Gompertz
∗ ∗∗Gompertz 1980~1990 ! " # $ % & ' (
Whittaker)Bayesian)Kernel * + , - . / (0 1 2 3 . /
4 5 6 7 (89: ; < - (Bootstrap)= > ? @ (A B ! " Gompertz C D (E % F G H I J K L M N O P Q R S T U . / VGompertz V VBootstrap- VW X Y Z
1.
[ \ ] ^ _ ` a b c )d e f g h i L j k l m n o p (W X Y Z q % r s t u v w x y z { | Q} ' ~ w W X (1992)(1990 s t 65 Y W X 3.3 W ( W X 53 6.2% ( 80 1.0%)V [ 2025 (65 W X % W X 9.7% (80 1.6%)V[ 2050 65 W X % 14.0% (80 3.0%)V Y W X q ` ( Q [ w ¡ W X Y Z ¢ £ r (H I ¤ ¥ ¦ § ¨ © ª « ¬ w x ® ¯ ° (± ² 1993 ³ H I ´ ¬ µ ¶ · Z ¸ ¹ (º » ¼ 65 W X 7% )([ ½ ¾ ¿ (65 W X À 1999 Á « Â8.3% ∗ ∗∗Ã O Ä Å w Æ Ç Æ È É W X Ê Ë (Ì Í Î Ï % 0.2%(W X Y Z a Ð Ñ s t 6 Ò Q} ' Ó Ô Õ d e Ö D × Ø ¹ W Ù Ú Û Ü (1999)( Ý 2025 (65 W X Þ ß % 16.6%(« à á â s t Ì Í 9.7%V2050 ã ä 24.0%(% s t Ì Í 14.0 % 1.7å Q 65 W X Þ ß æ ª (Ì Í ç M º è ^ _ b c é ê ë ì ( È I í î Âr # $ % ß ( ! " (ï " ) Ì Í ç M ð 1951 53.10(57.32) Â 1970 66.66(71.50)(8 Â1999 72.34(78.05 )QY W X Þ ß 8 Ì Í ç M ì ( 9F G Y ¢ £ ã % ñ ò (ó ô õ W (ö ÷ )L ø Ú Û f Ô ù ¸ ¹ ú û ü ý i þ £ ( w W [ p s ¢ £ Q W X " è W X ` a Y Z (è â d © é 2 # ( % ß (â v % % Gompertz ( é G « ¬ w x # $ ò GompertzC D þ 6 Ü (Kannisto, 1996)( ò ( ¹ Y ¢ £ G ! " # Q 1980Â1990 ! " # $ % $ % ( & . / Ê = > i , - ( ' ( GompertzC D ) (z º * + . / , - . / 0 ( E % H I F G J K L M N O P Q
2.
1 : # $ % 1980~1990 2 3 Ð ! " # $ ( % 80 Ý 109( % 80 Â100(4 5 6 N )1Q è % 7 8 (Exposure)[ 2 © (ß 9 U # $ [ 98 « © : 10,000W ( [ 90 © : 1,000W ( % ; < = Ý W â > é ? ³ @ A B C ( D E 9: 2 F : . / ,
- (9 Whittaker)Bayesian)Kernel - . / (0 1 2 3
Vaino Kannisto, “The advancing frontier of survival” 1980~1990 , Good quality data Acceptable quality data
. / 4 5 (8G H ! " : Gompertz C D Q
F : . / , - v 1 Miller(1946) IJ(:KÐ"(Fit)fÌ L "
(Smoothness)E %./ & ' ( KÐ"M N ./ O P µ # $ O Q R S Ð(Ì L " M N ./ O ¾ Ì L T U "Q p V W ¼ 9 : * + ./ , - 6 P X fY E , Z Q )[Whittaker./ - - z P \ KÐ"fÌ L "( ] 5 "^ , Z _ ( § Z % M = F + hS( FN KÐ"(` F a b
∑
= − = n x x x x v u w F 1 2 ) ( %KÐ" )(S N Ì L "(` F a b∑
