————‫ڀݚ‬࿦จ————————————————————
ςϯιϧղੳΛ༻͍ͨࢮҼผকདྷࢮ๢཰ͷಉ࣌ਪఆླ໦ɹ޹ଠ࿠ ∗দࢁɹ௚थ †

2020 ೥ 1 ݄ 6 ೔౤ߘ2020 ೥ 1 ݄ 22 ೔डཧ֓ཁຊ‫Ͱڀݚ‬͸ࢮҼผࢮ๢཰ͷಉ࣌ਪఆख๏ΛఏҊ͢ΔɽҰൠʹࢮҼผࢮ๢཰ਪఆ͸ࢮҼؒͷ૬‫ޓ‬࡞༻Λ൓өग़དྷͣɼ߹ࢉͨ͠શࢮҼࢮ๢཰ͷকདྷਪఆ͕ҙຯΛ࣋ͨͳ͍͜ͱ͕஌ΒΕ͍ͯΔɽ͜ͷͨΊࢮ๢཰ղੳͰඪ४తͳ Lee-Carter ϞσϧΛ४༻͠ɼ೥ྸɾ࣌ؒɾࢮҼɾੑผͷ 4 ࣠Λ΋ͭ 4 ֊ςϯιϧ‫ࣜܗ‬ͷσʔλʹରͯ͠ɼࢮ๢཰ͷվળ͕࣠ʹΑΒͣ୯Ұͷҩྍਫ४ͷਐาͰυϥΠϒ͞ΕΔͱΈͳͨ͠୯Ұ࣌‫ྻܥ‬ϞσϧΛఏҊ͢Δɽ͜ͷख๏ʹΑΓɼࢮҼؒͷ૬ؔؔ܎Λ໌Β͔ʹ͢Δͱ‫ʹڞ‬ɼશࢮҼࢮ๢཰ͱ੔߹తͳࢮҼผࢮ๢཰ͷ֎ૠΛ࣮‫ͨ͠ݱ‬ɽΩʔϫʔυɿ Lee-Carter ϞσϧɼࢮҼผࢮ๢཰ɼςϯιϧղੳɼWHO ࢮ๢཰σʔλϕʔε

1 ‫ڀݚ‬ͷഎ‫ܠ‬
1.1 ‫ࡁܦ‬Ձ஋ධՁͷཁ੥ۙ೥อ‫ݥ‬ձࣾ͸‫ࡁܦ‬Ձ஋ϕʔεͰͷอ‫ݥ‬ෛ࠴ධՁΛਪ͠ਐΊ͍ͯΔɽͦͷഎ‫ʹܠ‬͸֤ࠃͷ‫؂‬ಜ౰‫୲͕ہ‬೚͢
Δ৽͍͠ࢿຊ‫੍ن‬ͷଘࡏ͕͋ΓɼEU ʹ͓͚Διϧϕϯγʔ II ΍ IAIS ͷެදͨ͠ ICS ver2.0 ͳͲ͕୅දతͳ
‫͋Ͱ੍ن‬Δɽ͜͜Ͱ͍͏ෛ࠴ͷ‫ࡁܦ‬Ձ஋ͱ͸‫ʹ੍ن‬Αͬͯࡉ෦ͷఆ͕ٛҟͳΔ͕ɼ

1. ෛ࠴͕কདྷੜΈग़͢‫ظ‬଴ΩϟογϡϑϩʔͷׂҾ‫ࡏݱ‬Ձ஋
2. ͦͷ‫ظ‬଴஋Λ௒͑ΔϦεΫϚʔδϯ
ͷ 2 ͭΛ‫ج‬ຊͱͯ͠‫͞ࢉܭ‬ΕΔ֓೦ͱ͍͏఺Ͱ͸֓ͶҰக͢Δɽ͜ΕΒͷ‫ʹ੍ن‬ΑΓอ‫ݥ‬ձࣾͷࡒ຿ঢ়‫گ‬Λ‫؂‬ಜ‫׭‬ி͕ద੾ʹ೺Ѳ͠ɼอ‫ݥ‬ձࣾͷϦεΫ؅ཧͷߴ౓ԽΛΑΓҰ૚ਪਐͤ͞Δͱ͍͏ૂ͍͕͋Δɽ͜ͷΑ͏ͳࠃࡍతͳಈ޲Λड͚ɼ೔ຊͰ΋‫ࡁܦ‬Ձ஋ϕʔεͰͷιϧϕϯγʔ‫੍ن‬ͷಋೖͷಈ͖͕͋Δɽ‫ߦݱ‬ͷอ‫ݥ‬ձ‫Ͱܭ‬͸ɼࢮ๢཰΍ۚརΛఆ਺ͱΈͳ͢͜ͱͰ఻౷తͳอ‫਺ݥ‬ཧ‫ࢉܭ‬ͷख๏ʹଇͬͯɼຬ‫Ͱ·ظ‬ͷอ‫ݥ‬ෛ࠴Λ‫ܖ‬໿࣌఺ͰධՁ͢Δ͜ͱ͕ग़དྷͨɽҰํͰ‫ࡁܦ‬Ձ஋ϕʔεͰධՁ͢Δ৔߹ɼอ‫ݥ‬ෛ࠴ͷ࣌ՁධՁɼ
͢ͳΘͪධՁ࣌఺Ͱͷࢮ๢཰΍ۚརͷϦΞϧλΠϜͳ৘ใ͕ඞཁʹͳΓɼࢮ๢཰΍ۚརͷকདྷʹΘͨΔ௒௕‫ظ‬ͷ࣌ؒมಈΛద੾ʹϞσϦϯά͢Δ͜ͱ͕‫ߦݱ‬ձ‫ͱܭ‬ൺֱͯ͠ΑΓॏཁࢹ͞ΕΔΑ͏ʹͳΔɽ
͜͜Ͱ͍͏ʮద੾ͳϞσϦϯάʯʹ໌֬ͳఆٛ͸ͳ͍͕ɼ௒௕‫ظ‬ͷ֎ૠΛલఏͱͨ͠ϞσϦϯάͰ͸༧ଌ݁Ռ͕ଥ౰ͳ΋ͷͰ͋Δ͔ɼఆྔతͳ౷‫ܭ‬తϞσϧධՁࢦඪͷΈͳΒͣɼֶࡍతٞ࿦Ͱ͋ͬͨΓɼఆੑతͳ‫ݧܦ‬∗ ໌࣏େֶେֶӃ ઌ୺਺ཧՊֶ‫ڀݚ‬Պ E-mail: cs181011(at)meiji.ac.jp† ໌࣏େֶ ૯߹਺ཧֶ෦ E-mail: ma2yama(at)meiji.ac.jp

リスクと保険35ʹΑΔٞ࿦΋ඞཁෆՄܽͰ͋ΔɽͦͷͨΊෳࡶͳ౷‫ܭ‬Ϟσϧʢ΋͘͠͸‫ػ‬ցֶशͷΑ͏ͳΞϧΰϦζϜతख๏ʣʹΑΔߴ͍ਫ਼౓ͷ༧ଌΑΓ΋ղऍՄೳੑɾՄಡੑɾೲಘੑͱ͍ͬͨཁૉ΋Ճຯ͞ΕͨϞσϧͷํ͕๬·͍͠ɽ

1.2 ϦϨʔγϣφϧϞσϧͷ‫ط‬ଘ‫ڀݚ‬ຊ࿦จͰ͸কདྷࢮ๢཰༧ଌΛςʔϚʹѻ͏͕ɼকདྷࢮ๢཰༧ଌ͸্‫ه‬ͷ‫ࡁܦ‬Ձ஋ϕʔεͰͷอ‫ݥ‬ෛ࠴ධՁͷͨΊʹ‫ݶ‬Βͣɼอ‫ݥ‬ͷϓϥΠγϯά΍ࢮ๢ϦεΫͷ؅ཧɼ೥੍ۚ౓ӡӦɼਓ‫ޱ‬ਪ‫Ͳͳܭ‬ɼকདྷͷࢮ๢ʹΑͬͯมಈ͢ΔΩϟογϡϑϩʔͷ੍‫ޚ‬΍ҙࢥܾఆΛ͢Δ্ͰඇৗʹॏཁͰ͋ΓɼͦͷͨΊ͔ͶͯΑΓଟ͘ͷϞσϧ͕ఏҊ͞Ε͖ͯͨɽͦͷதͰ΋୅දతͳ౷‫ܭ‬Ϟσϧͷ 1 ͭͱͯ͠ Lee-Carter Ϟσϧ͕஌ΒΕ͍ͯΔɽ

Lee-Carter Ϟσϧ͸೥ྸ xɼྐྵ೥ t ͷࢮ๢཰ mx,t Λ
log mx,t = αx + βx κtͱ ද ‫ ͢ ݱ‬Δ Ϟ σ ϧ ͷ ૯ শ Ͱ ͋ Γ ɼ͍ ͘ ͭ ͔ ͷ ό Ϧ Τ ʔ γ ϣ ϯ ͕ ஌ Β Ε ͯ ͍ Δ ɽ‫ ݪ‬࿦ จ Ͱ ͋ Δ Lee andCarter[1992] ͰఏҊ͞ΕͨϞσϧ͸ɼ‫ ߲ࠩޡ‬εx,t Λ i.i.d. ͱͯ͠
log mx,t = αx + βx κt + εx,t
εx,t ∼ N(0, σ 2 )ͱ͍͏ਖ਼‫ن‬ϊΠζΛ൐ͬͨ૒ઢ‫ܗ‬ճ‫ܕؼ‬ͷϞσϧͰ͋Δɽ͔͠͠ɼ෼ࢄ͕೥ྸʹґΒͣҰఆͰ͋Γ෼ࢄෆ‫ۉ‬ҰੑʢheteroscedasticityʣΛද‫͔ͱ͍͜ͳ͖Ͱݱ‬Β Brouhns et al.[2002] Ͱ͸Dx,t ∼ Po(Ex,t mx,t )log mx,t = αx + βx κtͱ͍͏૒ઢ‫ܗ‬ϙΞιϯճ‫ܕؼ‬ͷϞσϧΛఏҊ͓ͯ͠Γɼͪ͜Β΋ Lee-Carter Ϟσϧͷ୅දతͳ‫ͯ͠ͱܗ‬஌ΒΕ͍ͯΔɽ͜͜Ͱ Dx,t ɼEx,t ͸ͦΕͧΕ೥ྸ xɼྐྵ೥ t ʹ͓͚Δࢮ๢ऀ਺ɼϦεΫରԠԆ΂೥਺ʢExposureʣ
Ͱ͋Δɽ

Lee-Carter Ϟσϧͷߏ଄͸ύϥϝʔλͷղऍՄೳੑͱਪఆͷ༰қੑʹ༏Ε͍ͯΔ͜ͱ͔Βɼ‫ݪ‬࿦จʹΑͬͯఏҊ͞Ε͔ͯΒ‫ࢸʹࡏݱ‬Δ·Ͱଟ͘ͷ࣮ূͱ֦ு͕ͳ͞Εɼ࣮຿ʹ͓͍ͯ΋ར༻͞Ε͍ͯΔɽ͜͜Ͱɼ֤ύϥϝʔλ͸ αx ͕ʮର਺ࢮ๢཰ͷฏ‫ۉ‬తͳਫ४ʯ
ɼκt ͕ʮࢮ๢཰ͷվળ౓ʯɼͦͯ͠ βx ͕ʮࢮ๢཰վળ κt ͷ೥ྸผ‫ײ‬Ԡ౓ʯͱղऍ͞ΕΔ͜ͱ͕ଟ͍ɽಛʹɼκt ͸௨ৗϊΠζΛ൐͍ͭͭ΋୯ௐʹԼ͕͍ͬͯ͘͜ͱ͔Βɼҩྍਫ४ͷ޲্Λද͢ͱղऍ͞ΕΔ৔߹͕ଟ͍ɽ
͔͠͠ͳ͕ΒɼLee-Carter Ϟσϧ͸Մಡੑ͕ߴ͍൓໘ɼ༧ଌਫ਼౓͕௿͍͜ͱ͕ࢦఠ͞Ε͍ͯΔɽ·ͨ৴པ۠ؒΛաখਪఆ͢Δ͜ͱ΋ใࠂ͞Ε͓ͯΓɼ‫ࡏݦ‬Խ͞ΕͨϞσϧϦεΫͷ 1 ͭͱͯ͠஌ΒΕ͍ͯΔʢྫ͑͹ɼLee[2000]ɼCairns[2000] Λࢀরʣɽ͜ͷ 2 ͭͷ໰୊఺ʹରͯͦ͠ΕͧΕղܾํ๏͕ఏҊ͞Ε͍ͯΔɽ
·ͣ༧ଌਫ਼౓ΛߴΊΔํ๏ͱͯ͠ɼ࣌ؒґଘ߲Λ૿΍͢Ϟσϧ͕ Renshaw and Haberman[2003] ʹΑΓఏҊ͞Ε͍ͯΔɽ͜ͷϞσϧ͸

log mx,t = αx +

K


(k)

