Abstract
We consider the procedure for small-sample estimation of reliability parameters. The main shortcomings of the classical methods and the Bayesian approach are analyzed. Models that find robust Bayesian estimates are proposed. The sensitivity of the Bayesian estimates to the choice of the prior distribution functions is investigated using models that find upper and lower bounds. The proposed models reduce to optimization problems in the space of distribution functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger J.O., (1984), The robust Bayesian viewpoint (with discussion), in Kadane J., (ed.), Robustness of Bayesian Analysis, North-Holland, Amsterdam.
J.O. Berger (1994) ArticleTitleAn overview of robust Bayesian analysis (with comments) Test 3 IssueID1 5–124
C. Carota F. Ruggeri (1994) ArticleTitleRobust Bayesian analysis given priors on partition sets Test 3 IssueID2 73–86
J.J. Deeley M.S. Tierney W.J. Zimmer (1970) ArticleTitleOn the usefulness of the maximum entropy principle in the Bayesian estimation of reliability IEEE Transactions on Reliability R-19 110–115
Golodnikov A.N., (1979), Extremum Seeking for a Linear Functional in the Class of Distribution Functions Satisfying Linear Inequality Constraints. Preprint IK AN UkrSSR, No. 79-13, Kiev.
A.N. Golodnikov L.S. Stoikova (1978) ArticleTitleNumerical method for estimation of some functionals characterizing reliability Kibernetika 2 73–77
E.T. Jaynes (1968) ArticleTitlePrior probabilities IEEE Transactions on Systems Science and Cybernetics SSC-4 227–241
H. Jeffreys (1961) Theory of Probability EditionNumber3 Clarendon Press Oxford
H.F. Martz R.A. Waller (1991) Bayesian Reliability Analysis Krieger Publishers Co. Malabar, FL
A. Mosleh G. Apostolakis (1982) ArticleTitleSome properties of distributions useful in the study of rare events IEEE Transactions on Reliability R-31 87–94
K. Pearson (1957) Tables of the Incomplete Gamma Functions, Biometrika Univ. College London
K. Pearson (1968) Table of the Incomplete Beta Functions Cambridge University Press Cambridge
H. Robins (1951) Asymptotically sub-minimax solutions to compound statistical decision problems, Proc. Second Berkeley Sympos. Math. Stat. Prob., Vol. 1 Univ. Calif. Press Berkeley
F. Ruggeri S. Sivaganesan (2000) ArticleTitleOn a global sensitivity measure for Bayesian inference Sankhya: The Indian Journal of Statistics 62 110–127
V.P. Savchuk H.F. Martz (1994) ArticleTitleBayes reliability estimation using multiple sources of prior information: binomial sampling IEEE Transactions on Reliability 43 138–144 Occurrence Handle10.1109/24.285128
Vidakovic B. (2000), Gamma-minimax: a paradigm for conservative robust Bayesians. In: Rios., Ruggeri (eds). Robustness of Bayesian Inference, Lecture Notes in Statistics 152, 241–259, Springer
R.A. Waller M.M. Johnson M.S. Waterman H.F. Martz (1977) Gamma prior distribution selection for Bayesian analysis of failure rate and reliability J.B. Fussel G.R. Burdick (Eds) Nuclear System Reliability Engineering and Risk Assessment SIAM Philadelphia
Waterman M.S., Martz H.F., Waller R.A., (1976), Fitting beta prior distributions in Bayesian reliability analysis, Los Alamos Scientific Laboratory Report LA-6395-MS, July.
H. Weiler (1965) ArticleTitleThe use of incomplete beta functions for prior distributions in binomial sampling Technometrics 7 335–347
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Golodnikov, A.N., Knopov, P.S. & Pepelyaev, V.A. Estimation of Reliability Parameters Under Incomplete Primary Information. Theor Decis 57, 331–344 (2004). https://doi.org/10.1007/s11238-005-3217-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11238-005-3217-9