Constrained Bayes and Empirical Bayes Estimator Applications in Insurance Pricing

Kim, Myung Joon,Kim, Yeong-Hwa The Korean Statistical Society 2013 Communications for statistical applications and me Vol.20 No.4

Bayesian and empirical Bayesian methods have become quite popular in the theory and practice of statistics. However, the objective is to often produce an ensemble of parameter estimates as well as to produce the histogram of the estimates. For example, in insurance pricing, the accurate point estimates of risk for each group is necessary and also proper dispersion estimation should be considered. Well-known Bayes estimates (which is the posterior means under quadratic loss) are underdispersed as an estimate of the histogram of parameters. The adjustment of Bayes estimates to correct this problem is known as constrained Bayes estimators, which are matching the first two empirical moments. In this paper, we propose a way to apply the constrained Bayes estimators in insurance pricing, which is required to estimate accurately both location and dispersion. Also, the benefit of the constrained Bayes estimates will be discussed by analyzing real insurance accident data.

The comparative studies on stress-strength reliability for Rayleigh distribution

오지은,손중권 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.4

This paper deals with the estimation of the stress-strength parameter R = P(Y < X) with three different methods for Rayleigh distribution. We assumed that the stress and strength variables are independent. We derive a maximum likelihood estimator of R and its an asymptotic distribution.We also compute an asymptotic confidence intervals. The bootstrap estimator and confidence intervals are also derived based on maximum likelihood estimator of R. We obtained the Bayes estimator based on inversed gamma priors on scale parameters and its most plausible set for constructing the confidence interval of R which is quiet similar to classical condence interval unlike the highest posterior density region We compare the performances of three estimators using the mean squared error (MSE) of each estimator. This paper deals with the estimation of the stress-strength parameter R = P(Y <X) with three different methods for Rayleigh distribution. We assumed that the stress and strength variables are independent. We derive a maximum likelihood estimator of R and its an asymptotic distribution. We also compute an asymptotic confidence intervals. The bootstrap estimator and confidence intervals are also derived based on maximum likelihood estimator of R. We obtained the Bayes estimator based on inversed gamma priors on scale parameters and its most plausible set for constructing the confidence interval of R which is quiet similar to classical confidence interval unlike the highest posterior density region We compare the performances of three estimators using the mean squared error (MSE) of each estimator.

Optimization of the Smoothing Parameter of the Adaptive Kernel Estimator used in Bayes Classifier - Application to Microarray Data Analysis

Yissam Lakhdar,El Hassan Sbai 보안공학연구지원센터 2015 International Journal of Software Engineering and Vol.9 No.3

In this work, we focus on nonparametric kernel methods for estimating the probability density function (pdf). The convergence of a kernel estimator depends crucially on the choice of the smoothing parameter. We present in this paper, a new method for optimizing the bandwidth of an estimator of the probability density function: the adaptive kernel estimator. This optimized estimator is used to construct the Bayes classifier. In this sense, we have proposed a new approach to optimize the pdf based on the statistical properties of the probability distributions of random variables. We adopt the maximum entropy principle (MEP) in order to determine the optimal value of the smoothing parameter used in the estimator. In the proposed criterion, the estimated probability density function is called optimal in the sense of having a minimum error rate of classifying data. Finally, we illustrate the robustness of our optimization process of the kernel estimation methods by using a set of DNA microarray data showing that our approach effectively improves the performance of the classification process.

Jensen’s Alpha Estimation Models in Capital Asset Pricing Model

Le Tan Phuoc 한국유통과학회 2018 The Journal of Asian Finance, Economics and Busine Vol.5 No.3

This research examined the alternatives of Jensen’s alpha (α) estimation models in the Capital Asset Pricing Model, discussed by Treynor (1961), Sharpe (1964), and Lintner (1965), using the robust maximum likelihood type m-estimator (MM estimator) and Bayes estimator with conjugate prior. According to finance literature and practices, alpha has often been estimated using ordinary least square (OLS) regression method and monthly return data set. A sample of 50 securities is randomly selected from the list of the S&P 500 index. Their daily and monthly returns were collected over a period of the last five years. This research showed that the robust MM estimator performed well better than the OLS and Bayes estimators in terms of efficiency. The Bayes estimator did not perform better than the OLS estimator as expected. Interestingly, we also found that daily return data set would give more accurate alpha estimation than monthly return data set in all three MM, OLS, and Bayes estimators. We also proposed an alternative market efficiency test with the hypothesis testing Ho: α = 0 and was able to prove the S&P 500 index is efficient, but not perfect. More important, those findings above are checked with and validated by Jackknife resampling results.

Generalized Bayes estimation for a SAR model with linear restrictions binding the coefficients

Chaturvedi, Anoop,Mishra, Sandeep The Korean Statistical Society 2021 Communications for statistical applications and me Vol.28 No.4

The Spatial Autoregressive (SAR) models have drawn considerable attention in recent econometrics literature because of their capability to model the spatial spill overs in a feasible way. While considering the Bayesian analysis of these models, one may face the problem of lack of robustness with respect to underlying prior assumptions. The generalized Bayes estimators provide a viable alternative to incorporate prior belief and are more robust with respect to underlying prior assumptions. The present paper considers the SAR model with a set of linear restrictions binding the regression coefficients and derives restricted generalized Bayes estimator for the coefficients vector. The minimaxity of the restricted generalized Bayes estimator has been established. Using a simulation study, it has been demonstrated that the estimator dominates the restricted least squares as well as restricted Stein rule estimators.