− = ∆ =n z x x z v S 1 2 ) ( %Ì L " )( c w `x F 7 8 n x · (h%d e f (u Nx P µ # $ O (v Nx ./ O (g N R V (Difference)2(h ð 9 ¤ i M j Z G k ë 3 ./ O Q ® [./ O a l ([z = 3 % m 3(Efficient)V h a b n B KÐ" Ì L " Þ ° (h o j N ./ O o p P µ O (h o N o ° Ì L " " # (2 X P q a b 7 8 (nx)Q r O (±ð À 7 8 Q R â à %ß (80 7 8 % 1,488,902W (é 100 7 7 8 [2,300W p (7 8 Ì Í O %280,000Ë (./ s t L ¥ 7 © Í u { | Q( Ì Í 7 8 % 2,300(º æ ª ¢ £ Q) %k 2 K O O ( v w x y 4 + © z h O U h = 10)1,000) 100,000)300,000( Ì Í 7 8 Q )Q (5 p z )[h = 10 (./ 7 [ 1027 t L { | m s ( ® { } © ( è ~ © P \ h = 10Vh[ À 1,0007 : { k ( a h100,000 h = 300,000 2 Ì L m ( h100,000 h = 300,000 ./ O ° ( Whittaker./ O b z = 3(h = 100,000Q (5 p z )v w h = 10)2000)5000( Ì Í 7 8 Q )(2 3 a l % z = 3(h = 2000QWhittaker- ./ 5 6 N Q 2(Difference operator, ) (Forward difference operator), ∆f(x)= f(x+1)− f(x)
3 Borgan(1979)
z )( ! "6 Whittaker./ O P µ Þ 2 ) Whittaker graduation(wx=nx)---Japan(male) Age M o rt a lit y r a te 80 85 90 95 100 105 110 0 .2 0 .4 0 .6 0 .8 1 .0 Crude data z=3,h=10 z=3,h=1000 z=3,h=100000 z=3,h=300000 z )( ! "6 Whittaker./ O P µ Þ 2 ) Whittaker graduation(wx=nx)---Singapore(male) Age M o rt a lit y r a te 80 85 90 95 100 0 .1 0 .2 0 .3 0 .4 0 .5 Crude data z=3,h=10 z=3,h=2000 z=3,h=5000
)[Bayesian ./ - , - % â d f E % â # (Prior information)( { [f 2 3 ë # $ (Data)( V 7 ë Ý (Posterior)( â # $ + { [# $ ã Q L M N . / %ß (O P â L M N ( { # $ ( X 1 â d Q R L M N QC D P µ O u x v ? q F V & ( ) , ~ ( ~ ~ | ~ t N t B u n ⇔ f(u~|~t)=k1exp[−(u~−~t)′B−1(u~−t~)/2]Q ~ % f O ( N â # (` F C Dt t~ : Nn(m~,A) V & (m~ %d O (An×n t~ y ? @ (Covariance matrix)VBn×nu yx ? @ ( N { [ # (k 1 % F (: 1 [(2 ) ] 1/2 − = B k π n Q { [# O Bn×n p G ² 2 3 O (è % % 0 V & (è ~ Bxx=Var(ux)=ux(1−ux)/nxQR À â # O a b ([m~ , v %[ P µ O u~¥ O (ß 9 m~ a L M N O E % O Q%k Z (An×n a b %C D 1 0 , 0 , 0 , > > ≤ < = − r p p r p p Axy x y x y x y ( r = 0.9 f r = 0.5Q x x p m a b B æxx ª (%Kannisto Ç * w # $ q J q Ç * w L M N Q v w d O mx 80 7 8 % 6,171,300W (à À P µ O 807 8 Ã 1,488,902W Ë ( A Q À Bé 2 j 4(./ O ¹ â d O V v w ´ Q z Q± d O t L ¢ £ U (5 p z * )[104)105 ./ O j À 103 ./ O ( M ® W ¡ ¡ } Ã P è %Ç * w L M N 4 5 © ¡ ¡ ( P µ O º M ~ } (é Bayesian ./ - ] 1 G ( ./ ² º M ~ } Ë Q ! " ./ 5 z ¢ (r = 0.5fr = 0.9 £ d O 4 5 ° ( ./ ¤ ¢ £ Qè ~ ( f ! " Bayesian ./ O â d (© Ì L f ¥ { } Q x y y x xy p p r A = − x x x x m m N p = (1− )/ x N