βx(k) κtk=1

ͱද‫͞ݱ‬ΕΔϞσϧͰ͋ΓɼK = 1 ͷͱ͖͸௨ৗͷ Lee-Carter ϞσϧʹҰக͢Δɽ͜Ε͸ Lee-Carter Ϟσϧ͕ߦྻͷಛҟ஋෼ղΛ༻͍ͨ௿ϥϯΫۙࣅΛϕʔεͱͨ͠ϞσϧͰ͋Δ͜ͱͷྨਪͰ͋Γɼୈ K ಛҟϕΫτ
ϧ·Ͱͷ৘ใΛ༻͍ͯߦྻͷۙࣅΛߦ͏͜ͱʹରԠ͍ͯ͠Δɽ
·ͨɼ৴པ۠ؒͷաখਪఆΛղܾ͢Δํ๏ʹؔͯ͠͸ɼա෼ࢄϞσϧʢLee-Carter Ϟσϧͷฏ‫ߏۉ‬଄Λอͪͭͭ෼ࢄͷΈେ͖͘͢ΔϞσϧʣ͕ఏҊ͞Ε͍ͯΔɽ͜Ε͸େ͖͘෼͚ͯ 2 ͭͷϞσϧ͕ఏҊ͞Ε͓ͯΓɼ136

テンソル解析を用いた死因別将来死亡率の同時推定

ͭ͸ Renshaw and Haberman[2006] ʹΑΓఏҊ͞ΕͨϞσϧͰɼ্ड़ͷϙΞιϯ‫ ܕ‬Lee-Carter Ϟσϧʹରͯ͠ V[Dx,t ] = φEx,t ͱ͍͏ߏ଄Λ෇Ճ͠ɼφ ʹΑͬͯ෼ࢄͷେ͖͞Λίϯτϩʔϧ͢ΔϞσϧͰ͋Δɽ΋͏ 1
ͭ͸ Li et al.[2009] ʹΑΓఏҊ͞ΕͨϞσϧͰɼ͜Ε΋ϙΞιϯ‫ ܕ‬Lee-Carter Ϟσϧʹର͢Δ֦ுͰ͸͋Δ
͕ɼ͜ͷ࿦จͰ͸೥ྸɾ࣌ؒҎ֎ͷཁૉͰࢮ๢཰͕ҟͳΔͱ͠ɼ

Dx,t | wx ∼ Po(wx Ex,t mx,t )wx ∼ Γ(θx−1 , θx−1 )log mx,t = αx + βx κtͱ͍͏฼਺ࠞ߹‫ܕ‬ϞσϧΛఏҊͨ͠ɽ͜͜Ͱ wx ͸ฏ‫ ۉ‬1ɼ෼ࢄ θx ͱͳΔΑ͏ʹબ͹Ε͓ͯΓɼ͜ΕʹΑͬͯ

Lee-Carter Ϟσϧͷฏ‫ߏۉ‬଄Λอ͍ͬͯΔɽ·ͨɼDx,t ͸ wx Λੵ෼ফ‫͢ڈ‬Δ͜ͱͰෛͷೋ߲෼෍ʹͳΔ͜ͱ͕஌ΒΕ͓ͯΓɼ͜ͷ͜ͱ͔Βෛͷೋ߲‫ܕ‬ͷ Lee-Carter Ϟσϧͱ‫ݺ‬͹ΕΔ͜ͱ΋͋Δɽ
͜ΕΒͷϞσϧɼ΋͘͠͸ͦͷ૊Έ߹ΘͤʹΑ্ͬͯड़ͷ༧ଌਫ਼౓ͱ৴པ۠ؒͷ໰୊͸ղܾ͞ΕΔ͕ɼ৽ͨͳ
໰୊఺͕ͦΕͧΕࢦఠͰ͖Δɽ·ͣ࣌ؒґଘ߲Λ૿΍͢Ϟσϧʹ͓͍ͯ͸ɼ௥Ճͷ࣌ؒґଘ߲͕ʢLee-CarterϞσϧͰਪఆ͞ΕΔ κt ͕֓Ͷઢ‫͋Ͱܕ‬Δ͜ͱͱൺֱͯ͠ʣඇৗʹඇઢ‫͋Ͱܕ‬Γ౷‫ܭ‬తʹ༧ଌ͕೉͘͠ɼ·࣮ͨ຿໘ʹ͓͍ͯ΋ͦͷղऍ͕೉͍͠ͱ͍͏ܽ఺͕͋Δɽ·ͨա෼ࢄϞσϧʹ͓͍ͯ͸ɼฏ‫ߏۉ‬଄͕มΘ͍ͬͯͳ͍ͨΊத୹‫ظ‬ͷ༧ଌʹؔͯ͠վળ͕ߦΘΕͳ͍ͱ͍͏఺͕ࢦఠͰ͖Δɽ͜ΕΒͷ໰୊͔Βɼ্ड़ͷʮద੾ͳϞσϦϯάʯͱ͍͏఺Ͱ͸֦ு͞Εͨ Lee-Carter ϞσϧΛอ‫࣮ݥ‬຿ʹ௚઀ద༻ͯ͠ྑ͍͔͸ٞ࿦ͷඞཁ͕͋Δɽ

1.3 ࢮҼผϞσϦϯάͷඞཁੑͱࠔ೉ੑ
ҰํͰɼ୯ͳΔࢮ๢཰ͷ༧ଌΛߦ͏͚ͩͰͳ͘ɼࢮҼผࢮ๢཰ͷ༧ଌΛߦ͏͜ͱ΋อ‫ݥ‬঎඼ͷ։ൃ΍ϓϥΠγϯάɼϦεΫ؅ཧͰ͋ͬͨΓɼ੓ࡦͷҙࢥܾఆͳͲʹ͓͍ͯ༗༻Ͱ͋Δɽैདྷͷͦ͏͍ͬͨࢮҼผࢮ๢཰ͷ෼ੳʹ͍ͭͯ͸ɼલड़ͷ Lee-Carter ϞσϧΛࢮҼผࢮ๢ऀ਺ͷσʔλʹ౰ͯ͸ΊɼύϥϝʔλͷղऍΛߦ͏ͱ͍ͬͨํ๏͕஌ΒΕ͍ͯΔɽ͢ͳΘͪɼલड़ͷࢮ๢཰ʹࢮҼ෼ྨ c Λ‫ؚ‬Ίͨ mx,t,c ʹରͯ͠log mx,t,c = αx,c + βx,c κt,cͱ͍͏ϞσϧΛ౰ͯ͸Ί͍ͯΔ͜ͱʹͳΔɽ͔͠͠ɼ͜ͷϞσϧʢࢮҼผ Lee-Carter Ϟσϧͱ‫Ϳݺ‬ʣ͸

1. ࢮҼʹΑͬͯ͸ Lee-Carter Ϟσϧͷ౰ͯ͸·Γ͕ѱ͍͜ͱɽ
2. ύϥϝʔλ૯਺ͷଟ͞ʹΑΓਪఆ͕ෆ҆ఆʹͳͬͯ͠·͏͜ͱɽ
3. σʔλ਺͕‫ʹ୺ۃ‬গͳ͍σʔλͰ͸ਪఆ͕ෆ҆ఆʹͳͬͯ͠·͏͜ͱɽ
4. ‫ݸ‬ผʹਪఆ͢Δ͜ͱʹΑΓࢮҼؒͷ૬‫༻࡞ޓ‬Λ൓өग़དྷͣɼ߹ࢉͨ͠શࢮҼࢮ๢཰ͷকདྷਪఆ͕ҙຯΛ࣋ͨͳ͍͜ͱɽ
ͷ 4 ఺͕ܽ఺ͱͯ͠‫͛ڍ‬ΒΕΔɽಛʹɼࢮҼผʹ‫ ʹ͖ͱͨݟ‬Lee-Carter Ϟσϧ͕౰ͯ͸·Βͳ͍໰୊͸ Girosi

and King[2007] Ͱ΋ࢦఠ͞Ε͓ͯΓɼஶऀΒ͸࿦จதͰࢮҼผʹ Lee-Carter ϞσϧΛ౰ͯ͸ΊΔ͜ͱʹରͯ͠ɼʮ೔ৗతʹ༻͍Δ΂͖Ͱ͸ͳ͍ʯͱද‫͍ͯ͠ݱ‬Δɽ
͜ΕΒͷ໰୊ͷதͰ΋ɼಛʹϞσϧͷ౰ͯ͸·ΓΛվળ͢Δํ๏ͱͯ͠ɼ্ड़ͷ࣌ؒґଘ߲Λ૿΍͢ϞσϧΛࢮҼผσʔλʹԉ༻͢Δͱ͍͏ํ๏͕‫͛ڍ‬ΒΕΔɽ͢ͳΘͪ

log mx,t,c = αx,c +

K


(k)

(k)
βx,c
κt,c

k=1

リスクと保険37ͱ͍͏ϞσϧͰ͋Δɽ͜ΕΛલड़ͷࢮҼผσʔλʹ౰ͯ͸ΊΕ͹౰ͯ͸·Γͷ໰୊͸ղܾ͞ΕΔɽ͔͠͠ɼ͜
ͷϞσϧʹରͯ͠΋

1. K ΛͲ͜·Ͱେ͖͘औΕ͹ྑ͍͔ͷ൑அ͕೉͍͜͠ͱɽ
2. Lee-Carter ϞσϧΑΓߋʹύϥϝʔλ਺͕ଟ͘ͳͬͯ͠·͏͜ͱɽ(k)

3. ਪఆ͞Εͨෳ਺ͷ κt,c(k = 1, . . . , K) ͷղऍ͕ࠔ೉Ͱ͋Δ͜ͱɽ

ͱ͍ͬͨܽ఺͕ࢦఠͰ͖Δɽ
ͱ͜ΖͰɼࢮҼผࢮ๢཰ਪఆ͸ա‫ʹڈ‬೔ຊͷެతͳਓ‫ޱ‬ਪఆʹ΋༻͍ΒΕ͍ͯͨͱ͍͏ա‫͕͋ڈ‬ΔʢੴҪ[2017] ࢀরʣɽ౰࣌͸౷‫ܭ‬Ϟσϧతͳख๏Ͱ͸ͳ͘ Gompertz ϞσϧͷΑ͏ͳ਺ֶతؔ਺ͷϑΟοςΟϯάʹ
ΑΔ༧ଌ͕ߦΘΕ͓ͯΓɼ‫͔͘ݹ‬ΒࢮҼผࢮ๢཰ͷ༧ଌ͕શࢮҼࢮ๢཰ͷ༧ଌʹ༗༻Ͱ͋Δͱೝࣝ͞Ε͍ͯͨͱ‫͑ݴ‬Δɽ‫ࡏݱ‬͸ࢮҼผࢮ๢཰ʹ‫ͮ͘ج‬ਓ‫ޱ‬ਪఆ͸ߦΘΕ͍ͯͳ͍͕ɼ͜Εʹ͸Լ‫ه‬ͷΑ͏ͳཧ༝͕͋͛ΒΕ͍ͯΔɽ

1. ύϥϝʔλͷଟ͞ͱσʔλͷগͳ͞ʹΑΓਪఆਫ਼౓͕ෆ҆ఆͰ͋Δ͜ͱɽ
2. Lee-Carter Ϟσϧʹ୅ද͞ΕΔ౷‫ܭ‬Ϟσϧ͕୆಄͖ͯͨ͜͠ͱɽ
3. ࢮҼผࢮ๢཰ਪఆ͸ࢮ๢཰վળΛաখਪఆ͢Δ͜ͱɽ
3 ͭΊʹؔͯ͠‫͢ٴݴ‬Δͱɼ͜Ε͸ Wilmoth[1995] ʹΑΓใࠂ͞Εͨ಺༰Ͱ͋Γɼ͍͔ͭ͘ͷԾఆͷ΋ͱͰকདྷ༧ଌ஋ͱ‫؍‬ଌ஋ͷؒʹਖ਼ͷόΠΞε͕ੜ͡Δ͜ͱΛཧ࿦తʹࣔͨ͠΋ͷͰ͋Δɽ

