American Journal of Theoretical and Applied Statistics

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Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach

Received: 07 January 2016    Accepted: 08 January 2016    Published: 27 January 2016
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Abstract

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.

DOI 10.11648/j.ajtas.s.2016050201.14
Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 2-1, March 2016)

This article belongs to the Special Issue Novel Ideas for Efficient Optimization of Statistical Decisions and Predictive Inferences under Parametric Uncertainty of Underlying Models with Applications

Page(s) 21-28
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Age Replacement, Parametric Uncertainty, Equivalent Transformed Problem, Optimization

References
[1] R. E. Barlow and L. C. Hunter, “Optimal preventive maintenance policies,” Operations Research, vol. 8, pp. 90–100, 1960.
[2] R. E. Barlow and F. Proschan. Mathematical Theory of Reliability. New York: Wiley, 1965.
[3] R. E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing: Probability Models. New York: Holt, Rinehart, and Winston, 1975.
[4] H. Ascher and H. Feingold, Repairable Systems Reliability: Modeling, Inference, Misconceptions and their Causes. New York: Marcel Dekker, Inc., 1984.
[5] 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.
[6] 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.
[7] R. I. Phelps, “Optimal policy for minimal repair,” Journal of the Operational Research Society, vol. 34, pp. 425–427, 1983.
[8] 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.
[9] G. J. Glasser, “The age replacement problem,” Technometrics, vol. 9, pp. 83–91, 1967.
[10] R. L. Scheaffer, “Optimum age replacement policies with an increasing cost factor,” Technometrics, vol. 13, pp. 139–144, 1971.
[11] M. B. Berg, “A marginal cost analysis for preventive maintenance policies,” European Journal of Operational Research, vol. 4, pp. 136–142, 1980.
[12] N. A. Nechval and E. K. Vasermanis, Improved Decisions in Statistics. Riga: Izglitibas soli, 2004.
Author Information
  • Department of Mathematics, Baltic International Academy, Riga, Latvia

  • Department of Management, University of Latvia, Riga, Latvia

  • Department of Marketing, University of Latvia, Riga, Latvia

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    Nicholas A. Nechval, Gundars Berzins, Vadims Danovics. (2016). Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach. American Journal of Theoretical and Applied Statistics, 5(2-1), 21-28. https://doi.org/10.11648/j.ajtas.s.2016050201.14

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    Nicholas A. Nechval; Gundars Berzins; Vadims Danovics. Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach. Am. J. Theor. Appl. Stat. 2016, 5(2-1), 21-28. doi: 10.11648/j.ajtas.s.2016050201.14

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    Nicholas A. Nechval, Gundars Berzins, Vadims Danovics. Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach. Am J Theor Appl Stat. 2016;5(2-1):21-28. doi: 10.11648/j.ajtas.s.2016050201.14

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  • @article{10.11648/j.ajtas.s.2016050201.14,
      author = {Nicholas A. Nechval and Gundars Berzins and Vadims Danovics},
      title = {Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {5},
      number = {2-1},
      pages = {21-28},
      doi = {10.11648/j.ajtas.s.2016050201.14},
      url = {https://doi.org/10.11648/j.ajtas.s.2016050201.14},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajtas.s.2016050201.14},
      abstract = {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.},
     year = {2016}
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    T1  - Optimization of Statistical Decisions for Age Replacement Problems via a New Pivotal Quantity Averaging Approach
    AU  - Nicholas A. Nechval
    AU  - Gundars Berzins
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    AB  - 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.
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