( Bayesian ) ( Bayesian ) Bayesian Graduation---Singapore(male) Age M o rt a lit y R a te 80 90 100 110 0 .1 0 .2 0 .3 0 .4 0 .5 Crude data Prior Data Posterior(r=0.9) Posterior(r=0.5) Bayesian Graduation---Japan(male) Age M o rt a lit y R a te 80 90 100 110 0 .2 0 .4 0 .6 0 .8 1 .0 Crude data Prior Data Posterior(r=0.9) Posterior(r=0.5)
Kernel ! " # $ % & ' () * + , - . ' / 0 ( + , 1 2 . ' (Density function)34f(x)5 X1 2 . ' (6 f
∑
= − = n i i h x x k nh f 1 ) ( 1 ˆ (7$.' k(x)8 9 :∫
∞ ∞ − =1 ) (x k ( ; < = > 5 < ? @ A N ) 2 1 ) ( ( 2 2 x N x e k = − π B Laplace.' 2 ) 1 ) ( (kL x e x L = − (C $.' D E F G H ( I J 0 < ? .' 3KL M N O P Q R S T U V W X / Y Z (7[ \ 5F ] ^ _ ` a b c ' d (e a f g (Outlier)h ] i j k l m n (C o I F J 0 o p qr T s Q R W t 0 u v w x y ' (ex) ' (dx)/ z Y Z ( { | } ~ U∑
∑
= = − − = n i i i n i i i x h x x k e h x x k d q 1 1 ) ( ) ( ˆ 7$ ' ˆi q 5 q i (h 5 (Bandwidth), 34 h (6 S x \ x i X fˆ x( )S D E q4 h (6 S x i X fˆ x( )S D E qh Whittaker $ .' F + h S$h (Ko = > h = 1, 2, 33 ( ~ ) H h ( Y 2 ( 6 H v ¡ ¢ £ ~ ¤ ¥ ( I O ¦ K§ ¨ © v ¡ W ª « ¬ Gompertz .' ( ® ¯ ° ± ² 5³ d ´ F µ Y ¶ · ~ ( ® ¯ ° h=15 3 ( ~ ¸ )( W ¯ h=13t 0 Kernel m n ¹ T 3 ® 5º ( Whittaker (» ¼ K 107~109 ½ S F ¾ (7m n ¿ À qÁ W C 5  e à z Y Z à ' Whittaker 5( ® K 107~109 ½ À X ! 5 Whittaker h h ! " # h=100000! $ % Whittaker & ' ( ) * (+ , -. / 0 1 ) 2 3 $ 4 5 - 4 6 7 8 9 : h ; < = > ? @ A ! h B $ & ' ( ) * ( ! . / 0 1 C D $ ) < 2Whittaker Ä ¨ 3 ® 4P Å Æ 107~109½ Ä ¨ Ç 2 (È
É ! P Ê Y (Whittaker- Ë Ì h ® h Í Î Ï
( Ð Ñ Ò Ó 2 (7Ô ° Õ Ö × Ø Ë ¢ ¨ 3
( ) Kernel graduation using Normal function---Japan(male)
Age M o rt a lit y R a te 80 85 90 95 100 105 110 0 .2 0 .4 0 .6 0 .8 1 .0 Original data Kernel with h=1 Kernel with h=2 Kernel with h=3 ¸ ( )
Kernel graduation using Normal function---Singapore(male)
Age M o rt a lit y R a te 80 85 90 95 100 0 .1 0 .2 0 .3 0 .4 0 .5 Original data Kernel with h=1 Kernel with h=2 Kernel with h=3
4. Gompertz
Ù § Ú © ¡ Û Ô ´ 5 Ü qForce of
mortality µxW ª 5Gompertz.' 3= > Û Ô µx9 : Gompertz.