2 ຊ‫ڀݚ‬ͷ໨తલষΛ౿·͑ɼຊ‫ڀݚ‬ͷ໨తΛ 2 ͭఏࣔ͢Δɽ1 ͭ͸ɼ‫ࡁܦ‬Ձ஋ධՁͷͨΊʹ‫ࢮͳ݈ؤ‬๢཰ͷ௕‫֎ظ‬ૠΛՄೳͱ͢Δ͜ͱͰ͋Δɽ΋͏ 1 ͭ͸ࢮҼผࢮ๢཰ΛશࢮҼࢮ๢཰ͱ੔߹తʹਪఆ͢Δ͜ͱͰ͋Δɽ࠷ॳͷ໨త͔Β࣌‫ྻܥ‬੒෼Λ 1 ͭʹ੍‫ ͠ݶ‬Lee-Carter Ϟσϧͷमਖ਼Λߦ͏͜ͱ͕ཁ੥͞Εɼ2 ͭ໨ͷ໨త͔Βςϯιϧ‫ܕ‬ϞσϧͰ͋Δ͜ͱ͕ཁ੥͞ΕΔɽຊ࿦จͰఏҊ͢ΔϞσϧ͸ɼLi et al.[2009] Ͱ‫͞ٴݴ‬Ε͍ͯΔʮ೥ྸɾྐྵ೥Ҏ֎ͷࢮ๢཰ܾఆཁҼʯͱͯ͠ࢮҼΛબ୒ͨ͠΋ͷͱߟ͑Δ͜ͱ΋ग़དྷΔͨΊɼઌߦ‫ڀݚ‬ͷ݁Ռͱ੔߹తͰ͋Δɽ·ͨςϯιϧ‫ܕ‬Ϟσϧ͸ɼຊ࿦จͰ͸۩ମతʹ೥ྸɾ࣌ؒɾࢮҼɾੑผͷ 4 ͭͷ࣠Λ΋ͭ 4 ֊ςϯιϧ‫ࣜܗ‬ͷσʔλʹର͢ΔϞσϧͰ͋Δɽຊ‫Ͱڀݚ‬͸ κt Λҩྍਫ४Λද͢ύϥϝʔλͱղऍ͠ɼࢮ๢཰ͷվળ͸௕‫ظ‬తʹ‫ݟ‬Δͱੑผ΍ࢮҼɼ೥ྸΛӽ͑ͯҩྍਫ४ͷ޲্ʹυϥΠϒ͞ΕΔͱԾఆ͠ɼੑผ΋‫ؚ‬Ίͨςϯιϧσʔλʹର͠୯Ұ࣌‫ྻܥ‬ͷϞσϧΛ౰ͯ͸Ίͨɽͨͩ͜͜͠Ͱʮ୯Ұ࣌‫ྻܥ‬ʯͱ͸ࢮ๢཰มԽͷϑΝΫλʔ͕ 1 ͭͱ͍͏ҙຯͰɼࢮҼผ΍೥ྸผͷࢮ๢཰τϨϯυ͕ಉҰͰ͋Δͱ͍͏Θ͚Ͱ͸ͳ͍͜ͱʹ஫ҙ͞Ε͍ͨɽͳ͓ɼςϯιϧ‫ࣜܗ‬ͷσʔλΛࢮ๢཰ਪఆʹ༻͍ͨઌߦ‫ͯ͠ͱڀݚ‬͸໺ଜ [2018] ͕‫͛ڍ‬ΒΕΔɽ໺ଜ [2018]͸খ஍Ҭ i ͝ͱͷࢮ๢཰ mx,t,i ʹରͯ͠log mx,t,i = αx +K(k) (k)βx(k) κt γi

k=1

Λ౰ͯ͸Ίͨɽ͜Ε͸ςϯιϧͷߴ࣍ಛҟ஋෼ղʢςϯιϧ෼ղʣͱ‫ݺ‬͹ΕΔख๏Λ༻͍͍ͯΔɽςϯιϧ෼ղʹ͸ CP ෼ղ΍ Tucker ෼ղ͕‫ج‬ຊతͳख๏ͱͯ͠஌ΒΕ͓ͯΓɼ໺ଜ [2018] ͸ͦͷதͰ΋ CP ෼ղʹΑΔ
ํ๏ΛఏҊ͍ͯ͠Δ͕ɼຊ‫Ͱڀݚ‬͸୯Ұ࣌‫ʹྻܥ‬ΑΔਪఆ‫ͼٴ‬ύϥϝʔλ਺ͷઅ໿ʹΑΔਪఆͷ҆ఆΛ໨ඪͱ
͍ͯ͠ΔͨΊɼςϯιϧ෼ղΑΓ΋γϯϓϧͳϞσϧΛఏҊ͢Δɽ
·ͨຊ࿦จͰ͸্‫ه‬ͷ Wilmoth[1995] ͷࣔͨ͠಺༰͕‫ࡏݱ‬ͷσʔλɾਪఆख๏ʹରͯ͠΋౰ͯ͸·Δ͔Ͳ͏͔֬ೝ͠ɼ‫͢ٴݴ‬Δɽ

38

テンソル解析を用いた死因別将来死亡率の同時推定3 ఏҊϞσϧ
3.1 ϞσϧͷఆࣜԽຊ‫Ͱڀݚ‬͸೥ྸ xɼྐྵ೥ tɼࢮҼ cɼੑผ s ͷࢮ๢཰ mx,t,c,s ʹରͯ͠Լ‫ه‬ͷϞσϧΛఏҊ͢Δɽ

log(1 − mx,t,c,s ) = βx,c,s κt + εx,t,c,s(1)

͜͜Ͱύϥϝʔλ͸ βx,c,s ɼκt Ͱ͋Γ࠷໬๏ʹΑΓਪఆ͢Δɽ·ͨ εx,t,c,s ͸͜ͷϞσϧͷ‫͋Ͱ߲ࠩޡ‬Δ͕ɼຊ
‫Ͱڀݚ‬͸

1. ೥ྸɾࢮҼɾੑผ͝ͱͷ૬ؔߏ଄Λಠཱʹ‫ه‬ड़Ͱ͖ɼಛʹࢮҼؒͷ૬ؔߏ଄ΛਪఆͰ͖Δ͜ͱ
2. χϡʔτϯ๏ʹ୅ද͞ΕΔޯ഑๏Λ༻͍ͳ͍ͨΊ҆ఆͨ͠ਪఆ͕ग़དྷΔ͜ͱɽͱ͍͏ 2 ఺͔Βɼ‫ࠩޡ‬෼෍ʹςϯιϧਖ਼‫ن‬෼෍ TN ΛԾఆͨ͠ʢςϯιϧਖ਼‫ن‬෼෍ʹ͍ͭͯ͸෇࿥ B ΛࢀরʣϞσϧΛςϯιϧද‫͢ه‬ΔͱԼ‫ه‬ͷ௨ΓͰ͋Γɼͪ͜Βͷද‫ه‬ͷํ͕‫ࠩޡ‬ͷߏ଄͕ଊ͑΍͍͢ͱ͍͏ಛ௃͕
͋Δɽ

Mt = Bκt + Et

Et ∼ TN(0, Σage , ΣCoD , Σsex )ͨͩ͠ɼM = (log(1 − mx,t,c,s ))ɼB = (βx,c,s )ɼEt = (εx,t,c,s ) Ͱ͋ΓɼΣage ɼΣCoD ɼΣsex ͸ͦΕͧΕ೥ྸʢageʣɾࢮҼʢCause of Deathʣɾੑผʢsexʣ͝ͱͷ෼ࢄ‫ڞ‬෼ࢄߦྻͰ͋Δɽ

ຊϞσϧ͸֎ૠੑೳΛॏࢹ͢Δ‫͔఺؍‬Β Lee-Carter Ϟσϧͷฏ‫ۉ‬ਫ४ αx ʹ͋ͨΔ߲Λ΋ͨͳ͍͕ɼཧ༝ʹ͍ͭͯ͸ୈ 6 અͰ‫ޙ‬ड़͢Δɽ·ͨɼ௨ৗͷ Lee-Carter Ϟσϧ΍ͦͷ֦ுϞσϧ͕ࢮ๢཰ͷର਺஋ʹର͢ΔϞσϧͰ͋Δͷʹରͯ͠ɼఏҊϞσϧ͸ͦͷࢮҼͷੜଘ཰ʹର͢ΔϞσϧͱͳ͍ͬͯΔɽ͜Ε͸ɼࢮҼผͷ‫؍‬ଌࢮ๢ऀ਺͕ 0 ͷͱ͖σʔλ͔Β௚઀‫͞ࢉܭ‬ΕΔૈର਺ࢮ๢཰͕ −∞ ʹͳͬͯ͠·͏ͨΊͰ͋Δɽ

3.2 ਪఆํ๏
ఏҊϞσϧ͸ࣜͷ‫͔ܗ‬Β β ͱ κ ʹΑͬͯ‫׬‬શʹม਺͕෼཭͞Ε͍ͯΔɽैͬͯʮ೥ྸɾࢮҼɾੑผʯͱʮྐྵ
೥ʯͷ 2 ͭͷ࣠ʹͳΔΑ͏ʹςϯιϧͷߦྻԽΛߦ͏͜ͱͰɼ‫ݪ‬࿦จͷ Lee-Carter Ϟσϧͱಉ͡ಛҟ஋෼ղΛ༻͍ͨਪఆ͕ՄೳͰ͋Δʢςϯιϧ‫ؔ͢ʹࢉܭ‬Δ‫߸ه‬͸෇࿥ A Λࢀরʣɽ͢ͳΘͪσʔλςϯιϧ M Λඪ४తͳಛҟ஋෼ղʹΑͬͯ

M(2) = U SV Tͱ෼ղ͠κt = U [t, 1]βq = S[1, 1]V [q, 1]ͱͨ͠‫ʹޙ‬Ϟσϧͷࣝผੑͷ໰୊͔Β β0 =q βq ͱͯ͠

κt = β0 κt
βq =

βq
β0

リスクと保険39ͱमਖ਼͠ɼϕΫτϧ βq Λ࠶౓ςϯιϧԽ͢Ε͹ྑ͍ɽ͜͜Ͱ q ͸ςϯιϧ βx,c,s ΛϕΫτϧԽͨ͠‫ޙ‬ͷఴ͑ࣈͰ͋Γɼ

q = x + (c − 1)Nage + (s − 1)Nage NCoDͱ͢Ε͹͍͍ɽͨͩ͠ Nage ɼNCoD ͸ͦΕͧΕ೥ྸɾࢮҼͷ֤࣠ʢϞʔυʣͷ࣍‫͋Ͱݩ‬Δɽ
·ͨɼ෼ࢄ‫ڞ‬෼ࢄߦྻ͸ಛఆͷཁૉʹ஫໨ͦ͠ͷଞͷ෼ࢄ‫ڞ‬෼ࢄߦྻ͸‫ط‬஌ͱ͢Δͱղੳతʹ࠷໬ਪఆྔ͕
‫·ٻ‬ΔͨΊɼσʔλ͔Βฏ‫ߏۉ‬଄ΛҾ͍ͨ࢒ࠩςϯιϧ Rt ʹରͯ࣍͠ͷΑ͏ͳަ‫࠷ޓ‬దԽͰ‫ٻ‬Ίͨʢಋग़͸෇࿥ B Λࢀরʣ
ɽ

T

Σi =

Ni 
T
Rt,(i) Σ−1¬i Rt,(i)N T t=1(i = age, CoD, sex)͜͜Ͱ Rt,(i) ͸࣠ i ʹର͢Δςϯιϧ Rt ͷߦྻԽͰ͋ΓɼΣ−1¬i ͸ͦΕͧΕ−1−1
Σ−1¬age = Σsex ⊗ ΣCoD ,

−1
−1
Σ−1
¬CoD = Σage ⊗ Σsex ,

−1
−1
Σ−1
¬sex = ΣCoD ⊗ ΣageͰ͋Γɼ⊗ ͸ߦྻͷΫϩωοΧʔੵͰ͋Δɽͨͩ͠ɼࠓճ͸ίʔϗʔτޮՌͳͲͷ೥ྸؒͷ૬ؔ΍ੑผؒͷ૬ؔ͸ߟ͑ͣɼΣage , Σsex ͸ର֯ߦྻͱ͠ɼΣCoD ͸Ұൠͷ෼ࢄ‫ڞ‬෼ࢄߦྻͱͨ͠ɽ

4 ਪఆ݁Ռ
4.1 ࢖༻σʔλຊ‫Ͱڀݚ‬͸ WHO ͷఏ‫͢ڙ‬Δ WHO mortality database ͔ΒσʔλΛ༻ҙͨ͠ɽྐྵ೥͸ 1995 ೥͔Β 2015
೥·Ͱɼ೥ྸ͸ 0 ࡀɼ1 ࡀ͔Β 4 ࡀɼ5 ࡀ͔Β 9 ࡀɼʢҎ߱ 5 ࡀ۠෼ʣɼ80 ࡀ͔Β 84 ࡀɼ85 ࡀҎ্ͱෆ‫ۉ‬Ұʹ·ͱΊΒΕ͍ͯΔɽࢮҼʹؔͯ͠͸‫ݩ‬ͷσʔλͰ͸ࠃࡍతͳࢮҼ෼ྨͰ͋Δ ICD10 ʹैͬͯ෼ྨ͞Ε͍ͯΔ͕ɼ
֤߲໨͕ඇৗʹৄࡉͰ͋ΔͨΊɼްੜ࿑ಇল͕ఏ͍ࣔͯ͠Δ ICD10 ४‫جڌ‬ຊ෼ྨදͷ 22 ۠෼ʹσʔλΛѹॖ্ͨ͠Ͱ෼ੳΛߦͬͨɽͦͷ۠෼͸ผදͷ௨ΓͰ͋Δɽͳ͓ɼಛघ໨త༻ͷίʔυͱͦͷ۠෼͕͋ΔͨΊ࣮ࡍͷࢮҼ͸ 19 ෼ྨ·Ͱѹॖ͞Εͨɽͦͷ݁Ռɼࠓճ࢖༻͢ΔσʔλͰ͸࣍‫ ͕ݩ‬19ʢ೥ྸʣ×21ʢྐྵ೥ʣ×19ʢࢮ

