The Credibility Estimators Of Risk Premium And Their Statistical Inferences

In the framework of credibility theory, the feature of a risk X (possible loss incurred under an insurance contract) is identified by an unknown risk parameterθ, due to the heterogeneity over policies in the concerned portfolio, all possible values ofθare then modeled by a random variable (?) following a probability distributionπ(θ) that is referred to as prior distribution in statistical community, and the object is to predict (estimate) a possible future loss (premium) of the risk X by the assistant of a sequence of its historical claims. A basic idea of credibility is to determine the premium of insurance contract, by taking into account the information provided by both the claim experience and the prior distribution. In classical Buhlmann's theory, the credibility premium has the form of weighted sum of the individual mean and the overall mean, and thus can easily be implemented in practice due to its simple form in mathematics and distribution-free fea-ture in statistics. It has been applied to broad insurance practices, such as automobile insurance, worker's compensation, loss reserving, etc.This thesis is concerned with a few aspects of the credibility estimators of risk pre-miums and their statistical inferences, described in detail as follows.The first part of thesis includes Chapters 2 and 3, in which we extend the the credibil-ity methods to a few premium calculation principles allowing safety loading. This research is motivated by the fact that almost all of the credibility theory have so far been devel-oped only for the net premium principle that allows zero safety loading and is not in fact practically applicable. Chapter 2 is devoted to the credibility models for the generalized weighted premium principle H(X)= a generalization of net premium principle in the sense that it minimizes the expectation of loss function L(X, P)= (v(X)-P)2h(X). It can be reduced to certain classical premium theory with positive safe loading, such as Esscher premium, modified premium principle, Kamp principle, conditional tail expecta-tion theory and so on, when v(X) and h(X) are specified to be certain special functions. In this chapter, Bayes premium, two versions of credibility estimators, i.e., Pan-type (Pan. et al(2008)) and Gerber-type (Gerber (1980)) credibility premiums, are derived. More-over, the consistency of those estimators is also examined:Bayes premium and pan-type credibility premium are of desired convergency property, but Gerber-type credibility pre-mium is not necessarily consistent unless a very much stringent condition is satisfied. In Chapter 3, the credibility estimate of exponential premium principle is established and the conditions under which the Bayes premiums coincide with credibility premiums are explored. Unlike in the case of weighted premium principle, the credibility estimator for exponential principle is easy to calculate and the consistency is ready to prove. In addi-tion, in multiple contract models, we get the homogeneous and inhomogeneous credibility estimators, discuss the estimates of the structure parameters, and prove the statistical properties of those estimators. Finally, we prove that the empirical Bayes estimators of the credibility premiums are asymptotically optimal.In Chapter 4, we propose a unified experience rating models for arbitrary premium calculation principles. The main idea is to estimate the distribution function of a fu-ture claim by the idea of credibility such that the credibility premiums under arbitrary premium principles are derived by inserting the estimated distribution functions into the premium calculation principles. In this chapter, the credibility estimator under such clas-sical premium principles as net, variance, standard deviation, conditional tail expected, Kamp, Esscher, exponential, Dutch, and distorion premium principles can be treated uni-formly. The consistency of these unified credibility estimators are also checked and the efficiency of our unified credibility estimators are compared with the other types of pre-mium estimators under net, exponential, and Esscher principles. The results show that, while for the premiums of which the credibility premiums has been established, although our method does not produce optimal empirical ratemakings, it does produce premiums that are acceptable in practical sense, for the other premiums of which the credibility pre-miums has not yet been investigated so far, we provide a unified method to construct their credibility-like experience ratemaking. Finally, in the multiple contracts models we derive the unbiased and consistent estimator of the structure parameters, and also prove that the empirical Bayes estimators of the credibility premiums are asymptotically optimal.Chapter 5 discusses credibility models with dependence structure over risks. In this chapter, we propose a conception of inhomogeneous and homogeneous orthogonal pro-jection in the Hibert space generated by random variables. We find the relationship be-tween orthogonal projection and credibility estimator, and derive the simplified formulae of credibility estimator by applying orthogonal projection technique. By the orthogonal projection technique, we also treat the counterparts of Buhlmann, Buhlmann-Straub, and Hachemeister credibility regression models.We conclude the text in Chapter 6 where some future research topics are also dis-cussed.