Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach
American Journal of Theoretical and Applied Statistics
Volume 5, Issue 2-1, March 2016, Pages: 21-28
Received: Jan. 7, 2016;
Accepted: Jan. 8, 2016;
Published: Jan. 27, 2016
Views 3633 Downloads 91
Nicholas A. Nechval, Department of Mathematics, Baltic International Academy, Riga, Latvia
Gundars Berzins, Department of Management, University of Latvia, Riga, Latvia
Vadims Danovics, Department of Marketing, University of Latvia, Riga, Latvia
Follow on us
Age replacement strategies, where a unit is replaced upon failure or on reaching a predetermined age, whichever occurs first, provide simple and intuitively attractive replacement guidelines for technical units. Within theory of stochastic processes, the optimal preventive replacement age, in the sense of leading to minimal expected costs per unit of time when the strategy is used for a sequence of similar units over a long period of time, is derived by application of the renewal reward theorem. The mathematical solution to the problem of what is the optimal age for replacement is well known for the case when the parameter values of the underlying lifetime distributions are known with certainty. In actual practice, such is simply not the case. When these models are applied to solve real-world problems, the parameters are estimated and then treated as if they were the true values. The risk associated with using estimates rather than the true parameters is called estimation risk and is often ignored. When data are limited and (or) unreliable, estimation risk may be significant, and failure to incorporate it into the model design may lead to serious errors. Its explicit consideration is important since decision rules that are optimal in the absence of uncertainty need not even be approximately optimal in the presence of such uncertainty. In the present paper, for efficient optimization of statistical decisions under parametric uncertainty, the pivotal quantity averaging (PQA) approach is suggested. This approach represents a new simple and computationally attractive statistical technique based on the constructive use of the invariance principle in mathematical statistics. It allows one to carry out the transition from the original problem to the equivalent transformed problem (in terms of pivotal quantities and ancillary factors) via invariant embedding a sample statistic in the original problem. In this case, the statistical optimization of the equivalent transformed problem is carried out via ancillary factors. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors. This approach allows one to eliminate unknown parameters from the problem and to find the better decision rules, which have smaller risk than any of the well-known decision rules. To illustrate the proposed approach, the numerical examples are given.
Age Replacement, Parametric Uncertainty, Equivalent Transformed Problem, Optimization
To cite this article
Nicholas A. Nechval,
Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach, American Journal of Theoretical and Applied Statistics. Special Issue: Novel Ideas for Efficient Optimization of Statistical Decisions and Predictive Inferences under Parametric Uncertainty of Underlying Models with Applications.
Vol. 5, No. 2-1,
2016, pp. 21-28.
R. E. Barlow and L. C. Hunter, “Optimal preventive maintenance policies,” Operations Research, vol. 8, pp. 90–100, 1960.
R. E. Barlow and F. Proschan. Mathematical Theory of Reliability. New York: Wiley, 1965.
R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models. New York: Holt, Rinehart, and Winston, 1975.
H. Ascher and H. Feingold, Repairable Systems Reliability: Modeling, Inference, Misconceptions and their Causes. New York: Marcel Dekker, Inc., 1984.
T. Nakagawa, “Summary of periodic replacement with minimal repair at failure,” Journal of the Operations Research Society of Japan, vol. 24, pp. 213–228, 1981.
C. Valdez-Flares and R. M. Feldman, “A survey of preventive maintenance models for stochastically deteriorating single-unit systems,” Naval Research Logistics, vol. 36, pp. 419–446, 1989.
R. I. Phelps, “Optimal policy for minimal repair,” Journal of the Operational Research Society, vol. 34, pp. 425–427, 1983.
A. Tahara and T. Nishida, “Optimal replacement policy for minimal repair model,” Journal of the Operations Research Society of Japan, vol. 18, pp. 113–124, 1975.
G. J. Glasser, “The age replacement problem,” Technometrics, vol. 9, pp. 83–91, 1967.
R. L. Scheaffer, “Optimum age replacement policies with an increasing cost factor,” Technometrics, vol. 13, pp. 139–144, 1971.
M. B. Berg, “A marginal cost analysis for preventive maintenance policies,” European Journal of Operational Research, vol. 4, pp. 136–142, 1980.
N. A. Nechval and E. K. Vasermanis, Improved Decisions in Statistics. Riga: Izglitibas soli, 2004.