Yilda ishonchlilik nazariyasi , ta'lim sohasi aktuar fan , Bühlmann modeli a tasodifiy effektlar modeli (yoki "variatsion komponentlar modeli" yoki ierarxik chiziqli model ) mosligini aniqlash uchun ishlatiladi premium sug'urta shartnomalari guruhi uchun. Model 1967 yilda birinchi marta tavsifini nashr etgan Xans Byuhlman nomidan olingan.[1]
Model tavsifi
Ko'rib chiqing men tarixiy ma'lumotlar tasodifiy yo'qotishlarni keltirib chiqaradigan xatarlar m so'nggi da'volar mavjud (indekslangan j ). Uchun mukofot men Talablarning kutilayotgan qiymatidan kelib chiqib, tavakkalchilik aniqlanishi kerak. O'rtacha kvadrat xatosini minimallashtiradigan chiziqli taxminchi izlanadi. Yozing
X ij uchun j - da'vo men -chinchi xavf (biz barcha da'volarni qabul qilamiz deb taxmin qilamiz men - xavf mustaqil va bir xil taqsimlangan ) X ¯ men = 1 m ∑ j = 1 m X men j {displaystyle stsenariysi {ar {X}} _ {i} = {frac {1} {m}} sum _ {j = 1} ^ {m} X_ {ij}} Θ men {displaystyle Theta _ {i}} m ( ϑ ) = E [ X men j | Θ men = ϑ ] {displaystyle m (vartheta) = operator nomi {E} chapda [X_ {ij} | Theta _ {i} = vartheta ight]} Π = E ( m ( ϑ ) | X men 1 , X men 2 , . . . X men m ) {displaystyle Pi = operator nomi {E} (m (vartheta) | X_ {i1}, X_ {i2}, ... X_ {im})} m = E ( m ( ϑ ) ) {displaystyle mu = operator nomi {E} (m (varteta))} s 2 ( ϑ ) = Var [ X men j | Θ men = ϑ ] {displaystyle s ^ {2} (varteta) = operator nomi {Var} chapda [X_ {ij} | Theta _ {i} = vartheta ight]} σ 2 = E [ s 2 ( ϑ ) ] {displaystyle sigma ^ {2} = operator nomi {E} chap [s ^ {2} (vartheta) ight]} v 2 = Var [ m ( ϑ ) ] {displaystyle v ^ {2} = operator nomi {Var} chap [m (vartheta) ight]} Eslatma: m ( ϑ ) {displaystyle m (varteta)} s 2 ( ϑ ) {displaystyle s ^ {2} (varteta)} ϑ {displaystyle vartheta}
Bühlmann modeli bu muammoning echimi:
a r g m men n a men 0 , a men 1 , . . . , a men m E [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) 2 ] {displaystyle {underset {a_ {i0}, a_ {i1}, ..., a_ {im}} {operator nomi {arg, min}}} operator nomi {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight]} qayerda a men 0 + ∑ j = 1 m a men j X men j {displaystyle a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij}} Π {displaystyle Pi} arg min ifodani minimallashtiradigan parametr qiymatlarini ifodalaydi.
Model echimi
Muammoning echimi:
Z X ¯ men + ( 1 − Z ) m {displaystyle Z {ar {X}} _ {i} + (1-Z) mu} qaerda:
Z = 1 1 + σ 2 v 2 m {displaystyle Z = {frac {1} {1+ {frac {sigma ^ {2}} {v ^ {2} m}}}}} Biz ushbu natijaga sharh bera olamiz: mukofotning Z qismi bizda o'ziga xos xavf tug'diradigan ma'lumotlarga asoslanadi va (1-Z) qismi butun aholi haqida ma'lumotga asoslanadi.