Application of Constrained Bayes Estimation under Balanced Loss Function in Insurance Pricing

Kim, Myung Joon,Kim, Yeong-Hwa The Korean Statistical Society 2014 Communications for statistical applications and me Vol.21 No.3

Constrained Bayesian estimates overcome the over shrinkness toward the mean which usual Bayes and empirical Bayes estimates produce by matching first and second empirical moments; subsequently, a constrained Bayes estimate is recommended to use in case the research objective is to produce a histogram of the estimates considering the location and dispersion. The well-known squared error loss function exclusively emphasizes the precision of estimation and may lead to biased estimators. Thus, the balanced loss function is suggested to reflect both goodness of fit and precision of estimation. In insurance pricing, the accurate location estimates of risk and also dispersion estimates of each risk group should be considered under proper loss function. In this paper, by applying these two ideas, the benefit of the constrained Bayes estimates and balanced loss function will be discussed; in addition, application effectiveness will be proved through an analysis of real insurance accident data.

Empirical and Hierarchical Bayes Estimator's in General Linear Models

Choi, Kuey Chung CHOSUN UNIVERSITY 1997 Basic Science and Engineering Vol.1 No.1

Consider the problem of estimating a vector of unknown regression coefficient or an unknown estimate function under the sum of squared error losses in general linear models. We first provide empirical and hierarchical Bayes estimator of an estimable function shrinking towards a fixed known point. We also propose empirical and hierarchical Bayes estimators of a vector of unknown regression coefficients shriking towards regression surface, and then show that the hierarchical Bayes estimator of an estimable function is identical to the least squares estimator of the same estimable function.

일반화된 조정 점진적 복합 중도절단에서 하프 로지스틱 분포의 추정

조수빈(Subin Cho),이경준(Kyeongjun Lee) 한국데이터정보과학회 2021 한국데이터정보과학회지 Vol.32 No.2

생존실험에서 조정 점진적 복합 중도절단 방법은 실험이 종료하기까지 시간이 오래 소요될 수 있다는 단점이 존재한다. 따라서 최근 일반화된 조정 점진적 복합 중도절단 방법이 소개되어졌다. 본 논문은 일반화된 조정 점진적 복합 중도절단에서 하프 로지스틱분포의 모수를 추정하고자 한다. 모수를 추정하는 방법으로 최대우도추정량과 테일러 급수 전개를 이용한 근사 최대우도추정량을 사용하였다. 또한, 대칭손실함수를 이용하여 하프 로지스틱분포의 모수를 추정하였다. 하지만 베이지안 추정량를 정확하게 계산할 수 없어 Tierney와 Kadane의 근사적인 방법을 사용하여 베이지안 추정량을 계산하였다. 그리고 다양한 일반화된 조정 점진정 복합 중도절단 상황에서 모의실험을 통하여 제안한 추정량들의 평균제곱오차와 편의를 계산하여 비교하였다. 마지막으로 사례 자료를 이용하여 제안한 추정량들을 계산하였다. One of the disadvantages of the adaptive progressive hybrid censoring scheme is that the time of the experiment can be very long if units are highly reliable. Therefore, generalized adaptive progressive hybrid censoring scheme was proposed. In this article, the estimation of the parameter of half-logistic distribution based on the generalized adaptive progressive hybrid censored sample has been considered. The parameter is estimated by maximum likelihood estimator and approximate maximum likelihood estimator using Taylor series expansion. The Bayes estimator for the parameter of the half-logistic distribution based on the squared error loss function, are also provided. The Bayes estimators cannot be obtained explicitly, and Tierney and Kadane approximation is used to obtain the Bayes estimator. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, a real dataset has been analyzed for illustrative purposes.

Bayes estimation of entropy of exponential distribution based on multiply Type II censored competing risks data

Lee, Kyeongjun,Cho, Youngseuk The Korean Data and Information Science Society 2015 한국데이터정보과학회지 Vol.26 No.6

In lifetime data analysis, it is generally known that the lifetimes of test items may not be recorded exactly. There are also situations wherein the withdrawal of items prior to failure is prearranged in order to decrease the time or cost associated with experience. Moreover, it is generally known that more than one cause or risk factor may be present at the same time. Therefore, analysis of censored competing risks data are needed. In this article, we derive the Bayes estimators for the entropy function under the exponential distribution with an unknown scale parameter based on multiply Type II censored competing risks data. The Bayes estimators of entropy function for the exponential distribution with multiply Type II censored competing risks data under the squared error loss function (SELF), precautionary loss function (PLF) and DeGroot loss function (DLF) are provided. Lindley's approximate method is used to compute these estimators.We compare the proposed Bayes estimators in the sense of the mean squared error (MSE) for various multiply Type II censored competing risks data. Finally, a real data set has been analyzed for illustrative purposes.

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

장동호,박지훈,김찬수 한국통계학회 2011 Journal of the Korean Statistical Society Vol.40 No.3

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.