Ҽʣ×2ʢੑผʣͷ 4 ֊ςϯιϧͱͳͬͨɽ

4.2 ղੳ݁Ռ
ఏҊϞσϧʹΑΓਪఆͨ݁͠ՌͷҰྫΛਤ 1-4 ʹද͢ɽ͜͜Ͱԣ࣠͸্ड़ͷ೥ྸ۠෼ɼॎ࣠͸ର਺ࢮ๢཰

log mx,t,c,s Ͱ͋ΔɽϞσϧ log(1 − mx,t,c,s ) ͱग़ྗ͕ҟͳΔ͜ͱʹ஫ҙ͞Ε͍ͨɽ

40

テンソル解析を用いた死因別将来死亡率の同時推定

Proposed Model

Raw Mortality

−4

1995
2005
2015

−10

−10

−8

−8

−6

−6

−4

1995
2005
2015

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

20−24

30−34

age

ਤ1

50−54

60−64

70−74

80−84

age

உੑɿ৽ੜ෺ʢ‫؍‬ଌ஋ʣ

ਤ2உੑɿ৽ੜ෺ʢਪఆ஋ʣ
Proposed Model

Raw Mortality
1995
2005
2015

−12

−12

−10

−10

−8

−8

−6

−6

−4

1995
2005
2015

−4

40−44

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

20−24

30−34

age

ਤ3

40−44

50−54

60−64

70−74

80−84

age

ঁੑɿ‫ܥثٵݺ‬ͷ࣬‫ױ‬ʢ‫؍‬ଌ஋ʣ

ਤ4ঁੑɿ‫ܥثٵݺ‬ͷ࣬‫ױ‬ʢਪఆ஋ʣ

͜ͷਤ͔ΒɼఏҊϞσϧ͕ࢮҼผࢮ๢཰ͷಛ௃Λྑ͘ද‫͍͖ͯͰݱ‬Δ͜ͱ͕Θ͔Δɽ࣮ࡍɼਪఆ͞Εͨฏ‫ۉ‬ߏ଄ͱσʔλͷဃ཭Λϑϩϕχ΢εϊϧϜͰଌΔͱɼnorm percent ͸ 91.06% Ͱ͋Γɼඇৗʹߴ͍ਫ਼౓ͰσʔλΛද‫͍͖ͯͰݱ‬Δ͜ͱ͕෼͔ͬͨɽ͜͜Ͱ norm percent ͸‫؍‬ଌ஋ Xobs ͱਪఆ஋ Xest ʹରͯ͠ҎԼͰఆٛ͞ΕΔɽ

norm percent =

1−|Xest − Xobs |F|Xobs |F



× 100

(2)

.

0.2

6
0

0.0

2

0.1

4

.

8

0.3

10

0.4

12

࣍ʹ෼ࢄߏ଄ͷਪఆ݁ՌΛਤ 5,6 ʹࣔ͢ɽ

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

M

age

ਤ5

೥ྸผඪ४ภࠩʢର਺εέʔϧʣ

F
sex

ਤ6

உঁผඪ४ภࠩʢର਺εέʔϧʣ͜͜ͰɼΣage , Σsex ͸ର֯ߦྻʹ੍‫͍ͯ͠ݶ‬ΔͨΊର֯੒෼Λϓϩοτ͍ͯ͠Δɽ·ͨςϯιϧਖ਼‫ن‬෼෍ʹ͓͚Δ෼ࢄ‫ڞ‬෼ࢄߦྻ͸εέʔϧҰҙੑΛ࣋ͨͣ஋Λ௚઀ධՁͰ͖ͣ૬ରධՁ͔͠Ͱ͖ͳ͍ͨΊɼͲͪΒͷਤ

リスクと保険41΋࠷খ஋Λ 1 ʹ‫ج‬४Խ্ͨ͠Ͱର਺εέʔϧͰϓϩοτ͍ͯ͠Δɽ૬ରධՁͱ͍͏‫͜Ͱ఺؍‬ͷਤΛ‫ݟ‬Δͱɼ0ࡀͱߴ೥ྸ૚ͷ෼ࢄ͕େ͖͍͜ͱ͕Θ͔ͬͨɽ͜Ε͸ਤ 1 ͷ 0 ࡀͱߴ೥ྸ૚Λ‫ͯݟ‬Θ͔ΔΑ͏ʹɼճ‫ࠩޡؼ‬ͷେ͖͞ͱ‫؍‬ଌσʔλͷϒϨͷେ͖͞ͷ 2 ͭʹ‫ݪ‬Ҽ͕͋Δͱߟ͑ΒΕΔɽ·ͨஉੑͷํ͕ঁੑΑΓ΋෼ࢄ͕େ͖
͍͜ͱ΋Θ͔ͬͨɽ
࣍ʹਪఆ͞Εͨ࣌‫ྻܥ‬੒෼ κt Λਤ 7 ʹද͢ɽ

0.18

0.20

0.22

0.24

0.26

kappa

1995

1997

1999

2001

2003

2005

2007

2009

2011

2013

2015

year

ਤ7

ਪఆ͞Εͨ κt୺఺ಉ࢜Λ఺ઢͰ‫͍Ͱ͍ܨ‬Δ͕ɼ͜ͷਤ͔Β κt ͸௨ৗ Lee-Carter Ϟσϧͷจ຺ͰԾఆ͞ΕΔΑ͏ͳ௚ઢత
Ͱ͸ͳ͘ɼచ‫ݮ‬తͳ‫ݮ‬গτϨϯυΛ࣋ͭ͜ͱ͕ࣔࠦ͞Εͨɽ࠷‫ʹޙ‬ఏҊϞσϧͱࢮҼผ Lee-Carter ϞσϧͷAIC Λද 1 ʹࣔ͢ɽ
ද1

AIC ͷൺֱϞσϧର਺໬౓ύϥϝʔλ਺

AIC

ఏҊϞσϧ

163577954-325246ࢮҼผ LC

4884

2204

-5360

͜ͷΑ͏ʹఏҊϞσϧ͸ࢮҼผ Lee-Carter Ϟσϧͱൺֱͯ͠ύϥϝʔλ਺΋࡟‫ ͭͭ͠ݮ‬AIC Λେ෯ʹ࡟‫ݮ‬͢Δ͜ͱ͕ग़དྷͨɽ

4.3 ࢮҼؒͷ૬ؔߏ଄͜͜Ͱ͸ςϯιϧਖ਼‫ن‬෼෍ͷਪఆʹΑΓಘΒΕΔࢮҼؒͷ૬ؔߏ଄ʹ͍ͭͯ‫͢ٴݴ‬Δɽςϯιϧਖ਼‫ن‬෼෍Λ༻͍ͣʹࢮҼؒͷ૬ؔΛಘΑ͏ͱͨ͠ͱ͖ɼಘΒΕͨ‫؍‬ଌର਺ࢮ๢཰͔Βฏ‫ߏۉ‬଄ΛҾ͍ͨ࢒ࠩςϯιϧ

rx,t,c,s ΛࢮҼͷ࣠ʹؔͯ͠ߦྻԽ͠ɼ֤ྻ͕ಠཱʹಘΒΕͨඪຊͩͱΈͳͤ͹෼ࢄ‫ڞ‬෼ࢄߦྻΛಘΔ͜ͱ͕ՄೳͰ͋Δɽ͔͠͠ɼ͜ͷํ๏͸೥ྸؒ΍ੑผؒͷ෼ࢄͷҧ͍Λແࢹ͍ͯ͠Δͱ͍͏ܽ఺͕͋ΓɼҰํͰ೥ྸɾ
ੑผ͝ͱʹ෼ࢄ‫ڞ‬෼ࢄߦྻΛ‫͢ࢉܭ‬Δͱɼͪ͜Β͸૬ରతͳσʔλͷগͳ͞ʹΑΓਪఆ͕ෆ҆ఆʹͳΔͱ͍͏
ܽ఺͕͋Δɽςϯιϧਖ਼‫ن‬෼෍͸ͦΕΒ 2 ͭͷܽ఺Λղফ͢Δํ๏ͱͯ͠༗༻Ͱ͋ΓɼຊϞσϧʹ࠾༻ͨ͠ཧ
༝ͷ 1 ͭͰ͋Δ͜ͱ͸લઅͰड़΂ͨ௨ΓͰ͋Δɽ
ਤ 8 ʹ ΣCoD ΑΓಘΒΕΔภ૬ؔߦྻΛද͢ɽภ૬ؔ͸஫໨͢Δ 2 ͭͷࢮҼҎ֎ͷӨ‫ڹ‬ΛऔΓআ͍ͨ૬ؔͰ
͋Γɼ୯ͳΔ૬ؔߦྻΑΓ΋ਅͷ૬ؔ֓೦ʹ͍ۙͨΊͪ͜ΒΛ༻͍ͨɽ·ͨʢภʣ૬ؔߦྻ͸෼ࢄ‫ڞ‬෼ࢄߦྻ42

テンソル解析を用いた死因別将来死亡率の同時推定ͷҰछͷ‫֨ن‬ԽͰ͋ΔͨΊɼ෼ࢄ‫ڞ‬෼ࢄߦྻͷҰҙੑͷ໰୊͸ͳ͘ɼେখൺֱͰ͸ͳ͘௚઀਺஋ΛධՁ͢Δ͜
ͱ͕ՄೳͰ͋Δɽ1

c1
0.33 c2

0.8

0.04 −0.07 c3
0.02 −0.1 0.03 c4

0.6

0.13 0.01 0.06 −0.1 c5−0.13 0.24 0.17 0.13 0.12 c6

0.4

0 −0.02−0.06−0.03−0.07 0.01 c7
00.01 −0.03 0 −0.01 0.03 −0.02 c8

0.2−0.09−0.02 0.26 0.31 −0.17−0.69 0.01 0.04 c9
0.09 0.04 0.08 0.07 0.07 0.07 0.01 0.04 0.28 c10

00.08 0.16 −0.02 0.3 0.24 0.06 0.01 −0.01 0.07 −0.06 c11−0.2−0.16 0.09 −0.03 0.02 0.03 0.14 0.02 0.03 0.05 −0.06−0.03 c12
0.25 −0.13 0.12 0.13 −0.15 0.13 −0.03 0.06 −0.1 0.02 −0.04 0.08 c13−0.4

0.02 0.04 0.06 0.17 0.25 0.11 0.01 −0.02 0.15 0.28 0.13 −0.01 0.06 c14−0.06 0.05 0.1 −0.01−0.02−0.06 0 −0.05−0.04 0 −0.02 0.04 0.12 0.04 c15
0.02 −0.01 0 −0.01−0.01 0−0.02−0.02 0.05 0.01

00.01 −0.01 0.01 −0.02−0.01 0

0

0.02

0

c16

−0.6

0.21 0.01 0.08 0.1 0.09 −0.09−0.07−0.05−0.07 0.01 0.27 c17−0.21−0.24−0.31−0.03 0.33 0.44

0 −0.05 0.26 −0.02 −0.3 0.12 0.07 0.14 −0.01 0 −0.09 c18

0.07 0.05 −0.09 0.12 0.02 0.13 −0.01 0.07 0.19 −0.11 0.05 −0.04 −0.1 −0.16−0.11−0.02 0.13 0.09 c20