Isbot Quyidagi dalil asl qog'ozdagi dalillardan bir oz farq qiladi. Bu yana umumiyroq, chunki u barcha chiziqli baholovchilarni hisobga oladi, asl dalil esa faqat o'rtacha da'voga asoslangan baholovchilarni hisobga oladi.[2]
Lemma. Muammoni muqobil ravishda quyidagicha ifodalash mumkin: f = E [ ( a men 0 + ∑ j = 1 m a men j X men j − m ( ϑ ) ) 2 ] → min {displaystyle f = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ^ {2} ight] o min} Isbot:
E [ ( a men 0 + ∑ j = 1 m a men j X men j − m ( ϑ ) ) 2 ] = E [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) 2 ] + E [ ( m ( ϑ ) − Π ) 2 ] + 2 E [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) ( m ( ϑ ) − Π ) ] = E [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) 2 ] + E [ ( m ( ϑ ) − Π ) 2 ] {displaystyle {egin {aligned} mathbb {E} left [left (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ^ {2} ight] & = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight] + mathbb {E } chap [chap (m (varteta) -Pi ight) ^ {2} ight] + 2mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ { ij} -Pi ight) chap (m (varteta) -Pi ight) ight] & = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight] + mathbb {E} chap [chap (m (vartheta) -Pi ight) ^ {2} ight] oxiri {hizalanmış}}} Oxirgi tenglama haqiqatdan kelib chiqadi
E [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) ( m ( ϑ ) − Π ) ] = E Θ [ E X [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) ( m ( ϑ ) − Π ) | X men 1 , … , X men m ] ] = E Θ [ ( a men 0 + ∑ j = 1 m a men j X men j − Π ) [ E X [ ( m ( ϑ ) − Π ) | X men 1 , … , X men m ] ] ] = 0 {displaystyle {egin {aligned} mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) chap (m (varteta) - Pi ight) ight] & = mathbb {E} _ {Theta} left [mathbb {E} _ {X} left.left [left (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij } X_ {ij} -Pi ight) (m (vartheta) -Pi) ight | X_ {i1}, ldots, X_ {im} ight] ight] & = mathbb {E} _ {Theta} chap [chap (a_) {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) left [mathbb {E} _ {X} left [(m (vartheta) -Pi) | X_ { i1}, ldots, X_ {im} ight] ight] ight] & = 0end {hizalanmış}}} Biz bu erda umumiy kutish qonuni va haqiqatdan foydalanamiz Π = E [ m ( ϑ ) | X men 1 , … , X men m ] . {displaystyle Pi = mathbb {E} [m (vartheta) | X_ {i1}, ldots, X_ {im}].}
Oldingi tenglamamizda biz minimallashtirilgan funktsiyani ikkita ifoda yig'indisida ajratamiz. Ikkinchi ifoda minimallashtirishda ishlatiladigan parametrlarga bog'liq emas. Shuning uchun funktsiyani minimallashtirish yig'indining birinchi qismini minimallashtirish bilan bir xil.
Funktsiyaning muhim nuqtalarini topaylik
1 2 ∂ f ∂ a 01 = E [ a men 0 + ∑ j = 1 m a men j X men j − m ( ϑ ) ] = a men 0 + ∑ j = 1 m a men j E ( X men j ) − E ( m ( ϑ ) ) = a men 0 + ( ∑ j = 1 m a men j − 1 ) m {displaystyle {frac {1} {2}} {frac {kısmi f} {qisman a_ {01}}} = mathbb {E} chap [a_ {i0} + sum _ {j = 1} ^ {m} a_ { ij} X_ {ij} -m (varteta) ight] = a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} mathbb {E} (X_ {ij}) - mathbb {E} ( m (varteta)) = a_ {i0} + chap (sum _ {j = 1} ^ {m} a_ {ij} -1ight) mu} a men 0 = ( 1 − ∑ j = 1 m a men j ) m {displaystyle a_ {i0} = chap (1-sum _ {j = 1} ^ {m} a_ {ij} ight) mu} Uchun k ≠ 0 {displaystyle keq 0}
1 2 ∂ f ∂ a men k = E [ X men k ( a men 0 + ∑ j = 1 m a men j X men j − m ( ϑ ) ) ] = E [ X men k ] a men 0 + ∑ j = 1 , j ≠ k m a men j E [ X men k X men j ] + a men k E [ X men k 2 ] − E [ X men k m ( ϑ ) ] = 0 {displaystyle {frac {1} {2}} {frac {qisman f} {qisman a_ {ik}}} = mathbb {E} chap [X_ {ik} chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ight] = mathbb {E} chap [X_ {ik} ight] a_ {i0} + sum _ {j = 1, jeq k} ^ {m} a_ {ij} mathbb {E} [X_ {ik} X_ {ij}] + a_ {ik} mathbb {E} [X_ {ik} ^ {2}] - mathbb {E} [X_ {ik} m (varteta)] = 0} Biz quyidagilarni ta'kidlab, lotinni soddalashtirishimiz mumkin.
