KIKUCHI Yasuki Nagasaki University, School of Health Science, Associate Professor, 医学部, 助教授 (10244140)
NOMAKUCHI Kentarou Kochi University, Faculty of Science, Professor, 理学部, 教授 (60124806)
KOJIMA Hirao Seinan Gakuin University, Department of Commerce, Professor, 商学部, 教授 (80170249)
|Budget Amount *help
¥3,600,000 (Direct Cost : ¥3,600,000)
Fiscal Year 2003 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 2002 : ¥2,300,000 (Direct Cost : ¥2,300,000)
For order restricted parameters, bootstrap estimators were investigated.
(1) Generally the maximum likelihood estimators(MLEs) for ordered parameters are not unbiased. Moreover, those estimators show a tendency. For example, in a case of 'simple order', we can recognize a relation such as majorization between the parameters and the order restricted MLEs. Geometrically, some properties of the order restricted MLE is characterized by the relation between the convex polyhedral cone and the dual cone. And it had been noticed that the bootstrap methods make the biases in the worse direction. Moreover, it had been found that if we make the size of resampling, say m, large, the estimators become very unstable for the parameters near the boundary of the set of order restricted parameters.
(2) When there is an order restriction on parameters, the Akaike's Information Criterion(AIC) contains unknown parameters even in a asymptotic case. Thus we need to estimate the bias-correction term through the
data. A direct application of bootstrap method that iteratively construct the corresponding statistics based on bootstrap samples causes biases. Contrarily we designed a simple method by means of indicator functions. This new method uses the number of different values of the MLEs as the estimator of the 'bias-correction term'. Thus we can construct an unbiased estimator. This property may be related to the method proposed by Kikuchi, Yanagawa and Nishiyama (1993, Statistical Sciences and Data Analysis, pp. 345-56). However, unfortunately, the estimator is unstable because of the variation. So we considered application of the bootstrap method to the new estimator. We expected the method is unbiased and stable. But, after all, by literature searching and deep considerations, we realized that the new estimator based on the bootstrap method may not be unbiased. An possible remedy for this problem will be to employ an correction term of the iterated logarithm order log log n against the original sample size n.
Thus, as mentioned above, since it had been found that our theory was not completely correct, our schedule had been delayed. Although our study term terminated, we would like to improve our method and publish the papers as soon as possible. Less