UNCERTAINTIES OF THE ESTIMATORS AND PARAMETERS OF DISTRIBUTION IN MEASUREMENTS WITH MULTIPLY OBSERVATIONS

2020;
: pp. 3-9
1
Національний університет “Львівська політехніка”
2
Rzeszow University of Technology, Poland

The article shows that the commonly used method of estimating the Type A uncertainty of measurements based on the standard deviation of estimators of population parameters does not meet the definition of uncertainty. For correct determination of the standard uncertainty, it is necessary to use the distribution of the corresponding population parameter at the values of population estimators determined from the experiment but not the probability distribution of the estimator. The joint probability distribution of population parameters can be derived by transforming the joint distribution of estimators using a Jacobian equal to the ratio of the scale parameter estimator to the population scale parameter itself. Independently on population distribution, the standard uncertainties of the location and scale parameters of the population depend on the number of observation n as a function of , i.e. can be determined when ≥ 4. When the number of observations is small then the uncertainty value calculated by the usual method may differ significantly from the correct value. The given numerical example confirms this statement.

 

[1] JCGM 100:2008 Evaluation of measurement data—Guide to the expression of uncertainty in measurement Joint Committee for Guides in Metrology, 2008.

[2] JCGM 200:2012 International vocabulary of metrology – Basic and general concepts and associated terms (VIM) 3rd edition, 2012.

[3] W. Bich, M. Cox, C. Michotte, “Towards a new GUM—an update, Metrologia, no.53, 2016.

https://doi.org/10.1088/0026-1394/53/5/S149

[4] R. Kacker, A. Jones, “On use of Bayesian statistics to make the Guide to the Expression of Uncertainty in Measurement consistent”, Metrologia, no.40, pp. 235–248, 2003.

https://doi.org/10.1088/0026-1394/40/5/305

[5] M. Cox, K. Shirono, “Informative Bayesian Type A uncertainty evaluation, especially applicable to a small number of observations”, Metrologia, no.54, pp.642–652, 2017.

https://doi.org/10.1088/1681-7575/aa787f

[6] K. Weise, W. Wöger, “A Bayesian theory of measurement uncertainty”, Meas. Sci. Technol., no.3, pp.1–11, 1992.

https://doi.org/10.1088/0957-0233/4/1/001

[7] J. Berger, Statistical Decision Theory and Bayesian Analysis. New York: Springer, 1985.

https://doi.org/10.1007/978-1-4757-4286-2

[8] I. Lira, W. Wöger, “Bayesian evaluation of the standard uncertainty and coverage probability in a simple measurement model”, Meas. Sci. Technol., no.12, pp.1172–9, 2001.

https://doi.org/10.1088/0957-0233/12/8/326

[9] I. Lira, W. Wöger, “Evaluation of Repeated Measurement from the Viewpoint of Conventional and Bayesian Statistics. Advanced Mathematical and Computational Tools in Metrology VII”, World Scientific Publishing Co., pp.73–84, 2006.

https://doi.org/10.1142/9789812774187_0007

[10] M. Fisz, Probability Theory and Mathematical Statistics. John Wiley & Sons, 1967.

[11] Handbook of Applicable Mathematics. Vol.VI: Statistics, part A, John Wiley and Sons, 1984.

[12] E. Lehmann, Theory of point estimation, Wiley, 1983.

https://doi.org/10.1007/978-1-4757-2769-2

[13] H. Cramer, Mathematical methods of statistics. Princeton University Press.

[14] M. Dorozhovets, “Forward and inverse problems of Type A uncertainty evaluation”, Measurement, vol.165, 2020.

https://doi.org/10.1016/j.measurement.2020.108072

[15] H. Jeffreys, Theory of probability, 3-d edition, Oxford University Press (Oxford), 1983.

[16] P. Christian, N. Chopin, J. Rousseau, H. Jeffreys’s, “Theory of Probability Revisited. Statistical Science, Institute of Mathematical Statistics”, vol.24, no.2, pp.141–172, 2009.

https://doi.org/10.1214/09-STS284

[17] M. Evans, N. Hastings, B. Peacock, Chi Distribution in Statistical Distributions, New York, Wiley, 2000.