E [ X men j X men k ] = E [ E [ X men j X men k | ϑ ] ] = E [ cov ( X men j X men k | ϑ ) + E ( X men j | ϑ ) E ( X men k | ϑ ) ] = E [ ( m ( ϑ ) ) 2 ] = v 2 + m 2 E [ X men k 2 ] = E [ E [ X men k 2 | ϑ ] ] = E [ s 2 ( ϑ ) + ( m ( ϑ ) ) 2 ] = σ 2 + v 2 + m 2 E [ X men k m ( ϑ ) ] = E [ E [ X men k m ( ϑ ) | Θ men ] = E [ ( m ( ϑ ) ) 2 ] = v 2 + m 2 {displaystyle {egin {hizalanmış} mathbb {E} [X_ {ij} X_ {ik}] & = mathbb {E} chap [mathbb {E} [X_ {ij} X_ {ik} | vartheta] ight] = mathbb { E} [{ext {cov}} (X_ {ij} X_ {ik} | vartheta) + mathbb {E} (X_ {ij} | vartheta) mathbb {E} (X_ {ik} | vartheta)] = mathbb { E} [(m (vartheta)) ^ {2}] = v ^ {2} + mu ^ {2} mathbb {E} [X_ {ik} ^ {2}] & = mathbb {E} chap [mathbb {E} [X_ {ik} ^ {2} | varteta] ight] = mathbb {E} [s ^ {2} (vartheta) + (m (vartheta)) ^ {2}] = sigma ^ {2} + v ^ {2} + mu ^ {2} mathbb {E} [X_ {ik} m (vartheta)] & = mathbb {E} [mathbb {E} [X_ {ik} m (vartheta) | Theta _ { i}] = mathbb {E} [(m (vartheta)) ^ {2}] = v ^ {2} + mu ^ {2} end {hizalanmış}}} Yuqoridagi tenglamalarni hisobga olgan holda va hosilaga qo'shib, bizda:
1 2 ∂ f ∂ a men k = ( 1 − ∑ j = 1 m a men j ) m 2 + ∑ j = 1 , j ≠ k m a men j ( v 2 + m 2 ) + a men k ( σ 2 + v 2 + m 2 ) − ( v 2 + m 2 ) = a men k σ 2 − ( 1 − ∑ j = 1 m a men j ) v 2 = 0 {displaystyle {frac {1} {2}} {frac {qisman f} {qisman a_ {ik}}} = chap (1-sum _ {j = 1} ^ {m} a_ {ij} ight) mu ^ { 2} + sum _ {j = 1, jeq k} ^ {m} a_ {ij} (v ^ {2} + mu ^ {2}) + a_ {ik} (sigma ^ {2} + v ^ {2 } + mu ^ {2}) - (v ^ {2} + mu ^ {2}) = a_ {ik} sigma ^ {2} -left (1-sum _ {j = 1} ^ {m} a_ { ij} ight) v ^ {2} = 0} σ 2 a men k = v 2 ( 1 − ∑ j = 1 m a men j ) {displaystyle sigma ^ {2} a_ {ik} = v ^ {2} chap (1-sum _ {j = 1} ^ {m} a_ {ij} ight)} O'ng tomon bog'liq emas k . Shuning uchun, barchasi a men k {displaystyle a_ {ik}}
a men 1 = ⋯ = a men m = v 2 σ 2 + m v 2 {displaystyle a_ {i1} = cdots = a_ {im} = {frac {v ^ {2}} {sigma ^ {2} + mv ^ {2}}}} Uchun echimdan a men 0 {displaystyle a_ {i0}}
a men 0 = ( 1 − m a men k ) m = ( 1 − m v 2 σ 2 + m v 2 ) m {displaystyle a_ {i0} = (1-ma_ {ik}) mu = chap (1- {frac {mv ^ {2}} {sigma ^ {2} + mv ^ {2}}} ight) mu} Va nihoyat, eng yaxshi taxminchi
a men 0 + ∑ j = 1 m a men j X men j = m v 2 σ 2 + m v 2 X men ¯ + ( 1 − m v 2 σ 2 + m v 2 ) m = Z X men ¯ + ( 1 − Z ) m {displaystyle a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} = {frac {mv ^ {2}} {sigma ^ {2} + mv ^ {2}}} {ar {X_ {i}}} + chap (1- {frac {mv ^ {2}} {sigma ^ {2} + mv ^ {2}}} ight) mu = Z {ar {X_ {i}} } + (1-Z) mu} Adabiyotlar
Iqtiboslar Manbalar Frits, E.V .; Yosh, V.R .; Luo, Y. (1999). "Ishonchlilik modellarining uzunlamasına ma'lumotlarni tahlil qilish talqini". Sug'urta: Matematika va iqtisodiyot . 24 (3): 229–247. doi :10.1016 / S0167-6687 (98) 00055-9 .