ਤ8

−0.8

−1

ࢮҼؒͷภ૬ؔϓϩοτਤ 8 ʹΑΔͱɼࢮҼؒʹ͸ਖ਼ͷ૬ؔͱෛͷ૬͕ؔ఺ࡏ͍ͯ͠Δ͕ɼൺֱతߴ૬ؔͱ‫͑ݴ‬Δ΋ͷ͸ࢮҼ෼ྨͷ

6 ൪໨ͱ 9 ൪໨Ͱ͋Δʮਆ‫ܥܦ‬ͷ࣬‫ױ‬ʯͱʮ॥‫ܥث؀‬ͷ࣬‫ױ‬ʯͷ૬ؔ (−0.69) ͷΈͰ͋ͬͨɽʮਆ‫ܥܦ‬ͷ࣬‫ױ‬ʯ͸਷ບԌ΍೴ԌɼύʔΩϯιϯපɼΞϧπϋΠϚʔපɼͯΜ͔Μɼ‫ے‬δετϩϑΟʔɼ೴ੑຑᙺͳͲΛ‫ؚ‬ΉࢮҼͰ͋Γɼʮ॥‫ܥث؀‬ͷ࣬‫ױ‬ʯ͸Ϧ΢Ϛν೤΍৺ෆશɼ‫݂ڏ‬ੑ৺࣬‫ױ‬ɼ೴݂؅࣬‫Ͳͳױ‬Λ‫ؚ‬ΉࢮҼͰ͋Δɽ
͜ͷΑ͏ͳ݁Ռ͕ಘΒΕͨ‫ݪ‬ҼΛௐ΂ΔͨΊʹ࢒ࠩͷ෼ੳΛߦ͏ɽਪఆ஋ͱ‫؍‬ଌ஋ͷ࢒ࠩΛϓϩοτͨ͠ਤ
͕ਤ 9 Ͱ͋Γɼ͔֬ʹத৺෇ۙͰ͸ӈԼ͕Γͷ܏޲͕ಡΈऔΕͨɽ·ͨ͜ͷ࢒ࠩͷ࣌ؒਪҠΛϓϩοτͨ͠ਤ
͕ਤ 10 Ͱ͋Δɽ2 ͭͷ࢒ࠩ͸ 2007 ೥Λ‫্ʹڥ‬Լ͕൓స͠ɼͦͷ‫ޙ‬ʮਆ‫ܥܦ‬ͷ࣬‫ױ‬ʯͷ࢒ࠩʢc6ʣ͸ෛͷํ޲ʹ֦େ͠ɼʮ॥‫ܥث؀‬ͷ࣬‫ױ‬ʯͷ࢒ࠩʢc9ʣ͸ਖ਼ͷํ޲ʹ֦େ͍ͯ͠Δ͜ͱ͕Θ͔ͬͨɽ͜ͷΑ͏ͳ‫΁޲ํٯ‬ͷಈ͖͕૬ؔʹ‫د‬༩͍ͯ͠Δͱߟ͑ΒΕΔ͕ɼ͜ΕΒͷࢮҼؒʹෛͷ૬͕ؔൃੜ͢Δఆੑతͳཧ༝ɾղऍΛ༩͑Δ͜ͱ͸Ͱ͖ͳ͔ͬͨɽ͜ͷ෦෼ʹؔͯ͠͸ҩֶతࢹ఺͔Βͷ‫͕ڀݚ‬๬·ΕΔɽͨͩ͠ࠓճࢮҼͷ෼ྨʹ༻͍ͨ ICD10 ͸ɼ2007 ೥ʹʢICD ͷόʔδϣϯΛม͑ͳ͍ϚΠφʔνΣϯδͷதͰʣେ͖ͳมߋ͕Ճ͑ΒΕ͓ͯΓɼͦͷӨ‫ʹڹ‬Αͬͯ 2007 ೥લ‫ޙ‬ͷτϨϯυมԽ͕‫͍͖ͯى‬ΔՄೳੑ͕͋Δ͕ɺ࢒ࠩͷ τϨϯυͷ޲͖΍େ͖͞ͷཧ༝ɾղऍΛಘΔ͜ͱ͸Ͱ͖ͳ͔ͬͨɽ
ͳ͓͜͜Ͱ͸೥ྸɾੑผͷӨ‫ڹ‬Λഉআ͢ΔͨΊɼ‫؍‬ଌ஋͔Βਪఆ஋ΛҾ͍ͨ࢒ࠩςϯιϧ Rt ʹରͯͦ͠Ε−1ͧΕͷ෼ࢄ‫ڞ‬෼ࢄߦྻͷ‫ྻߦٯ‬Λ͔͚ͨ RCoD
= Rt ×1 Σ−1tage ×3 Σsex ͱ͍͏ม‫׵‬Λߦ͍ͬͯΔɽਤ 9 ͸ͦ͏ͯ͠ಘΒΕͨ࢒ࠩ RCoDʢ͔Βʮਆ‫ܥܦ‬ͷ࣬‫ױ‬ʯͱʮ॥‫ܥث؀‬ͷ࣬‫ױ‬ʯΛநग़ͨ͠΋ͷʣΛͦͷ··ϓϩοτt

͍ͯ͠Δ͕ɼਤ 10 Ͱ͸ࢮҼؒͷൺֱͷͨΊʹͲͪΒ΋ฏ‫ ۉ‬0ɼ෼ࢄ 1 ʹεέʔϦϯά্ͨ͠Ͱɼ೥ྸ࣠ͱੑผ
࣠ͷฏ‫ۉ‬Λऔͬͨ࢒ࠩͷϓϩοτΛ͍ͯ͠Δɽ

リスクと保険

43

1.0
.

0.0

0.5

4e+06

−0.5
−1.0

c9

0e+00
−4e+06c6
c9−3e+06−2e+06−1e+06

0e+00

1e+06

2e+06

1995

1997

1999

2001

2003

c6

ਤ9

2005

2007

2009

2011

2013

2015

time

࢒ࠩϓϩοτਤ 10ฏ‫ؒ࣌ࠩ࢒ۉ‬ਪҠϓϩοτ

5 কདྷਪఆ΁ͷద༻
5.1 κt ͷ֎ૠલઅͰɼఏҊϞσϧ͕ൺֱର৅ͱͨ͠ࢮҼผ Lee-Carter Ϟσϧͱൺ΂ύϥϝʔλ਺Λ࡟‫͕ͳ͠ݮ‬Βߴਫ਼౓ͳਪఆ͕ߦ͍͑ͯΔ͜ͱ͕෼͔ͬͨɽ͜ͷઅͰ͸ɼਪఆͨ͠ κt ͷ֎ૠʹΑΓࢮ๢཰ͷকདྷਪఆΛߦ͏ɽ͔͠
͠ɼͦͷ··কདྷͷ֎ૠΛߦ͏ͱ֬཰ͱ͸‫͍ͳ͑ݴ‬஋ʹൃࢄͯ͠͠·͏ͨΊɼ੍໿৚݅Λ՝͢ɽࢮҼผࢮ๢཰ͷຬͨ͢ੑ࣭ 0 ≤ mx,t,c,s < 1 ΑΓɼ‫߲ࠩޡ‬Λແࢹ͢Δͱ

0 ≤ mx,t,c,s < 1

⇐⇒

⇐⇒

⇐⇒

0 < 1 − mx,t,c,s ≤ 1

log (1 − mx,t,c,s ) ≤ 0
βx,c,s κt ≤ 0(3)ͱ͍͏͕ࣜಘΒΕΔɽύϥϝʔλ (βx,c,s , κt ) ؒʹ͸ −1 ഒͷࣗ༝౓ɼ͢ͳΘͪ�
βx,c,s= −βx,c,s ,κ�t = −κt�ͱ͍͏ม‫Ͱ׵‬ಘΒΕͨ (βx,c,s, κ�t ) ΋Ϟσϧͷ‫ܗ‬Λม͑ͳ͍ͨΊɼ(3) ࣜ͸ βx,c,s ≤ 0ɼκt > 0 ͱͯ͠΋ҰൠੑΛଛͳΘͳ͍ɽͦ͜Ͱຊ‫Ͱڀݚ‬͸ಘΒΕͨ κt ͷର਺ΛͱΓɼlog κt ʹରͯ͠ ARIMA ϞσϧΛ౰ͯ͸ΊΔɽ

AIC ʹ‫ ͍ͯͮج‬ARIMA ϞσϧͷϞσϧબ୒Λߦͬͨͱ͜Ζ࣍ͷ ARIMA(0,1,1) ͕બ୒͞Εͨɽlog κt = log κt−1 + d + �t + θ�t−1ਪఆ͞ΕͨύϥϝʔλΛද 3 ʹࣔ͢ɽ
ද2ਪఆύϥϝʔλύϥϝʔλ਺஋ඪ४‫ࠩޡ‬

d-0.0188

0.0033

θ-0.52200.2628‫ࠩޡ‬෼ࢄ

0.0008265-

ਪఆ͞Εͨ κt ͷ༧ଌฏ‫ۉ‬Λਤ 8 ʹࣔ͢ɽ44

テンソル解析を用いた死因別将来死亡率の同時推定

(4)

0.05

0.10

0.15

0.20

0.25

forecasted kappa

1995

2015

2035

2065

2115

year

ਤ 11

ਪఆ͞Εͨ κt ͱͦͷ֎ૠ

͜͜Ͱɼ2015 ೥·Ͱ͸ (1) ࣜͰఆٛ͞ΕΔఏҊϞσϧʹΑͬͯਪఆͨ͠΋ͷͰ͋Γਤ 7 ͱಉ͡΋ͷͰ͋Δɽ
ͦͯ͠ 2015 ೥͔Β 100 ೥‫ޙ‬ͷ 2115 ೥·Ͱ্͕‫( ه‬4) ࣜͰਪఆͨ͠΋ͷͰ͋ΓɼLee and Carter[1992] ͱಉ༷ʹೋஈ֊ਪఆͷϓϩηεΛ࠾͍ͬͯΔɽ

5.2 ֎ૠ͞Εͨ κt Λ༻͍ͨকདྷࢮ๢཰ͷਪఆ࣍ʹ֎ૠʹΑΓকདྷਪఆ͞Εͨ κt Λ༻͍ͨকདྷࢮ๢཰ͷਪఆ஋Λਤ 12,13 ʹࣔ͢ɽͨͩ͠ਤ 1-4 ͱಉ༷ɼԣ
࣠͸೥ྸ۠෼ɼॎ࣠͸ର਺ࢮ๢཰ log mx,t,c,s Ͱ͋Δɽ
Proposed Model

Proposed Model

−14

−12

1995
2015
(2025)(2065)(2115)

−14

−18

−12

−16

−10

−8

−6

1995
2015
(2025)(2065)(2115)

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

age

ਤ 12

உੑɿ಺෼ൻɼӫཆ‫ͼٴ‬୅ँ࣬‫ױ‬

20−24

30−34

40−44

50−54

60−64

70−74

80−84

age

ਤ 13

ঁੑɿ೛৷ɼ෼ส‫᧘࢈ͼٴ‬͜ͷΑ͏ʹɼఏҊϞσϧ͸௒௕‫ظ‬ͷ༧ଌʹ͓͍ͯ΋άϥϑͷ֓‫ܗ‬Λ่͢͜ͱͳ͘ਪఆ͕ߦ͑Δ͜ͱ͕Θ͔ͬͨɽ͜͜Ͱਤ 13 ͷए೥ྸ૚͸ɼࢮ๢཰͕ 0 Ͱ͋Γର਺ࢮ๢཰͕‫͍ͨͳ͖Ͱࢉܭ‬Ίදࣔ͞Ε͓ͯΒͣɼ‫؍‬ଌࢮ๢཰ͷಛ௃Λྑ͘ଊ͍͑ͯΔͱ‫͑ݴ‬Δɽ

5.3 ࢮҼผࢮ๢཰ͷաখਪఆʹؔͯ͠࠷‫ʹޙ‬ɼ๯಄Ͱड़΂ͨ௨Γ Wilmoth[1995] ͷࣔͨ͠ʮࢮҼผকདྷਪఆ͸ࢮ๢཰վળΛաখਪఆ͢Δʯͱ͍͏಺༰ʹؔͯ֬͠ೝ͢Δɽ͜͜Ͱ͸ 2010 ೥·ͰͷσʔλΛ༻͍ͯɼอ͍࣋ͯ͠Δσʔλͷ࠷৽೥Ͱ͋Δ 2015 ೥·ͰͷਪఆΛߦ͍ɼਪఆ݁Ռͱ‫؍‬ଌ஋ͷൺֱΛߦ͏ɽͨͩ͠ɼෳ਺ͷࢮҼؒͰͷൺֱΛߦ͏ͨΊɼ2010 ೥ͷࢮ

リスクと保険45๢཰Λ 1 ʹ‫ج‬४Խ্ͨ͠ͰൺֱΛߦ͍ɼੑผ͸உੑͷΈͱͨ͠ɽͦͷ݁Ռ͕Լ‫ه‬ͷਤ 14-17 Ͱ͋Δɽ·ͨ͜ͷ

1.5

1.6

ൺֱํ๏͸ੴҪ [2017] ʹଈͨ͠ํ๏Ͱ͋Δɽ

OBS
EST

.

1.1

1.2

1.2

0.9

0.8

1.0

1.0

.

1.3

1.4

1.4

OBS
EST

1995

1997

1999

2001

ਤ 14

2003

2005

2007

2009

2011

2013

2015

1995

1997

1999

‫ײ‬છ঱‫دͼٴ‬ੜ஬঱

2001

2003

ਤ 15

2007

2009

2011

2013

2015

৽ੜ෺ʻजᙾʼ

1.4

OBS
EST

.

1.1
0.8

0.9

0.9

1.0

1.0

1.1

.

1.2

1.2

1.3

1.3

1.4

OBS
EST

2005

1995

1997

1999

ਤ 16

2001

2003

2005

2007

2009

2011

2013

2015

1995

1997

ઌఱ‫ܗح‬ɼม‫ͼٴܗ‬છ৭ମҟৗ

1999

2001

ਤ 17

2003

2005

2007

2009

2011

2013

2015

ইප‫ࢮͼٴ‬๢ͷ֎Ҽ

Wilmoth[1995] ͷओு௨ΓͰ͋Ε͹ɼͲͷࢮҼʹؔͯ͠΋ਪఆ݁ՌΑΓ΋‫؍‬ଌ஋ͷํ͕খ͘͞ͳΔʢ͢ͳΘ࣮ͪઢ͕ԼʹҐஔ͢Δʣ͸ͣͰ͋Δ͕ɼͦͷΑ͏ͳ݁Ռ͸ಘΒΕͳ͔ͬͨɽ

6 ࣌ؒฏ‫߲ۉ‬ͷऔѻ͍ʹؔ͢Δ‫ݕ‬౼ຊ‫ڀݚ‬͸ Lee-Carter ϞσϧͷΞΠσΞΛςϯιϧσʔλʹԠ༻ͨ͠ϞσϧΛఏҊ͕ͨ͠ɼఏҊϞσϧʹ͸

Lee-Carter Ϟσϧͷ࣌ؒฏ‫ ߲ۉ‬αx ʹରԠ͢Δ߲Λઃ͚͍ͯͳ͍ɽຊઅͰ͸࣌ؒฏ‫߲ۉ‬ͷ༗ແʹΑΔਪఆ݁Ռͷࠩʹ͍ͭͯ‫͢ٴݴ‬Δɽ
ຊઅͰൺֱ͢ΔϞσϧ͸

log(1 − mx,t,c,s ) = αx,c,s + βx,c,s κt + εx,t,c,sͱ͍͏ࣜͰද͞ΕΔϞσϧͰ͋Γɼαx,c,s ͷ༗ແҎ֎͸͢΂ͯಉ৚݅ͱ͢Δɽਪఆ݁ՌΛද 3 ʹࣔ͢ɽ
ද3

