Project/Area Number 
09680314

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Statistical science

Research Institution  Osaka Prefecture University 
Principal Investigator 
NAGAO Hisao College of Engineering, Osaka Prefecture University, Professor, 工学部, 教授 (80033869)

CoInvestigator(Kenkyūbuntansha) 
KOYAMA Hideyuki College of Engineering, Osaka Prefecture University, Assistant Professor, 工学部, 講師 (20109888)
HAYAKAWA Kantaro College of Engineering, Osaka Prefecture University, Professor, 工学部, 教授 (10028201)
SHIRASAKI Manabu College of Engineering, Osaka Prefecture University, Assistant Professor, 工学部, 講師 (80226331)
KURIKI Shinji College of Engineering, Osaka Prefecture University, Associate Professor, 工学部, 助教授 (00167389)

Project Period (FY) 
1997 – 1998

Project Status 
Completed (Fiscal Year 1998)

Budget Amount *help 
¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)

Keywords  prior distribution / covariance matrix / martingale / urivariate mutiparameter exponential distribution / multivariate normal / martingale / optinal theorem / conjugate / A.P.O.rule 
Research Abstract 
Let a multivariate normal distribuion have mean mu and covariance matrix SIGMA and we assume both parameters are unknown. We consider the estimating problem of mean mu. Its loss function is the sum of squared loss and cost x no. of sample. As the prior distribution, we take a conjugate distribution. At this time we want to find the esimation of/and stopping rule which minimizes the expectation loss. It is difficult to find the stopping rule. So when c * 0, we define A.P.O.rule which is nearly optimal rule. When we choose this rule, we give the asymptotic expansion of the risk. It can be expressed with the power of ROO<c>. To get it, we considered it from three points. (1) When the covariance matrix has some structure, we assume that the matrix can be expressed as the sum of symmmetric matrix. This assumption has been used in the author's paper. We can get the expression of the loss. (2) We consider the case that the covariance matrix is completely unknown and the same problem as (1). As prior, we choose a conjugate distribution. Then we got the similar results as in (2). From (1) and (2), we find that the result (1) can get putting covariance structure in (2) as if it has such a structure. That shows interresting. Also we get the similar results for multinomnial distribution. The method for caluculating bases on martingale theory and derivatives of matrices. (3) We consider univariate multiparameter exponential distribuition. We choose any distribution as prior. Under this assumption, we consider the same problem as (1) and (2). We want to find how the risk can be expressed. After all, we find whether the posteria variance of some function is uniformly integrable. However, we can see it. So we can get the results for general case.
