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Amari–Chentsov structure on the statistical manifold of models for accelerated life tests
Fode Zhang1,3; Hon Keung Tony Ng2; Yimin Shi3; Ruibing Wang3
2019-03
Source PublicationTEST
ISSN1133-0686
Volume28Issue:1Pages:77-105
Abstract

The invariant geometric structures on the statistical manifold under sufficient statistics have played an important role in both statistical inference and information theory. In this paper, we focus on one of the commonly used invariant geometric structures, the Amari-Chentsov structure, on a statistical manifold. The manifold is derived from statistical models for accelerated life tests (ALTs) with censoring based on the exponential family of distributions. The constant-stress ALTs and step-stress ALTs are considered. We show that the statistical manifold still belongs to the exponential family of distributions, but the cumulant generating function depends on a random variable related to the experimental design of the ALT, which is different from the usual situation. We also investigate the Bregman divergence and Riemannian metric. The relationships between the Riemannian metric and the expected Fisher information metric are studied. The dual coordinate system is studied by using the Legendre transformation. Then, the Amari-Chentsov structure is derived based on the two different coordinate systems. The methodologies are illustrated by using two distributions, the exponential and gamma distributions, in the exponential family of distributions. Finally, using the results of Fisher information metric, optimal designs of the two types of ALTs are presented with different optimal criteria. Finally, numerical examples are provided to demonstrate the practical applications of the results developed in this paper..

KeywordAmari–chentsov Structure Statistical Manifold Accelerated Life Tests Censoring Data Bregman Divergence
DOI10.1007/s11749-018-0587-1
Indexed BySCIE
Language英语
WOS Research AreaMathematics
WOS SubjectStatistics & Probability
WOS IDWOS:000461252700008
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Cited Times [WOS]:3   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionInstitute of Chinese Medical Sciences
Corresponding AuthorHon Keung Tony Ng
Affiliation1.Center of Statistical Research, School of Statistics, Southwestern University of Finance andEconomics, Chengdu 611130, Sichuan, People’s Republic of China
2.Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072,Shaanxi, People’s Republic of China
3.Department of Statistical Science, Southern Methodist University, Dallas, TX 75275-0332, USA
Recommended Citation
GB/T 7714
Fode Zhang,Hon Keung Tony Ng,Yimin Shi,et al. Amari–Chentsov structure on the statistical manifold of models for accelerated life tests[J]. TEST,2019,28(1):77-105.
APA Fode Zhang,Hon Keung Tony Ng,Yimin Shi,&Ruibing Wang.(2019).Amari–Chentsov structure on the statistical manifold of models for accelerated life tests.TEST,28(1),77-105.
MLA Fode Zhang,et al."Amari–Chentsov structure on the statistical manifold of models for accelerated life tests".TEST 28.1(2019):77-105.
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