AIC ͷൺֱϞσϧର਺໬౓ύϥϝʔλ਺

AIC

ఏҊϞσϧ

163577954-325246ൺֱϞσϧ

1685001676-333648͜ͷද͔ΒɼఏҊϞσϧͷ AIC ΑΓ΋ൺֱϞσϧͷ AIC ͷํ͕஋͕খ͍͞ɽ͢ͳΘͪ AIC ʹΑΔఆྔతͳ൑அʹ‫ ͱͮ͘ج‬αx,c,s Λ‫ؚ‬ΉൺֱϞσϧͷํ͕༏Ε͍ͯΔͱ݁࿦෇͚ΒΕΔɽ͔͠͠ɼ͜ͷൺֱϞσϧ͔
46

テンソル解析を用いた死因別将来死亡率の同時推定Βਪఆ͞Εͨকདྷࢮ๢཰Λਤ 18-21 ʹࣔ͢ɽͨͩ͠ਤ 1-4,12,13 ͱಉ༷ɼԣ࣠͸೥ྸ۠෼ɼॎ࣠͸ର਺ࢮ๢཰

log mx,t,c,s Ͱ͋Δɽ−6

Proposed Model

Comparison Model

−10

−8

−6

1995
2015
(2025)(2065)(2115)

−18

−18

−16

−16

−14

−14

−12

−12

−10

−8

1995
2015
(2025)(2065)(2115)

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

20−24

30−34

age

ਤ 18

உੑɿਫ਼ਆ‫ߦͼٴ‬ಈͷো֐ʢఏҊϞσϧʣ

ਤ 19

60−64

70−74

80−84

உੑɿਫ਼ਆ‫ߦͼٴ‬ಈͷো֐ʢൺֱϞσϧʣComparison Model
1995
2015
(2025)(2065)(2115)

−10
−12

−12

−10

−8

−6

1995
2015
(2025)(2065)(2115)

−6

50−54

age

Proposed Model

−8

40−44

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

20−24

30−34

age

ਤ 20

40−44

50−54

60−64

70−74

80−84

age

ঁੑɿইප‫ࢮͼٴ‬๢ͷ֎ҼʢఏҊϞσϧʣਤ 21ঁੑɿইප‫ࢮͼٴ‬๢ͷ֎ҼʢൺֱϞσϧʣ͜ͷΑ͏ʹɼαx,c,s Λ‫ؚ‬Ίͨ৔߹ͷকདྷࢮ๢཰͸ɼ‫؍ࡏݱ‬ଌ͞Ε͍ͯΔάϥϑͷ֓‫ܗ‬Λେ่͖ͯ͘͠͠·͏͜ͱ͕Θ͔ͬͨɽಛʹɼࢮ๢཰্͕ঢ͠ଓ͚Δ೥ྸ૚ͱ‫ݮ‬গ͠ଓ͚Δ೥ྸ૚ʹ෼͔Εͯ͠·͏‫ݱ‬৅͕֬ೝͰ͖ɼ
ਤʹ͍ࣔͯ͠ͳ͍ࢮҼ΋‫ؚ‬Ί΄ͱΜͲͷࢮҼͰ 50 ೥‫ޙ‬ͷ༧ଌͷஈ֊Ͱࢮ๢཰͕ 0 ʹऩଋͯ͠͠·ͬͨɽͨ͠
͕ͬͯ͜ͷϞσϧʹΑΔΩϟογϡϑϩʔ༧ଌ΍ҙࢥܾఆ͸߹ཧతͰͳ͍ͱ൑அͨ͠ɽ
͜͏ͳͬͯ͠·͏‫ݪ‬Ҽͱͯ͠ βx,c,s ͷූ߸҆ఆੑ͕‫͛ڍ‬ΒΕΔɽਤ 20,21 ͷਪఆʹ༻͍ͨ βx,c,s Λϓϩοτ

.

.

0

0

͢ΔͱͦΕͧΕਤ 22,23 ͷΑ͏ʹͳΔɽ

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

age

ਤ 22

ঁੑɿইප‫ࢮͼٴ‬๢ͷ֎ҼʢఏҊϞσϧͷ β ʣ

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

age

ਤ 23

ঁੑɿইප‫ࢮͼٴ‬๢ͷ֎ҼʢൺֱϞσϧͷ β ʣਤΛ‫໌ͯݟ‬Β͔ͳΑ͏ʹఏҊϞσϧ͸ βx,c,s ͕ৗʹෛͰ͋Δͷʹର͠ɼൺֱϞσϧͰ͸ βx,c,s ͷҰ෦ʹਖ਼ͷ

リスクと保険47෦෼͕‫ݱ‬ΕΔɽκt ͸֓Ͷ୯ௐ‫ݮ‬গͰ͋ΔͨΊɼࢮ๢཰ͷ௿Լɾ্ঢ͸ βx,c,s ͷූ߸ʹґଘ͠ɼβx,c,s ͕ਖ਼ͷ෦෼͸ࢮ๢཰͕೥ʑ্ঢ͍ͯ͘͠ͱਪఆ͞ΕΔʢ͜Ε͸ਤ 21,23 Λ‫ͯݟ‬΋໌Β͔Ͱ͋Δʣɽ͕ͨͬͯ͠ βx,c,s ͷූ߸͕ෆ҆ఆͰ͋Δ͔ΒൺֱϞσϧ͸‫ܗ‬ঢ়͕҆ఆ͠ͳ͍ɼ‫͑׵͍ݴ‬Δͱ βx,c,s ͷූ߸҆ఆੑ͕ఏҊϞσϧͷ‫ܗ‬ঢ়‫݈ؤ‬ੑΛߴΊ͍ͯΔཧ༝ͷ 1 ͭͩͱ‫͑ݴ‬Δɽ

7 શࢮҼࢮ๢཰ͱ Lee-Carter ϞσϧͱͷൺֱຊઅͰ͸ఏҊϞσϧʹΑΓಘΒΕͨࢮҼผਪఆ஋Λ߹ࢉ͠શࢮҼࢮ๢཰ʹ্ͨ͠Ͱɼ௨ৗͷ Lee-Carter ϞσϧͱͷൺֱΛߦ͏ɽ͢ͳΘͪਪఆ͞ΕͨࢮҼผࢮ๢཰ mx,t,c,s ʹରͯ͠ log

c mx,t,c,s Λ Lee-Carter Ϟσϧͱൺֱ͢Δɽ͔͠͠ɼ‫߲ࠩޡ‬Λ‫ؚ‬ΊΔͱ෼෍͕ཅʹදͤͳ͍ͨΊฏ‫ߏۉ‬଄ͷΈͰൺֱΛߦͬͨɽਪఆ݁ՌΛਤ 24-26 ʹɼ‫؍‬ଌ஋ͱͷ norm percent ʹΑΔൺֱ݁ՌΛද 4 ʹද͢ɽ͜͜Ͱ norm percent ͷఆٛ͸ࣜ (2)ͱಉ༷Ͱ͋Δɽ
Raw Mortality

−12

−10

−8

−6

−4

−2

1995
2015

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

age

ਤ 24

‫؍‬ଌ஋

Lee−Carter

Proposed Model

−10

−8

−6

−4

−2

1995
2015
(2025)(2065)(2115)

−12

−12

−10

−8

−6

−4

−2

1995
2015
(2025)(2065)(2115)

0

1−4

10−14

20−24

30−34

40−44

50−54

60−64

70−74

80−84

0

1−4

10−14

20−24

30−34

age

ਤ 25

50−54

age

Lee-Carter Ϟσϧ

ਤ 26

ද4

48

40−44

norm percent ͷൺֱϞσϧnorm percent

ఏҊϞσϧ

75.49

Lee-Carter Ϟσϧ84.37

テンソル解析を用いた死因別将来死亡率の同時推定

ఏҊϞσϧ

60−64

70−74

80−84

͜ΕΒͷਤɾද͔ΒɼఏҊϞσϧ͸શମࢮ๢཰ʹରͯ͠௨ৗͷ Lee-Carter Ϟσϧ΄Ͳͷσʔλઆ໌ྗ͸࣋ͨͳ͍΋ͷͷ 75% ͱ͍͏ൺֱతߴ͍਺஋ΛಘΔ͜ͱ͕ग़དྷͨɽ·ͨকདྷ༧ଌͷ‫Ͱ఺؍‬͸ɼLee-Carter Ϟσϧ͕લઅͷൺֱϞσϧͱಉ͡Α͏ʹ௒௕‫ظ‬ͷ༧ଌʹ͓͍ͯάϥϑͷ֓‫ܗ‬Λ่ͯ͠͠·͏ͷʹରͯ͠ɼఏҊϞσϧ͸શࢮҼࢮ๢཰ͷ༧ଌʹ͓͍ͯ΋άϥϑͷ֓‫ܗ‬Λ่ͣ҆͞ఆͨ͠ਪఆ஋ΛಘΔ͜ͱ͕ग़དྷͨɽ

8 ·ͱΊͱ՝୊
ຊ‫Ͱڀݚ‬͸ Lee-Carter ϞσϧͱςϯιϧղੳͷΞΠσΞΛ‫ʹج‬ɼ೥ྸɾྐྵ೥ʹՃ͑ࢮҼɾੑผͷσʔλΛ಺แͨ͠ 4 ֊ςϯιϧͷσʔλʹରͯ͠৽ͨͳࢮ๢཰ϞσϧͷఏҊΛߦͬͨɽࢮҼผࢮ๢཰ΛಠཱͰ͸ͳ͘ಉ࣌ʹਪఆ͢Δ͜ͱͰ೥ྸɾੑผʹΑΒͳ͍ࢮҼؒͷ૬ؔؔ܎Λ‫ؚ‬ΜͩਪఆΛ͢Δ͜ͱ͕Ͱ͖ɼશࢮҼࢮ๢཰ͱ੔߹తͳ݁ՌΛಘΔ͜ͱ͕ग़དྷͨɽՃ͑ͯ௕‫ظ‬ͷ֎ૠʹ͓͚Δ‫ܗ‬ঢ়‫݈ؤ‬ੑΛ֬อ͢Δͱͱ΋ʹɼWilmoth ͷࢦఠ͢ΔࢮҼผࢮ๢཰վળͷաখਪఆ໰୊Λճආ͢Δ͜ͱ͕ग़དྷͨɽ͜ΕʹΑΓࢮҼ͝ͱͷτϨϯυ΍কདྷ༧ଌͱಉ࣌ʹશࢮҼࢮ๢཰ͷτϨϯυ΍কདྷ༧ଌΛߦ͏͜ͱ͕Ͱ͖ɼอ‫ݥ‬঎඼ͷ։ൃ΍ࢮ๢཰Λ༻͍Δ‫ۀا‬ͷҙࢥܾఆʹԠ༻ग़དྷΔͱߟ͍͑ͯΔɽ
কདྷ༧ଌ΍શࢮҼࢮ๢཰ͷൺֱͰ͸ɼຊ‫ڀݚ‬͸ฏ‫ߏۉ‬଄ʹͷΈ஫໨͕࣮ͨ͠ࡍ͸෼ࢄߏ଄ʹΑΓଟগͷζϨ͕ੜ͡ΔͷͰɼγϛϡϨʔγϣϯ౳Λ௨ͯͦ͡ͷζϨ͕ͲͷΑ͏ͳӨ‫ڹ‬Λ༩͑Δ͔Λௐ΂Δඞཁ͕͋ΔɽಛʹɼࢮҼผͷର਺ࢮ๢཰ log(1 − mx,t,c,s ) ͸ਖ਼‫ن‬෼෍ͱԾఆ͍ͯ͠Δ΋ͷͷɼࢮҼ߹ࢉ‫ޙ‬ͷର਺ࢮ๢཰

log



c mx,t,c,s ͸΋͸΍ਖ਼‫ن‬෼෍ʹै͍ͬͯͳ͍ͷͰɼ੄ͷॏ͞౳ͷ෼෍‫ؔ͢ʹܗ‬Δ৘ใ΋ௐ΂Δඞཁ͕͋ΔͷͰࠓ‫ޙ‬ͷ՝୊ͱ͍ͨ͠ɽ·ͨࠓճ͸ਪఆΞϧΰϦζϜͷ҆ఆੑɾ‫ࢉܭ‬଎౓ʹؑΈͯςϯιϧਖ਼‫ن‬෼෍ΛԾఆ͕ͨ͠ɼෳࡶͳ૬ؔΛද‫͢ݱ‬Δํ๏ͱͯ͠पล෼෍ʹϙΞιϯ෼෍΍ೋ߲෼෍Λ༻͍ɼ૬ؔߏ଄ʹίϐϡϥΛ
༻͍Δํ๏΋ߟ͑ΒΕΔͷͰͦ͏͍ͬͨϞσϧ΋‫ݕ‬౼͍͖͍ͯͨ͠ɽͨͩͦ͠͏͍ͬͨෳࡶͳϞσϧ͕෼෍ͷ‫ܗ‬ঢ়‫݈ؤ‬ੑΛอ͔ͭͲ͏͔ɼ࣮຿ʹ଱͑͏ΔղऍՄೳੑ΍‫ࢉܭ‬଎౓ɼਪఆͷ҆ఆੑΛ΋͔ͭͲ͏͔͸ະ஌ͷ෦෼ͳͷͰɼࠓ‫ޙ‬ͷ՝୊ͱͯͦͪ͠Β΋ௐ͍ࠪͨ͠ɽ

