

    60 65 70 75 80 85 90 95

    4.5

    3.5

    2.5

    x vs. x

    age

    60 65 70 75 80 85 90 95

    0.020

    0.030

    0.040

    x(1) vs. x

    age

    1980 1990 2000 2010

    50

    5

    t(1)

    vs. t

    year

    1890 1910 1930 19501

    .00

    .50.0

    0.5

    tx vs. tx

    cohort

    . , ..

  • - M3

    1980 1985 1990 1995 2000 2005 2010

    3.0

    2.9

    2.8

    2.7

    t(1)

    vs. t

    year

    1980 1985 1990 1995 2000 2005 2010

    0.096

    0.100

    0.104

    0.108

    t(2)

    vs. t

    year

    1980 1985 1990 1995 2000 2005 2010

    3.5

    3.4

    3.3

    3.2

    3.1

    3.0

    t(1)

    vs. t

    year

    1980 1985 1990 1995 2000 2005 2010

    0.120

    0.125

    0.130

    0.135

    0.140

    t(2)

    vs. t

    year

    . , ..

  • 1

    2

    3

    . , ..

  • I

    I

    I

    ,

    . , ..

  • - M1

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 2010

    32

    10

    12

    3calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 2010

    32

    10

    12

    3

    calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    I

    . , ..

  • - M2

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 2010

    32

    10

    12

    3calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 2010

    32

    10

    12

    3

    calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    . , ..

  • - M3

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 20103

    21

    01

    23

    calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    60 65 70 75 80 85 90 95

    32

    10

    12

    3

    age

    resid

    uals

    1980 1990 2000 2010

    32

    10

    12

    3

    calendar year

    resid

    uals

    1890 1910 1930 1950

    32

    10

    12

    3

    year of birth

    resid

    uals

    I ( )I

    . , ..

  • I AIC = 2k 2 log L, k

    I AIC (c) = AIC +2k(k + 1)

    n k 1, n

    I BIC = (log n)k 2 log L

    AIC(c) BICM1 -5,472.27 98 11,162(2) 11,623(3)M2 -5,080.40 151 10,516(1) 11,207(1)M3 -5,573.81 60 11,275(3) 11,563(2)

    M1 -6,234.97 98 12,687(2) 13,149(2)M2 -5,171.16 151 10,697(1) 11,388(1)M3 -7,923.67 60 15,975(3) 16,263(3)

    . , ..

  • I AIC = 2k 2 log L, k

    I AIC (c) = AIC +2k(k + 1)

    n k 1, n

    I BIC = (log n)k 2 log L

    AIC(c) BICM1 -5,472.27 98 11,162(2) 11,623(3)M2 -5,080.40 151 10,516(1) 11,207(1)M3 -5,573.81 60 11,275(3) 11,563(2)

    M1 -6,234.97 98 12,687(2) 13,149(2)M2 -5,171.16 151 10,697(1) 11,388(1)M3 -7,923.67 60 15,975(3) 16,263(3)

    . , ..

  • I AIC = 2k 2 log L, k

    I AIC (c) = AIC +2k(k + 1)

    n k 1, n

    I BIC = (log n)k 2 log L

    AIC(c) BICM1 -5,472.27 98 11,162(2) 11,623(3)M2 -5,080.40 151 10,516(1) 11,207(1)M3 -5,573.81 60 11,275(3) 11,563(2)

    M1 -6,234.97 98 12,687(2) 13,149(2)M2 -5,171.16 151 10,697(1) 11,388(1)M3 -7,923.67 60 15,975(3) 16,263(3)

    . , ..

  • I AIC = 2k 2 log L, k

    I AIC (c) = AIC +2k(k + 1)

    n k 1, n

    I BIC = (log n)k 2 log L

    AIC(c) BICM1 -5,472.27 98 11,162(2) 11,623(3)M2 -5,080.40 151 10,516(1) 11,207(1)M3 -5,573.81 60 11,275(3) 11,563(2)

    M1 -6,234.97 98 12,687(2) 13,149(2)M2 -5,171.16 151 10,697(1) 11,388(1)M3 -7,923.67 60 15,975(3) 16,263(3)

    . , ..

  • I M1 M2 (M1 M2)

    I H0 : M1 M2H1 : M2

    I LR = 2 logL2

    L1( 2(n2n1),)