【摘要】：在保險實踐中,一項重要的工作是確定足夠的保費應(yīng)對風(fēng)險。到目前為止,已出現(xiàn)許多保險價格確定的方法。其中信度厘定方法是一種被廣泛認(rèn)可的重要的技術(shù),它通過風(fēng)險集的過去索賠數(shù)據(jù),來預(yù)測風(fēng)險集未來的索賠。信度理論最早出現(xiàn)在二十世紀(jì)初,到二十世紀(jì)六十年代,Buhlmann模型則奠定了現(xiàn)代信度理論的基礎(chǔ),它通過最小二乘法確定的信度保費解決了無需分布的問題,并把表示成個體平均和集合平均二者的加權(quán)平均。以后又出現(xiàn)了Buhlmann-Straub模型、回歸信度模型、分層模型等重要信度模型。但在實踐中,Biihlmann模型的風(fēng)險間獨立的假設(shè)不一定滿足,風(fēng)險相關(guān)情形常常發(fā)生,具有風(fēng)險相關(guān)結(jié)構(gòu)的Biihlmann模型逐步受到重視。 在經(jīng)典的決策理論,損失函數(shù)往往只關(guān)注估計的精度,但擬合度也是一個重要的標(biāo)準(zhǔn)。Zellner首先在一般線性模型中引入了反映精度與擬合度的平衡損失函數(shù),而在平衡損失函數(shù)下的信度厘定方法在最近幾年也被開始研究。另外加權(quán)保費原理下的信度模型也不斷被研究,這主要是由于加權(quán)損失函數(shù)可以減少純保費原理中的負(fù)安全負(fù)載,如出現(xiàn)了如Esscher保費原理,調(diào)整保費原理等。 本文主要研究二部分內(nèi)容,其一是不同于經(jīng)典信度模型中風(fēng)險結(jié)構(gòu)的信度模型研究；其二是不同于經(jīng)典信度模型中損失函數(shù)的信度模型研究。 本文的第一部分內(nèi)容在第二章中研究。主要把風(fēng)險相依、時間上條件獨立的信度模型拓展為風(fēng)險相依和時間相關(guān)的信度模型。首先給出了正交投影和信度估計的關(guān)系和由正交投影表示的信度估計公式：通過正交投影的方法,得到具有風(fēng)險相依、時間相關(guān)Buhlmann和Buhlmann-Straub模型的風(fēng)險保費的齊次與非齊次的信度估計；最后推導(dǎo)了具有共同效應(yīng)風(fēng)險結(jié)構(gòu)的信度保費和風(fēng)險等相關(guān)結(jié)構(gòu)下的信度保費。 在第二部分,研究把經(jīng)典信度模型中二次損失函數(shù)修改為平衡損失函數(shù)和加權(quán)平衡損失函數(shù)時信度保費問題。在第三章中,首先給出在二次損失函數(shù)和平衡損失函數(shù)下信度保費的關(guān)系；然后推導(dǎo)了平衡損失函數(shù)下風(fēng)險相依、時間相關(guān)一般信度保費表達(dá)式,具有共同效應(yīng)風(fēng)險結(jié)構(gòu)的齊次與非齊次信度保費被重點研究,同時還給出信度公式中相關(guān)參數(shù)估計；最后平衡損失函數(shù)下指數(shù)保費原理的信度保費得到討論。 在第四章中,首先給出了在平衡損失函數(shù)下具有風(fēng)險相依結(jié)構(gòu)的線性回歸信度保費的表達(dá)式；然后得到當(dāng)平衡損失函數(shù)中目標(biāo)估計取特殊估計時的信度保費；最后討論了二種特殊風(fēng)險結(jié)構(gòu)即風(fēng)險等相關(guān)和具有共同風(fēng)險下的回歸信度模型。 在第五章中,考慮到安全負(fù)載缺失往往是由于損失函數(shù)造成,將經(jīng)典信度模型中的二次損失函數(shù)用加權(quán)平衡二次損失函數(shù)代替。首先定義一種關(guān)于新測度的新期望；然后討論了幾類常見加權(quán)平衡損失函數(shù)下的簡單Biihlmann模型信度保費,討論了信度估計的相合性。研究一般平衡損失函數(shù)Lρ,ω,δ0(θ,δ)=ωρ(δ0,δ)+(1-ω)ρ(θ,δ)下的保費問題,最后通過泰勒展開式得到了在平衡熵?fù)p失函數(shù)和平衡Linex損失函數(shù)下的信度保費。
[Abstract]:In insurance practice, an important task is to determine sufficient premiums to cope with risks. So far, there have been many methods to determine the insurance price. Among them, the reliability determination method is a widely accepted and important technology. It predicts the future claims of the risk set through the past claim data of the risk set. Now in the early twentieth century, until the 1960s, Buhlmann model has laid the foundation of modern reliability theory. It solves the problem of no distribution by using the least square method to determine the premium of reliability, and expresses it as the weighted average of individual average and collective average. However, in practice, the assumption of risk independence in Biihlmann model is not necessarily satisfied. Risk-related situations often occur. Biihlmann model with risk-related structure is gradually taken seriously.
In classical decision theory, the loss function only focuses on the accuracy of estimation, but the fitness is also an important criterion. Zellner first introduced the balance loss function which reflects the accuracy and fitness in the general linear model, and the reliability determination method under the balance loss function has also been studied in recent years. The reliability model under premium principle is also studied continuously, mainly because the weighted loss function can reduce the negative security load in pure premium principle, such as Esscher premium principle, adjusting premium principle and so on.
This paper mainly studies two parts: one is the study of the reliability model which is different from the classical reliability model in risk structure; the other is the study of the reliability model which is different from the loss function in the classical reliability model.
The first part of this paper is studied in the second chapter. The risk-dependent and time-dependent reliability model is extended to the risk-dependent and time-dependent reliability model. The relationship between orthogonal projection and reliability estimation and the reliability estimation formula expressed by orthogonal projection are given. Homogeneous and inhomogeneous reliability estimates of risk premiums in time-dependent Buhlmann and Buhlmann-Straub models are given. Finally, reliability premiums with common effect risk structures and risk-related structures are deduced.
In the second part, we study the reliability premium problem when the quadratic loss function in the classical reliability model is modified into the balanced loss function and the weighted balanced loss function. The general reliability premium expression, homogeneous and inhomogeneous reliability premiums with common effect risk structure are studied emphatically, and the estimation of relevant parameters in the reliability formula is given. Finally, the reliability premium of exponential premium principle under the equilibrium loss function is discussed.
In Chapter 4, firstly, the linear regression reliability premium with risk-dependent structure under the balanced loss function is given; secondly, the reliability premium when the target estimator in the balanced loss function takes a special estimate is obtained; finally, the regression confidence under two special risk structures, i.e. risk equivalence correlation and common risk, is discussed. Degree model.
In Chapter 5, considering that the loss function is often responsible for the loss of security load, the quadratic loss function in the classical reliability model is replaced by the weighted balanced quadratic loss function. In this paper, we discuss the consistency of the reliability estimates. We study the premium problem under the general equilibrium loss function L_, _, _0 (theta, delta) = __ (delta 0, delta) + (1-_) Rho (theta, delta). Finally, we obtain the reliability premium under the equilibrium entropy loss function and the equilibrium Linex loss function by Taylor expansion.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2013
【分類號】：F840.3;O212.1

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本文編號：2247110