ँࣙຊ࿦จͷ౤ߘʹ͋ͨΓɼ༗ӹͳΞυόΠεΛ௖͍ͨಗ໊ࠪಡऀʹ‫͢ँײ‬Δɽ

リスクと保険49ࢀߟจ‫ݙ‬[1] ੴҪଠ [2017], ʮ೔ຊͷকདྷਪ‫ܭ‬ਓ‫ޱ‬ʯ, http://www.actuaries.jp/lib/meeting/reikai29-7-siryo.pdf,
ʢ2020 ೥ 1 ݄ 5 ೔Ӿཡʣ[2] ໺ଜढ़Ұ [2018], ʮςϯιϧ෼ղΛར༻ͨ͠ Lee-Carter Ϟσϧͷ֦ுͱ౎ಓ෎‫ݝ‬ผੜ໋දղੳʯ, ʰJARIP‫ൃڀݚ‬දେձ 2018 ༧ߘूʱ[3] Booth, H., Maindonald, J., and Smith, L. [2002], ”Applying Lee-Carter under conditions of variablemortality decline”, Population studies, 56(3), 325-336.
[4] Brouhns, N., Denuit, M., and Vermunt, J. K. [2002], ”A Poisson log-bilinear regression approach tothe construction of projected lifetables”, Insurance: Mathematics and economics, 31(3), 373-393.
[5] Cairns, A. J. [2000], ”A discussion of parameter and model uncertainty in insurance”, Insurance:
Mathematics and Economics, 27(3), 313-330.
[6] Cairns, A. J., Blake, D., and Dowd, K. [2006], ”A two ʖ factor model for stochastic mortality withparameter uncertainty: theory and calibration”, Journal of Risk and Insurance, 73(4), 687-718.
[7] Girosi, F., and King, G. [2007], ”Understanding the Lee-Carter mortality forecasting method”, Gking.
Harvard. Edu.
[8] Hunter, G., and Luis, C. [2018], ”An expectation–maximization algorithm for the matrix normal
distribution with an application in remote sensing”, Journal of Multivariate Analysis, 167, 31-48.
[9] Kolda, T. G., and Bader, B. W. [2009], ”Tensor decompositions and applications”, SIAM review,
51(3), 455-500.
[10] Lee, R. D., and Carter, L. R. [1992], ”Modeling and forecasting US mortality”, Journal of theAmerican statistical association, 87(419), 659-671.
[11] Lee, R. [2000], ”The Lee-Carter method for forecasting mortality, with various extensions and applications”, North American actuarial journal, 4(1), 80-91.
[12] Li, J. S. H., Hardy, M. R., amd Tan, K. S. [2009], ”Uncertainty in mortality forecasting: an extension
to the classical Lee-Carter approach”, ASTIN Bulletin: The Journal of the IAA, 39(1), 137-164.
[13] Manceur, A. M., and Dutilleul, P. [2013], ”Maximum likelihood estimation for the tensor normal
distribution: Algorithm, minimum sample size, and empirical bias and dispersion”, Journal of Computational and Applied Mathematics, 239, 37-49.
[14] Renshaw, A. E., and Haberman, S. [2003], ”Lee–Carter mortality forecasting with age-speciﬁc enhancement”, Insurance: Mathematics and Economics, 33(2), 255-272.
[15] Renshaw, A. E., and Haberman, S. [2006], ”A cohort-based extension to the Lee–Carter model for
mortality reduction factors”, Insurance: Mathematics and economics, 38(3), 556-570.
[16] Wilmoth, J. R. [1995], ”Are mortality projections always more pessimistic when disaggregated bycause of death?”, Mathematical Population Studies, 5(4), 293-319.

50テンソル解析を用いた死因別将来死亡率の同時推定෇࿥ Aςϯιϧͷੑ࣭

A.1 ςϯιϧͷಋೖຊ࿦จͰ͸ςϯιϧΛϓϩάϥϛϯάʹ͓͚Δଟ࣍‫ݩ‬഑ྻͱಉ༷ʹѻ͏ɽ͢ͳΘͪɼϕΫτϧ͕ 1 ࣍‫ݩ‬഑ྻɼߦྻ͕ 2 ࣍‫ݩ‬഑ྻͱͯ͠ѻΘΕΔ͜ͱͷྨਪͱͯ͠ n ࣍‫ݩ‬഑ྻʹରԠ͢Δςϯιϧ X Λ

X = (xi1 ,i2 ,...,in ) ∈ RN1 ×N2 ×···×Nnͱͯ͠ѻ͏ɽ͜Ε͸ 2 × 2 ߦྻ A Λ A = (ai,j ) ΍ A ∈ R2×2 ͱද‫͢ه‬Δ͜ͱʹରԠ͓ͯ͠Γɼn Λςϯιϧͷ֊਺ɼ(N1 , N2 , . . . Nn ) Λςϯιϧͷ࣍‫͍͍ͱݩ‬ɼ֤ఴࣈΛςϯιϧͷ࣠·ͨ͸Ϟʔυͱ͍͏ɽςϯιϧΛ਺ֶతʹ‫ʹີݫ‬ఆٛ͢Δʹ͸ςϯιϧੵʹΑΔํ๏΍ଟॏઢ‫૾ࣸܕ‬Λ༻͍ͨํ๏ͳͲ͕஌ΒΕ͍ͯΔ͕ɼຊ࿦จͰ͸ςϯιϧͷ‫ͳີݫ‬ఆٛ·Ͱ͸‫͍ͳ͠ٴݴ‬ɽ

A.2 ςϯιϧͷԋࢉςϯιϧͷࣜม‫Ͱܗ‬සൟʹ༻͍ΒΕΔϕΫτϧԽ (vectorization) ࡞༻ૉͱߦྻԽ (matricization) ࡞༻ૉΛ
ఆٛ͢Δɽུͨͩ͠‫ ͯ͠ͱه‬N =�nk=1 Nk ɼN¬m = N/Nm Λ༻͍ɼ૯࿨ͷൣғ΋লུ͢Δ͜ͱ͕͋Δɽ·ͣςϯιϧͷϕΫτϧԽ࡞༻ૉ vec : RN1 ×N2 ×···×Nn → RN ͸

xi1 ,i2 ,...,in = [vec X ]i
ͨͩ͠

i=1+

n
�

k=1

Λຬͨ͢Α͏ʹఆٛ͞ΕΔɽͨͩ͠ k = 1 ͷͱ͖(ik − 1)

�0

k−1
�

Nl

l=1

l=1 Nl = 1 ͱղऍ͢Δɽ·ͨ‫ʹٯ‬ɼi ͱ (i1 , i2 , . . . , in )͸ҰରҰରԠͰ͋ΔͨΊɼʢఴࣈͷಈ͖͏ΔൣғͰͷʣ೚ҙͷ i ʹ͍ͭͯ [vec X ]i = xi1 ,i2 ,...,in Λຬͨ͢(i1 , i2 , . . . , in ) ͕ଘࡏ͢Δɽ͜Ε͸ϕΫτϧԽ࡞༻ૉ͕ςϯιϧΛҰྻʹฒͼସ͍͑ͯΔ͚ͩͰ͋Δ͜ͱ͔Β΋໌Β͔Ͱ͋Δɽ࣍ʹςϯιϧͷϞʔυ m ʹؔ͢ΔߦྻԽ࡞༻ૉ •(m) : RN1 ×N2 ×···×Nn → RNm ×N¬m ͸ɼϕΫτϧԽ࡞༻ૉΛ༻͍ͯX(m) = (vec(Xim =1 ), vec(Xim =2 ), . . . , vec(Xim =Nm ))Tͱఆٛ͢Δɽ͜͜Ͱ Xim =k ͸ςϯιϧͷ࣠ m ʹؔ͢ΔεϥΠεͰ͋Γ��Xim =k = xi1 ,...im−1 ,k,im+1 ,...,in ∈ RN¬m

Ͱ͋Δɽ͜Ε͸ϓϩάϥϛϯάʹ͓͍ͯσʔληοτͷಛఆͷྻΛநग़͢Δ͜ͱʹରԠ͍ͯ͠Δɽ࣍ʹςϯιϧͷϊϧϜΛఆٛ͢Δɽςϯιϧͷϑϩϕχ΢εϊϧϜ � • �F : RN1 ×N2 ×···×Nn → R ͸⎛

�X �F = ⎝�i1 ,i2 ,...,in⎞ 12x2i1 ,i2 ,...,in ⎠ͱఆٛ͞Εɼߦྻͷϑϩϕχ΢εϊϧϜͱಉ༷ʹશཁૉͷ 2 ৐࿨ͷฏํࠜͰ͋Δɽࠓ·Ͱ͸ 1 ͭͷςϯιϧʹର͢Δ࡞༻ૉͰ͕͋ͬͨɼςϯιϧͱߦྻͷؒͷੵͱͯ͠ςϯιϧͷϞʔυੵͱ͍͏ԋࢉ͕஌ΒΕ͍ͯΔɽςϯιϧͷϞʔυ m ʹର͢ΔϞʔυੵ • ×m • : RN1 ×···Nm ···×Nn × RRm ×Nm →RN1 ×···Rm ···×Nn ͸࣍ͷࣜͰ༩͑ΒΕΔԋࢉͰ͋Δɽ

[X ×m A]i1 ,...,rm ,...,in =Nm�arm ,im xi1 ,...im ...,in

im =1

リスクと保険51

·ͨɼ‫߸ه‬×ni=1 Ai Λ ×1 A1 ×2 · · · ×n An ͷུ‫͢ͱه‬Δɽ

͜͜ͰςϯιϧͷϞʔυੵͱϕΫτϧԽ࡞༻ૉΛ݁ͼ͚ͭΔ࣍ͷެ͕ࣜଘࡏ͢Δɽͨͩ͠ ⊗ ͸ߦྻͷΫϩωοΧʔੵͰ͋Δɽ
໋୊ 1.


 n
vec X × Ai = (AN ⊗ · · · ⊗ A1 ) vec X
i=1

(k)

[ূ໌] ςϯιϧ X ×i=1 Ai ͷఴࣈΛ੔ཧ͢Δɽߦྻ Ak ͷཁૉΛ ark ,ik Ͱද͢ͱɼϞʔυੵͷఆٛΑΓnn

X × Aii=1

=r1 ,...,rn



(1)

(n)

xi1 ,...,in ar1 ,i1 · · · arn ,in

i1 ,...,in࣍ʹΫϩωοΧʔੵͷఆ͔ٛΒ
(1)

(n)

ar1 ,i1 · · · arn ,in = [AN ⊗ · · · ⊗ A1 ]r,i
ͨͩ͠

r =1+

n


k=1

i=1+

n


k=1

(rk − 1)
(ik − 1)

k−1


Rl ,

l=1

k−1


Nl

l=1

·ͨɼ্ࣜͷ i ͱ (i1 , . . . , in )ɼͦͯ͠ r ͱ (r1 , . . . , rn ) ͕ͦΕͧΕҰରҰରԠͰ͋Δ͜ͱΛ༻͍Δͱఴࣈͷಈ
͖͏Δൣғͷ೚ҙͷ r ʹରͯ͠[(AN ⊗ · · · ⊗ A1 ) vec X ]r =

i=[AN ⊗ · · · ⊗ A1 ]r,i [vec X ]i(1)i1 ,...,in(n)xi1 ,...,in ar1 ,i1 · · · arn ,inΛຬͨ͢ (r1 , . . . , rn ) ͕ଘࡏ͢ΔɽΑͬͯ೚ҙͷ r Ͱ[(AN ⊗ · · · ⊗ A1 ) vec X ]r =

i1 ,...,in



 n
(1)
(n)
xi1 ,...,in ar1 ,i1 · · · arn ,in = vec X × Ai
i=1

Ͱ͋Γɼ͢ͳΘͪ
 nvec X × Ai = (AN ⊗ · · · ⊗ A1 ) vec X
i=1

͕੒Γཱͭɽ
ಉ༷ʹϞʔυੵͱߦྻԽ࡞༻ૉͷؒʹ΋Լ‫ه‬ͷΑ͏ͳؔ܎͕੒Γཱ͕ͭূ໌͸লུ͢Δɽ
໋୊ 2.

