Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models

Nicholas A. Nechval; Natalija Ribakova; Gundars Berzins

doi:doi:10.11648/j.ajtas.s.2016050201.17

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Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models

Nicholas A. Nechval, Natalija Ribakova, Gundars Berzins

Published in American Journal of Theoretical and Applied Statistics (Volume 5, Issue 2-1)

Received: 31 January 2016 Accepted: 2 February 2016 Published: 23 February 2016

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Abstract

Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.

Published in	American Journal of Theoretical and Applied Statistics (Volume 5, Issue 2-1) 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
DOI	10.11648/j.ajtas.s.2016050201.17
Page(s)	49-55
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), 2016. Published by Science Publishing Group

Keywords

Future Outcomes, Parametric Uncertainty, Predictive Inferences

References

[1]	H. Scarf, “Bayes solutions of statistical inventory problem,” Ann. Math. Statist., vol. 30, pp. 490508, 1959.
[2]	S. Karlin, “Dynamic inventory policy with varying stochastic demands,” Management Sci., vol. 6, pp. 231258, 1960.
[3]	K. S. Azoury, “Bayes solution to dynamic inventory models under unknown demand distribution,” Management Sci., vol. 31, pp. 11501160, 1985.
[4]	S. A. Conrad, “Sales data and the estimation of demand,” Oper. Res. Quart., vol. 27, pp. 123127, 1976.
[5]	L. H. Liyanage and J. G. Shanthikumar, “A practical inventory control policy using operational statistics,” Oper. Res. Lett., vol. 33, pp. 341348, 2005.
[6]	K. S. Kaminsky and P. I. Nelson, “Prediction of order statistics,” in Handbook of Statistics-17: Order Statistic: Applications, N. Balakrishnan and C. R. Rao (Eds.). Elsevier Science, 1998, pp. 431450.
[7]	J. Aitchison and D. Sculthorpe, “Some problems of statistical prediction,” Biometrika, vol. 52, pp. 469483, 1965.
[8]	J. F. Lawless, Statistical Models and Methods for Lifetime Data. New York: John Wiley, 1982.
[9]	R. A. Fisher, “Two new properties of mathematical likelihood,” Proc. Roy. Statist. Soc., vol. A 144, pp. 285307, 1934.
[10]	N. A. Nechval, K. N. Nechval, and E. K. Vasermanis, “Statistical models for prediction of the fatigue crack growth in aircraft service,” in Fatigue Damage of Materials 2003, A. Varvani-Farahani and C. A. Brebbia (Eds.). Southampton, Boston: WIT Press, 2003, pp. 435445.
[11]	D. N. P. Murthy, M. Xie, and Y. Jiang, Weibull Models. New York: John Wiley and Sons Inc., 2004.
[12]	N. A. Nechval, K. N. Nechval, and E. K. Vasermanis, “Effective state estimation of stochastic systems,” Kybernetes (An International Journal of Systems & Cybernetics), vol. 32, pp. 666 678, 2003.
[13]	N. A. Nechval, G. Berzins, M. Purgailis, and K. N. Nechval, “Improved estimation of state of stochastic systems via invariant embedding technique,” WSEAS Transactions on Mathematics, vol. 7, pp. 141159, 2008.
[14]	N. A. Nechval, K. N. Nechval, and M. Purgailis, “Prediction of future values of random quantities based on previously observed data,” Engineering Letters, vol. 9, pp. 346359, 2011.
[15]	N. A. Nechval, M. Purgailis, K. N. Nechval, and V. F. Strelchonok, “Optimal predictive inferences for future order statistics via a specific loss function,” IAENG International Journal of Applied Mathematics, vol. 42, pp. 40 51, 2012.
[16]	N. A. Nechval, M. Purgailis, K. Cikste, G. Berzins, and K. N. Nechval, “Optimization of statistical decisions via an invariant embedding technique,” in Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering 2010 (WCE 2010). London, pp. 17761782, 2010.
[17]	G. Hadley, and T. M. Whitin, Analysis of Inventory Systems. New Jersey: Prentice-Hall Inc., 1963.

