Earticle

이 연구는 미지의 대각선 공분산 행렬을 갖는 다변량 정규분포에서 평균 벡터를 추정하는 문제를 다룬다. 이를 위해, 균형 잡힌 손실 함수에 기반하여 두 가지 유형의 추정량을 제안하고, 이들의 추정량을 비교하여 보다 유효한 추정 방법을 제시하고자 한다. 연구에서는 최대우도추정량(MLE)을 기반으로 첫 번째 유형의 추정량을 도 입하고, 이 추정량이 MLE를 개선할 수 있는 충분조건을 도출한다. 그 후, MLE와 Lindley형 추정량을 결합하여 새로운 유형의 추정량을 제시하고, 이들이 적절한 조건 하에서 Lindley형 위험 함수보다 더 작은 위험 함수를 갖 는다는 것을 증명한다. 제안된 두 가지 추정량은 기존의 MLE나 Lindley형 추정량에 비해 보다 유효한 추정 방법 을 제공한다. 수치 실험을 통해 제안된 추정량들이 기존 추정량들과 비교하여 더 우수한 추정량임을 확인할 수 있 었다. 본 연구에서 제안한 추정량은 다변량 정규분포에서 평균 벡터 추정의 정확도를 높일 수 있는 가능성을 시사 한다. 제시된 방법들은 이론적으로 우수함을 입증했으며, 실제 데이터에 대한 적용 가능성도 높을 것으로 기대된 다. 향후 연구에서는 다른 유형의 손실 함수나 다변량 분포에 대해서도 유사한 접근을 확장할 수 있을 것이다. 마 지막으로, 제안된 추정량들의 유효성 평가를 위해 수치 실험을 수행하고 그 결과를 제시하였다.

In this paper we estimate the mean vector of a multivariate normal distribution with unknown diagonal covariance matrix. Two classes of estimators for the mean vector is presented by their performance according to the the balanced loss function. First, we introduce a class of estimators derived from the maximum likelihood estimator(MLE) and provide a sufficient condition on the shrinkage function to improve upon the MLE. Then, from the MLE and the Lindley type estimator we build a new class of estimators, and under a simple practical condition, we show that their risks are less than those of Lindley type estimator, which explains why they perform better. We conclude the paper with numerical results that confirm the performance of the proposed estimators.

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.

Zellner(1994) introduced the notion of a balanced loss function in the context of a general liner model to reflect both goodness of fit and precision of estimation. We study the perspective of unifying a variety of results both frequentist and Bayesian from Poisson distributions. We show that frequentist and Bayesian results for balanced loss follow from and also imply related results for quadratic loss functions reflecting only precision of estimation. Several examples are given for Poisson distribution.

In this paper we develop Bayes and empirical Bayes estimators of the finite population mean with the assumption of posterior linearity rather than normality of the superpopulation under the balanced loss function. We compare the performance of the optimal Bayes estimator with ones of the classical sample mean and the usual Bayes estimator under the squared error loss with respect to the posterior expected losses, risks and Bayes risks when the underlying distribution is normal as well as when they are binomial and Poisson.

In this research, the performance of widely used Bayesian estimators such as Bayes estimator, empirical Bayes estimator, constrained Bayes estimator and constrained empirical Bayes estimator are compared by means of a measurement under balanced loss function for the typical normal-normal situation. The proposed measurement is a weighted sum of the precisions of first and second moments. As a result, one can gets the criterion according to the size of prior variance against the population variance.

In decision theoretic estimation, the loss function usually emphasizes precision of estimation. However, one may have interest in goodness of fit of the overall model as well as precision of estimation. From this viewpoint, Zellner(1994) proposed the balanced loss function which takes account of both "goodness of fit" and "precision of estimation". This paper considers estimation of the parameter of a Bernoulli distribution using Zellner's(1994) balanced loss function. It is shown that the sample mean $\overline{X}$, is admissible. More general results, concerning the admissibility of estimators of the form $a\overline{X}+b$ are also presented. Finally, minimax estimators and some numerical results are given at the end of paper,at the end of paper.

In this paper, we develop Bayesian inference of the finite population mean with the assumption of posterior linearity rather than normality of the superpopulation in the presence of auxiliary information under the balanced loss function. We compare the performance of the optimal Bayes estimator under the balanced loss function with ones of the classical ratio estimator and the usual Bayes estimator in terms of the posterior expected losses, risks and Bayes risks.