' (È x x =BC µ (B>0(C>1(C 5
∫
= − − − = 1 + 0 ln ( 1)] 1 exp[ ] exp[ BC C C dt px µx t x ( Ý Þ ° ß à X ' h á C p p x x+ = ln ln 1 3C o = 4µx5 Gompertz.' (6 lnpx 1 ln px+ â ã ¨ 3® ~ @ ä ® å æ ç è (Bootstrap)é ê x p ln lnpx+1 ë ì (C KÂ V Ù $Whittaker m n ¬ í î ï P Ê (® ~ ç è $ ð ñ ® Whittaker 5 3 ~ ò - ó @ ä 5 - - Whittaker BayesianKernel Ê ô lnpx ln px+1 ( l ¢ õ ( x x p p ln ln +1 K ö v ¡ ¨ 3 ( ò ) 1 x x p p ln ln +1 K 80~97 ½ À ÷ V ø ¨ (97 ½ C x y ' F : ¢ ù ú N ~ û ü ý þ ( W Û Ô µxK 97½ ® Â « ¬ Gompertz.' = > ( Ë Â X © ¡ m n Q 63 2 Whittaker á ô x x p p ln ln +1 ø ¨ 3 3 Bayesian x x p p ln ln +1 À (K 80~97 ½ ø ¨ (97½ ® j 3 x µ ! " # $ % Gompertz& ' ( 96 # ) * + , $ % Gompertz& ' - . / 0 1 " # 2 3 4 2 1 / 0 1 5 # 6 7 8-4 x x p p ln ln +1 K 80~97½ ø ¨ ( 97½ N ~ û ü ý þ 5 Ä ¨ 3 ò ( - WhittakerBayesian )
Is the force of mortality the Gompertz function?---Japan(male)
Age R a ti o 80 85 90 95 100 105 110 1 .0 1 .5 2 .0 Original data Whittaker with z=3 h=100000 Bayesian with r=0.9 Kernel with h=1 ó ( - WhittakerBayesian )
Is the force of mortality the Gompertz function?---Japan(male)
Age R a ti o 85 90 95 100 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0 Original data Whittaker with z=3 h=2000 Bayesian with r=0.9 Kernel with h=1
( ó ) 1 x x p p ln ln +1 K 80~95½ Y ( F ø ¨ (j ¥ ¶ å )(95½ ® j 3 2 Whittaker K 80~96½ ø ¨ ( 7 ý N ( V § ¨ 7 G H 3 3 Bayesian K80~97½ ø ¨ 3 4 K80~93½ ø ¨ (93½ ® j 3
Bootstrap
t 0 å æ ] / 5 ( ~ Í t 0 (Monte Carlo method)$ ç è ô x x p p ln ln +1 j '(®" § ¨ p | © ¡ Wª 9 : Gompertz .'= > 3ç è *Efron (1982) ô ( ! W" # S + , ÷ / W& l (Population)¢ X 7 $ % & (Resampling)(®Ê ° C b c F : ¢ % j '3î ï h 0 ! § ¨ x x p p ln ln +1 Wª « ¬ ' V ø ¨ < '(È ® Gompertz.' 5 $ Ô (N ~ ( É ! $ Ô ) Ý * ² + () * & á ô )( 4 , KV - Y . / 0 ; 1 N ~ ( $ Ô 2 3 4 F 5 ô N ~ ( $ Ô 2 3 (6 K95%6 7 8 ² 7~ (î ï F 9 : Û Ô µxW Gompertz. '= > 3®~ @ ä X . ê ; 3 1® . ç è U < ® - + , Æ b c 5 = (= > > '? @ T à @ A 8( & V A B ( M ` B x x p p ln ln +1 (C Ê ô j '3 * D $ m n h E ô , KV - ¿ ; 1 N ~ ( $ Ô 2 3 ( » ¼ K 87½ S F G H ¾ ( æ Û Ô µx« ¬ Gompertz. '( æ ( $ Whittaker )m n F 93¢ ( I ) h J G ! ) , ( ~ x x x b n q d " # b$ % & " x n $ ' ( ) * " x q $ ' ( + 9 , - . / 0 1 2 3 4 5 3 6 7 8 9 : ; < = > ? ; @ ( < = > ? ) * A ) " ; B = > ? ; A ( B = > ? ) * C ) D E F G H I J K L M N O P µxQ R