![{displaystyle m (vartheta) = operator nomi {E} chapda [X_ {ij} | Theta _ {i} = vartheta ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e93ffc728592598aed3dca03b38e3d8a91b1caf)
![{displaystyle s ^ {2} (varteta) = operator nomi {Var} chapda [X_ {ij} | Theta _ {i} = vartheta ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f9332433a5e057987f5066f77dc23c02e90794d)
![{displaystyle sigma ^ {2} = operator nomi {E} chap [s ^ {2} (vartheta) ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4db39345e36665026759e38b8e555d93aa27cc)
![{displaystyle v ^ {2} = operator nomi {Var} chap [m (vartheta) ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/073b42dd1170743acb688a79021d203a169b67f8)
![{displaystyle {underset {a_ {i0}, a_ {i1}, ..., a_ {im}} {operator nomi {arg, min}}} operator nomi {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d98dbf12cd7f0353b6a5e9b2bebad556e770d52d)
![{displaystyle f = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ^ {2} ight] o min}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be9fffcc0f7973c35516392fdc65093bc4445c2)
![{displaystyle {egin {aligned} mathbb {E} left [left (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ^ {2} ight] & = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight] + mathbb {E } chap [chap (m (varteta) -Pi ight) ^ {2} ight] + 2mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ { ij} -Pi ight) chap (m (varteta) -Pi ight) ight] & = mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) ^ {2} ight] + mathbb {E} chap [chap (m (vartheta) -Pi ight) ^ {2} ight] oxiri {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c94c458eb7f34d6039cd9e722abb164217b439b)
![{displaystyle {egin {aligned} mathbb {E} chap [chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) chap (m (varteta) - Pi ight) ight] & = mathbb {E} _ {Theta} left [mathbb {E} _ {X} left.left [left (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij } X_ {ij} -Pi ight) (m (vartheta) -Pi) ight | X_ {i1}, ldots, X_ {im} ight] ight] & = mathbb {E} _ {Theta} chap [chap (a_) {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -Pi ight) left [mathbb {E} _ {X} left [(m (vartheta) -Pi) | X_ { i1}, ldots, X_ {im} ight] ight] ight] & = 0end {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb00af5d102e9463a2b86fc78b44fab1ea40ec10)
![{displaystyle Pi = mathbb {E} [m (vartheta) | X_ {i1}, ldots, X_ {im}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/620217d4422158509323e1511c7ce48d13f8d6fd)
![{displaystyle {frac {1} {2}} {frac {kısmi f} {qisman a_ {01}}} = mathbb {E} chap [a_ {i0} + sum _ {j = 1} ^ {m} a_ { ij} X_ {ij} -m (varteta) ight] = a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} mathbb {E} (X_ {ij}) - mathbb {E} ( m (varteta)) = a_ {i0} + chap (sum _ {j = 1} ^ {m} a_ {ij} -1ight) mu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c073d6483862cdbd421d4485c68695f368fd631)
![{displaystyle {frac {1} {2}} {frac {qisman f} {qisman a_ {ik}}} = mathbb {E} chap [X_ {ik} chap (a_ {i0} + sum _ {j = 1} ^ {m} a_ {ij} X_ {ij} -m (vartheta) ight) ight] = mathbb {E} chap [X_ {ik} ight] a_ {i0} + sum _ {j = 1, jeq k} ^ {m} a_ {ij} mathbb {E} [X_ {ik} X_ {ij}] + a_ {ik} mathbb {E} [X_ {ik} ^ {2}] - mathbb {E} [X_ {ik} m (varteta)] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca5557712ee024b29c8e321c388f898af01b94f)
![{displaystyle {egin {hizalanmış} mathbb {E} [X_ {ij} X_ {ik}] & = mathbb {E} chap [mathbb {E} [X_ {ij} X_ {ik} | vartheta] ight] = mathbb { E} [{ext {cov}} (X_ {ij} X_ {ik} | vartheta) + mathbb {E} (X_ {ij} | vartheta) mathbb {E} (X_ {ik} | vartheta)] = mathbb { E} [(m (vartheta)) ^ {2}] = v ^ {2} + mu ^ {2} mathbb {E} [X_ {ik} ^ {2}] & = mathbb {E} chap [mathbb {E} [X_ {ik} ^ {2} | varteta] ight] = mathbb {E} [s ^ {2} (vartheta) + (m (vartheta)) ^ {2}] = sigma ^ {2} + v ^ {2} + mu ^ {2} mathbb {E} [X_ {ik} m (vartheta)] & = mathbb {E} [mathbb {E} [X_ {ik} m (vartheta) | Theta _ { i}] = mathbb {E} [(m (vartheta)) ^ {2}] = v ^ {2} + mu ^ {2} end {hizalanmış}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fac1d5cae6dd90e1a8a36783973e1eb54f2020cf)