ͨͩ͠

52



n

X × Aii=1(m)= Am X(m) AT¬mA¬m = An ⊗ · · · ⊗ Am+1 ⊗ Am−1 ⊗ · · · ⊗ A1

テンソル解析を用いた死因別将来死亡率の同時推定

r

෇࿥ B

ςϯιϧਖ਼‫ن‬෼෍

B.1 ςϯιϧਖ਼‫ن‬෼෍ͷఆٛX Λ n ֊ςϯιϧͷۭؒ RN1 ×N2 ×···×Nn ʹ஋ΛͱΔ֬཰ม਺ͱ͠ɼ࣍ͷΑ͏ͳີ౓ؔ਺Λ࣋ͭͱ͢Δɽ


2 
n
n


N
1
−
−N
2N

i exp −(X − M) × Ui f (X ) = (2π) 2
|Σi |


2
i=1
F
i=1

͜ͷͱ͖ X ͸ฏ‫ ۉ‬Mɼ෼ࢄ Σ1 , Σ2 , . . . , Σn ͷςϯιϧਖ਼‫͍ݴͱ͏ैʹن‬ɼX ∼ TN(M, Σ1 , Σ2 , . . . , Σn )ͱද͢ɽ͜͜Ͱ Ui ͸ UiT Ui = Σ−1i Λຬͨ͢ߦྻͰ͋Δʢςϯιϧਖ਼‫ن‬෼෍ͷৄࡉ͸ Manceur et al.[2013] Λࢀরʣɽ

B.2 ଟมྔਖ਼‫ن‬෼෍ͱͷؔ܎ςϯιϧਖ਼‫ن‬෼෍͸ଟมྔਖ਼‫ن‬෼෍Λ༻͍ͯ࣍ͷΑ͏ʹ‫ه‬ड़Ͱ͖Δ͜ͱ͕஌ΒΕ͍ͯΔɽ໋୊ 3.

X ∼ TN(M, Σ1 , Σ2 , . . . , Σn )ͨͩ͠x = vec X ,m = vec M,⇐⇒

x ∼ N(m, Σ)(B.1)

Σ = Σ n ⊗ · · · ⊗ Σ1[ূ໌] ·ͣϑϩϕχ΢εϊϧϜΛϞʔυੵͱΫϩωοΧʔੵͷੑ࣭Λ༻͍ͯม‫͢ܗ‬Δͱ

2

2

n
n




(X − M) × Ui  = vec (X − M) × Ui 




i=1

F

i=1

F
2

= �(Un ⊗ Un−1 ⊗ . . . ⊗ U1 ) vec (X − M)�F
T

= (x − m)T (Un ⊗ Un−1 ⊗ . . . ⊗ U1 ) (Un ⊗ Un−1 ⊗ . . . ⊗ U1 ) (x − m)T= (x − m)T UnT Un ⊗ Un−1
Un−1 ⊗ . . . ⊗ U1T U1 (x − m)


−1
−1
= (x − m)T Σ−1(x − m)n ⊗ Σn−1 ⊗ . . . ⊗ Σ1
= (x − m)T Σ−1 (x − m)

(B.2)

Αͬͯςϯιϧਖ਼‫ن‬෼෍ͷΧʔωϧʢਖ਼‫ن‬Խ߲Λআ͍ͨ෦෼ʣΛ‫͢ࢉܭ‬Δͱɼ্ࣜΑΓ


2 


n

1
1
T −1

=
exp
−
(X
−
M)
(x
−
m)
exp − 
U
Σ
(x
−
m)
× i
2
2
i=1
F


ӈล͸ଟมྔਖ਼‫ن‬෼෍ N(m, Σ) ͷΧʔωϧͰ͋Δɽ͕ͨͬͯ͠

X ∼ TN(M, Σ1 , Σ2 , . . . , Σn )⇐⇒x ∼ N(m, Σ)͕੒Γཱͭɽ

リスクと保険53

B.3 ύϥϝʔλͷ࠷໬ਪఆྔ
X ∼ TN(M, Σ1 , Σ2 , . . . , Σn ) ΛԾఆ͢Δɽ‫؍‬ଌσʔλ X1 , . . . , XT ͷର਺໬౓͸


2 
T
n
n



N
1
N

log L =−
log |Σi | − (Xt − M) × Ui 
− log 2π +

2
2Ni
2
i=1
F
t=1
i=1
2
n
T 
n




NT
NT
1
(Xt − M) × Ui =−log 2π +−log |Σi | −


2
2N
2
i
i=1
F
t=1
i=1

(B.3)

(B.4)

ͱॻ͚Δɽ

B.3.1 ฏ‫ۉ‬ϕΫτϧԽ্ͨ͠Ͱ໬౓ํఔࣜΛ‫ٻ‬ΊΔɽ‫߸ه‬͸ (B.1) ͱ‫ڞ‬௨ͱ͠ɼ(B.2) Λ༻͍Δͱ

2
T 
n

∂
1 ∂ 
(Xt − M) × Ui log L = −∂m
2 ∂m t=1
i=1
F
=−

T
1 ∂ 
(xt − m)T Σ−1 (xt − m)
2 ∂m t=1= −Σ−1
͕ͨͬͯ͠∂log L = 0
∂m

T

t=1

(xt − m)

T

⇐⇒

m=

1
xt
T t=1

T

⇐⇒

M=

͢ͳΘͪɼςϯιϧਖ਼‫ن‬෼෍ʹ͓͍ͯ΋ඪຊฏ‫࠷͕ۉ‬໬ਪఆྔʹͳΔɽ

1
Xt
T t=1

B.3.2 ෼ࢄ‫ڞ‬෼ࢄߦྻςϯιϧਖ਼‫ن‬෼෍͸෼ࢄ‫ڞ‬෼ࢄߦྻΛ࣠ͷ਺͚ͩ࣋ͭͱ͜Ζ͕ଟมྔਖ਼‫ن‬෼෍ͱͷେ͖ͳҧ͍Ͱ͋Γɼશͯͷ෼ࢄ‫ڞ‬෼ࢄߦྻͷ໬౓ํఔࣜΛ࿈ཱͤͨ͞ղΛղੳతʹ‫ٻ‬ΊΔ͜ͱ͕ग़དྷͳ͍ɽͦ͜Ͱަ‫࠷ޓ‬దԽͷ࿮૊ΈΛద༻͠ɼଞͷύϥϝʔλΛ‫ݻ‬ఆͨ͠͏͑Ͱɼ1 ͭͷ෼ࢄ‫ڞ‬෼ࢄߦྻʹରͯ͠໬౓ํఔࣜΛղ͘͜ͱΛߟ͑Δɽ

Rt = Xt − M ͱͯ͠ɼΣk ͷ໬౓ํఔࣜΛ‫ٻ‬ΊΔͨΊʹϞʔυ k ͰͷߦྻԽΛߦ͏ͱ
2
T 
n

∂
NT ∂
1 ∂ 

Rt × U i log L = −log |Σk | −∂Σk2Nk ∂Σk
2 ∂Σk t=1  i=1 F
=−
=−
=−

T

N T −T1 ∂ T 2
Uk Rt,(k) U¬k
Σk −
F
2Nk
2 t=1 ∂ΣkT
N T −T1 ∂TΣk −Tr U¬k RTt,(k) UkT Uk Rt,(k) U¬k
2Nk
2 t=1 ∂ΣkT
N T −T1 ∂TΣk −Tr U¬k RTt,(k) Σ−1
R
U
t,(k)
¬k
k
2Nk
2 t=1 ∂ΣkT
N T −T1  −TT=−Σ +Σ Rt,(k) U¬kU¬k RTt,(k) Σ−Tk
2Nk k
2 t=1 k=−54

T
N T −T1  −T−TTΣk +Σ Rt,(k) Σ−1¬k Rt,(k) Σk
2Nk
2 t=1 kテンソル解析を用いた死因別将来死亡率の同時推定͕ͨͬͯ͠

∂log L = 0
∂Σk

⇐⇒

T
N T −T1  −T−TTΣk =Σ Rt,(k) Σ−1¬k Rt,(k) Σk
2Nk
2 t=1 k

⇐⇒

Σk

⇐⇒

Nk 
T
Σk =
Rt,(k) Σ−1¬k Rt,(k)N T t=1

TNT
T=
Rt,(k) Σ−1¬k Rt,(k)Nkt=1T͜ͷࣜͱద౰ʹ༩͑ͨॳ‫ظ‬஋ʹΑͬͯ෼ࢄ‫ڞ‬෼ࢄߦྻΛஞ࣍తʹ‫ٻ‬Ί͍ͯ͘ɽ͔͠͠ɼςϯιϧਖ਼‫ن‬෼෍ͷ෼ࢄ‫ڞ‬෼ࢄߦྻ͸ੵԋࢉʹΑͬͯ݁͹Ε͍ͯΔͨΊɼεέʔϧม‫׵‬ͷҰҙੑΛ࣋ͨͳ͍ɽͦ͜Ͱ෼ࢄ‫ڞ‬෼ࢄߦྻͷ஋Λ n − 1 ‫ݻݸ‬ఆͯ͠ʢྫ͑͹ Σ1 [1, 1] = 1ɼΣ2 [1, 1] = 1 ͱ͢Δ౳ʣͦͷ‫ޙ‬x ∼ N(m, sΣ)Λ M, Σ1 , . . . , Σn ͕‫ط‬஌ͷ΋ͱͰ s ʹ͍ͭͯ࠷దԽ͢Δͱ͍͏ํ๏͕஌ΒΕ͍ͯΔʢHunter and Luis[2018]Λࢀরʣɽ͜ͷͱ͖ͷ s ͷਪఆྔ͸∂log L = 0
∂s

⇐⇒

s=

Ͱ༩͑ΒΕΔɽ


2
n

1 
 Rt × U i 


NT
i=1
F

·ͨɼຊ࿦จͰ͸෼ࢄ‫ڞ‬෼ࢄߦྻΛର֯ߦྻʹ੍‫͠ݶ‬ਪఆΛߦ͍ͬͯΔ͕ɼࠓճ͸ͦͷ৔߹ͷਪఆࣜΛಋग़
͢Δ͜ͱ͸ͤͣʹɼ্‫ه‬ͷ‫܁‬Γฦ͠‫ࢉܭ‬ͷ్தͰඇର֯੒෼ʹ 0 Λ୅ೖ͢Δ͜ͱʹΑͬͯର֯ߦྻΛಘͨɽࣜͰॻ͘ͱ͢Ε͹Լ‫ه‬ͷ௨ΓͰ͋ΔɽΣk = diag



T

Nk 
T
Rt,(k) Σ−1¬k Rt,(k)N T t=1



リスクと保険

55

ผද
ผද 1

56

ICD10 ४‫جڌ‬ຊ෼ྨදୈ1ষ‫ײ‬છ঱‫دͼٴ‬ੜ஬঱

ୈ2ষ

৽ੜ෺

ୈ3ষ

݂ӷ‫ͼٴ‬଄݂‫ث‬ͷ࣬‫ױ‬ฒͼʹ໔Ӹ‫ߏػ‬ͷো֐

ୈ4ষ಺෼ൻɼӫཆ‫ͼٴ‬୅ँ࣬‫ױ‬ୈ5ষਫ਼ਆ‫ߦͼٴ‬ಈͷো֐ୈ6ষਆ‫ܥܦ‬ͷ࣬‫ױ‬ୈ7ষ‫෇ͼٴ؟‬ଐ‫ث‬ͷ࣬‫ױ‬ୈ8ষࣖ‫ͼٴ‬ೕ༷ಥ‫ى‬ͷ࣬‫ױ‬ୈ9ষ॥‫ܥث؀‬ͷ࣬‫ױ‬

ୈ 10 ষ

‫ܥثٵݺ‬ͷ࣬‫ױ‬

ୈ 11 ষ

ফԽ‫ܥث‬ͷ࣬‫ױ‬

ୈ 12 ষ

ൽෘ‫ͼٴ‬ൽԼ૊৫ͷ࣬‫ױ‬

ୈ 13 ষ

‫͓ܥ֨ࠎے‬Αͼ݁߹૊৫ͷ࣬‫ױ‬

ୈ 14 ষ

ਛ೘࿏ੜ৩‫ܥث‬ͷ࣬‫ױ‬

ୈ 15 ষ

೛৷ɼ෼ส‫͡࢈ͼٴ‬ΐ͘ʻ᧘ʼୈ 16 ষ

प࢈‫ൃʹظ‬ੜͨ͠පଶ

ୈ 17 ষ

ઌఱ‫ܗح‬ɼม‫ͼٴܗ‬છ৭ମҟৗ

ୈ 18 ষ

঱ঢ়ɼ௃ީ‫ͼٴ‬ҟৗྟচॴ‫ݟ‬ɾҟৗ‫Ͱݟॴࠪݕ‬ଞʹ෼ྨ͞Εͳ͍΋ͷ

ୈ 19 ষ

ଛইɼதಟ‫ͦͼٴ‬ͷଞͷ֎ҼͷӨ‫ڹ‬

ୈ 20 ষ

ইප‫ࢮͼٴ‬๢ͷ֎Ҽ

ୈ 21 ষ

݈߁ঢ়ଶʹӨ‫ڹ‬Λ‫͢΅ٴ‬ཁҼ‫ͼٴ‬อ݈αʔϏεͷར༻ୈ 22 ষ

ಛघ໨త༻ίʔυ

テンソル解析を用いた死因別将来死亡率の同時推定

Joint estimation of cause-speciﬁc future mortality using tensor
analysis
Kotaro Suzuki

Naoki Matsuyama

Abstract
In this article, we propose a new method for joint estimation of cause-speciﬁc mortalities. In generally,
estimation of cause-speciﬁc mortalities cannot reﬂect interactions between causes of death, therefore
the total of them does not make sense. For this reason, we propose a Lee-Carter type cause-speciﬁcmortalities model using single time development factor and tensor structure with four modes: age, time,cause of death, and sex. This method clariﬁes the correlation between causes of death and achieves robust
extrapolation of cause-speciﬁc mortalities consistent with all-cause mortality.

リスクと保険57