Eô,KV-Y./0;1N~($Ô23(æ Û
Ô µx« ¬ Gompertz. ' ( æ ( ¸ Whittaker )m n F
103
D ( Gompertz. ' N~() The upper and lower bound of Gompertz function ratio---Japan(male)
Age G o m p e rt z f u n c ti o n r a ti o 80 85 90 95 100 105 110 0 .0 0 .5 1 .0 1 .5 2 .0
Crude data ratio
The mean of simulation ratio
The upper of simulation ratio(twice deviation) The lower of simulation ratio(twice deviation)
I ( Gompertz. ' N~()
The upper and lower bound of Gompertz function ratio---Singapore(male)
Age G o m p e rt z f u n c ti o n r a ti o 85 90 95 100 0 1 2 3
4 Crude data ratio
The mean of simulation ratio
The upper of simulation ratio(twice deviation) The lower of simulation ratio(twice deviation)
Gompertz D ( Gompertz )N
10
(80 10,468! )" # $ % (80 1,488,902! ) D E F &
2® .ç è U ® - b c 5 & l (= > > ' ? @ T Ã @ A ( & VA B (` B m n C ® Whittaker (à C M x x p p ln ln +1 (Ê ô j ' 3* I V ( )- I T ()h EôK ,KVY.-;1N~($Ô23(æ - Û Ô µx« ¬ Gompertz . ' ( 1 m ; À 3 I V- I T L 0(Gompertz. ' ) º ! D - I L 0 º (h J 0 h) M C x y ' + ¢ N Ï ² + 1 ý þ 11 O h* ò - ó û ü á Æ P Q (C o x x p p ln ln +1 j ¥ Ç 2 (R 7 W KS e a v ¡ j )3 I V( Gompertz. ' N~() — z=3, h=100000
Upper and lower bound of Gompertz function
ratio after the Whittaker(z=3,h=100000) graduation---Japan(male)
Age G o m p e rt z f u n c ti o n r a ti o 80 85 90 95 100 105 110 1 .0 0 1 .0 5 1 .1 0 1 .1
5 Graduation crude data ratio
The mean of graduation simulation ratio Upper bound of simulation ratio(twice deviation) Lower bound of simulation ratio(twice deviation)
11 Whittaker (z=3 h ) ! " # " $ % & '
I T ( Gompertz. ' N~() — z=3, h=2000
Upper and lower bound of Gompertz function
ratio after the Whittaker(z=3,h=2000) graduation---Singapore(male)
Age G o m p e rt z f u n c ti o n r a ti o 85 90 95 100 1 .0 0 1 .0 5 1 .1 0 1 .1 5 1 .2 0 1 .2 5
Graduation crude data ratio
The mean of graduation simulation ratio Upper bound of simulation ratio Lower bound of simulation ratio
V - Û Ô µxW ª « ¬Gompertz. ' T 1 2Whittaker 3Bayesian 4Kernel Bootstrap Gompertz Bootstrap Whittaker ! Gompertz " # 197$ % & 2' ( ) 397$ % & 4107$ % & & z=3, h=100000& * + , 1- & 2' ( ) 393$ % & 497$ % & & z=3, h=100000&
5.