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Nicholas A. Nechval, Natalija Ribakova, Gundars Berzins. (2016). Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. American Journal of Theoretical and Applied Statistics, 5(2-1), 49-55. https://doi.org/10.11648/j.ajtas.s.2016050201.17

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ACS Style

Nicholas A. Nechval; Natalija Ribakova; Gundars Berzins. Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. Am. J. Theor. Appl. Stat. 2016, 5(2-1), 49-55. doi: 10.11648/j.ajtas.s.2016050201.17

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AMA Style

Nicholas A. Nechval, Natalija Ribakova, Gundars Berzins. Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. Am J Theor Appl Stat. 2016;5(2-1):49-55. doi: 10.11648/j.ajtas.s.2016050201.17

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@article{10.11648/j.ajtas.s.2016050201.17,
  author = {Nicholas A. Nechval and Natalija Ribakova and Gundars Berzins},
  title = {Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {5},
  number = {2-1},
  pages = {49-55},
  doi = {10.11648/j.ajtas.s.2016050201.17},
  url = {https://doi.org/10.11648/j.ajtas.s.2016050201.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2016050201.17},
  abstract = {Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.},
 year = {2016}
}

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TY  - JOUR
T1  - Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models
AU  - Nicholas A. Nechval
AU  - Natalija Ribakova
AU  - Gundars Berzins
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DO  - 10.11648/j.ajtas.s.2016050201.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
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UR  - https://doi.org/10.11648/j.ajtas.s.2016050201.17
AB  - Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.
VL  - 5
IS  - 2-1
ER  -

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Author Information

Nicholas A. Nechval

Department of Mathematics, Baltic International Academy, Riga, Latvia
Natalija Ribakova

Department of Marketing, University of Latvia, Riga, Latvia
Gundars Berzins

Department of Management, University of Latvia, Riga, Latvia

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APA Style

Nicholas A. Nechval, Natalija Ribakova, Gundars Berzins. (2016). Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. American Journal of Theoretical and Applied Statistics, 5(2-1), 49-55. https://doi.org/10.11648/j.ajtas.s.2016050201.17

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ACS Style

Nicholas A. Nechval; Natalija Ribakova; Gundars Berzins. Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. Am. J. Theor. Appl. Stat. 2016, 5(2-1), 49-55. doi: 10.11648/j.ajtas.s.2016050201.17

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AMA Style

Nicholas A. Nechval, Natalija Ribakova, Gundars Berzins. Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models. Am J Theor Appl Stat. 2016;5(2-1):49-55. doi: 10.11648/j.ajtas.s.2016050201.17

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@article{10.11648/j.ajtas.s.2016050201.17,
  author = {Nicholas A. Nechval and Natalija Ribakova and Gundars Berzins},
  title = {Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {5},
  number = {2-1},
  pages = {49-55},
  doi = {10.11648/j.ajtas.s.2016050201.17},
  url = {https://doi.org/10.11648/j.ajtas.s.2016050201.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2016050201.17},
  abstract = {Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Efficient Predictive Inferences for Future Outcomes Under Parametric Uncertainty of Underlying Models
AU  - Nicholas A. Nechval
AU  - Natalija Ribakova
AU  - Gundars Berzins
Y1  - 2016/02/23
PY  - 2016
N1  - https://doi.org/10.11648/j.ajtas.s.2016050201.17
DO  - 10.11648/j.ajtas.s.2016050201.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 49
EP  - 55
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.s.2016050201.17
AB  - Predictive inferences (predictive distributions, prediction and tolerance limits) for future outcomes on the basis of the past and present knowledge represent a fundamental problem of statistics, arising in many contexts and producing varied solutions. In this paper, new-sample prediction based on a previous sample (i.e., when for predicting the future outcomes in a new sample there are available the observed data only from a previous sample), within-sample prediction based on the early data from a current experiment (i.e., when for predicting the future outcomes in a sample there are available the early data only from that sample), and new-within-sample prediction based on both the early data from that sample and the data from a previous sample (i.e., when for predicting the future outcomes in a new sample there are available both the early data from that sample and the data from a previous sample) are considered. It is assumed that only the functional form of the underlying distributions is specified, but some or all of its parameters are unspecified. In such cases ancillary statistics and pivotal quantities, whose distribution does not depend on the unknown parameters, are used. In order to construct predictive inferences for future outcomes, the invariant embedding technique representing the exact pivotal-based method is proposed. In particular, this technique can be used for optimization of inventory management problems. A practical example is given.
VL  - 5
IS  - 2-1
ER  -

Copy | Download