I ® U V 1980~1990v © ¡ b c 5 = ( K p W (@ ä ® WhittakerBayesian © v ¡ (7 $ Whitakker m n S ¬í X - Y P Ê q KY ¯ Ó ÷ h S F Z [ n q Bayesian C \ Ú x y ' 1 (t e í É ! \ Ú (» ] S ¬Ó 1^ \ Ú 5 _ ` (F a b t 0 Bayesian 3 ¢ K§ Ú W ª « ¬ Gampertz = > p W (\ 1 å æ- ç è Ú Q ( - © ¡ % ; W t 0 b c B W Whittaker (Z F 9 : « ¬ Gompertz . ' = > 3* ! c d e f g h i # j k l .m (> n o p ¦ g g q ý (L r © ¡ > n - ú s 8 Ê g t u 3v w x y (> n o hb z { k l S Ý k (C o I á ô © ¡ « ¬ Gompertz . ' m ; (F © ¡ t 0 | ' = > (º } U GompertzWeibull É < R )h] ( h/ 5 c d } Í © v ¡ ú s | 3 I $t 0 Bootstrap ~ à K S V + , ¶ å ~(_ à h Mô x x p p ln ln +1 j ' ( C ! x y ' v ¡ © Ë 1M (N Ï ² + 1 ( t § ¨ µ 3 ¾ ( I m ; 5 - KÓ÷ ~(F9:Û Ô µx« ¬ Gompertz . ' = > ( W (h ® M ô Gompertz . ' | ' ( C I á ô , KV - ( o - ] V (} ¯ ° Ó÷< ' ( } Í C é ê 36 Borgan, O. (1979), “On the theory of moving average graduation,”
Scandinavian Actuarial Journal, 62, 83-105.
7 Efron, B. and Tibshirani, R. J. (1993), An introduction to the Bootstrap, New York: Chapman and Hall.
8 Kannisto, V. (1996), The advancing frontier of survival life tables for old
age, Odense University Press.
9 Olshansky, S. J. and Carnes, B. A. (1997), “Ever Since Gompertz,”
Demography, 34(1), 1-15.
10 United Nations (1992), Long-range World population Projections, Department of International Economics and Social Affairs, United Nations, New York.
:
¹ V (80 109 80 100 ) ( ) I k ( ) ( ) x nx d x qx nx d x qx nx d x qx v ¡ x y ' ' x y ' ' x y ' ' 80 1488902 129059 0.08668 6171300 602450 0.09762 10468 974 0.09305 81 1297356 124697 0.09612 5386660 575263 0.10679 8873 900 0.10143 82 1118653 118437 0.10587 4637269 539856 0.11642 7395 825 0.11156 83 948822 109827 0.11575 3935705 498531 0.12667 6013 697 0.11592 84 801055 101416 0.12660 3300704 453904 0.13752 4894 624 0.12750 85 665127 92095 0.13846 2718510 405671 0.14923 3921 516 0.13160 86 540835 81528 0.15074 2203932 357107 0.16203 3101 447 0.14415 87 428401 71832 0.16767 1755371 309010 0.17604 2379 370 0.15553 88 329268 58712 0.17831 1367630 258807 0.18924 1756 275 0.15661 89 247987 48086 0.19391 1048111 213666 0.20386 1305 224 0.17165 90 183667 38599 0.21016 790000 173994 0.22025 976 170 0.17418 91 133629 30014 0.22461 586574 138558 0.23622 737 147 0.19946 92 94682 22923 0.24211 426200 107969 0.25333 518 119 0.22973 93 65001 16576 0.25501 302759 82024 0.27092 347 65 0.18732 94 43645 11898 0.27261 209593 60627 0.28926 245 59 0.24082 95 28379 8297 0.29236 142067 43595 0.30686 166 37 0.22289 96 18018 5524 0.30658 93741 30306 0.32330 113 30 0.26549 97 11186 3633 0.32478 60141 20295 0.33746 67 19 0.28358 98 6755 2253 0.33353 37828 13209 0.34919 40 11 0.27500 99 3987 1440 0.36117 23170 8784 0.37911 23 11 0.47826 100 2223 822 0.36977 13066 5486 0.41987 11 6 0.54545 101 1210 484 0.40000 7048 2979 0.42267 102 625 244 0.39040 3732 1606 0.43033 103 333 148 0.44444 1958 914 0.46680 104 155 63 0.40645 963 428 0.44444 105 84 40 0.47619 482 216 0.44813 106 38 17 0.44737 236 108 0.45763 107 16 6 0.37500 109 54 0.49541 108 9 6 0.66667 50 28 0.56000 109 2 2 1.00000 18 14 0.77778¹ T (- u s p m n )
( ) ( )
Whittaker Bayesian Kernel Whittaker Bayesian Kernel Age qx (z=3,h=100000) (r=0.9) (h=1) Age qx (z=3,h=2000) (r=0.9) (h=1) 80 0.08668 0.11245 0.09277 0.09094 80 0.09305 0.107463 0.09762 0.09682 81 0.09612 0.11578 0.10117 0.09632 81 0.10143 0.110905 0.10679 0.10163 82 0.10588 0.12008 0.11013 0.10458 82 0.11156 0.114709 0.11641 0.10863 83 0.11575 0.12534 0.11979 0.11456 83 0.11592 0.118882 0.12666 0.11630 84 0.12660 0.13157 0.13012 0.12519 84 0.12750 0.123452 0.13751 0.12414 85 0.13846 0.13877 0.14135 0.13669 85 0.13160 0.128475 0.14921 0.13224 86 0.15075 0.14694 0.15371 0.14931 86 0.14415 0.134061 0.16202 0.14140 87 0.16768 0.15610 0.16732 0.16288 87 0.15553 0.140375 0.17601 0.15054 88 0.17831 0.16625 0.18002 0.17635 88 0.15661 0.147657 0.18920 0.15824 89 0.19391 0.17741 0.19412 0.19033 89 0.17165 0.156225 0.20381 0.16664 90 0.21016 0.18959 0.20988 0.20533 90 0.17418 0.166475 0.22017 0.17775 91 0.22461 0.20282 0.22513 0.22064 91 0.19946 0.178872 0.23611 0.19360 92 0.24211 0.21714 0.24139 0.23591 92 0.22973 0.193943 0.25317 0.20778 93 0.25501 0.23257 0.25792 0.25102 93 0.18732 0.212251 0.27067 0.21230 94 0.27261 0.24916 0.27511 0.26688 94 0.24082 0.234377 0.28890 0.21983 95 0.29236 0.26696 0.29155 0.28386 95 0.22289 0.260868 0.30633 0.23305 96 0.30658 0.28601 0.30662 0.30049 96 0.26549 0.292205 0.32256 0.25074 97 0.32478 0.30638 0.31926 0.31602 97 0.28358 0.32877 0.33634 0.27168 98 0.33353 0.32811 0.32914 0.33100 98 0.27500 0.370826 0.34759 0.30590 99 0.36117 0.35128 0.35654 0.34764 99 0.47826 0.418511 0.37693 0.37908 100 0.36977 0.37593 0.39347 0.36507 100 0.54546 0.471863 0.41615 0.46636 101 0.40000 0.40213 0.39319 0.38201 102 0.39040 0.42992 0.39788 0.39682 103 0.44444 0.45937 0.43288 0.41288 104 0.40645 0.49050 0.41108 0.42671 105 0.47619 0.52337 0.42158 0.44046 106 0.44737 0.55799 0.44591 0.44882 107 0.37500 0.59438 0.51667 0.45657 108 0.66667 0.63255 0.65490 0.52485 109 1.00000 0.67251 1.00000 0.67031
