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Entropy 2018, 20(12), 966; https://doi.org/10.3390/e20120966

Likelihood Ratio Testing under Measurement Errors

1
Faculté de Mathématiques, Laboratoire de Probabilité, Statistique et Modélisation, Université Pierre et Marie Curie (Sorbonne Université), 4 place Jussieu, 75252 Paris CEDEX 05, France
2
Institute of Information Theory and Automation, The Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08 Prague 8, Czech Republic
3
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
4
Institute of Computer Science, The Czech Academy of Sciences, Pod Vodárenskou věží 2, 182 07 Prague 8, Czech Republic
*
Author to whom correspondence should be addressed.
Received: 13 November 2018 / Revised: 6 December 2018 / Accepted: 7 December 2018 / Published: 13 December 2018
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Abstract

We consider the likelihood ratio test of a simple null hypothesis (with density f 0 ) against a simple alternative hypothesis (with density g 0 ) in the situation that observations X i are mismeasured due to the presence of measurement errors. Thus instead of X i for i = 1 , , n , we observe Z i = X i + δ V i with unobservable parameter δ and unobservable random variable V i . When we ignore the presence of measurement errors and perform the original test, the probability of type I error becomes different from the nominal value, but the test is still the most powerful among all tests on the modified level. Further, we derive the minimax test of some families of misspecified hypotheses and alternatives. The test exploits the concept of pseudo-capacities elaborated by Huber and Strassen (1973) and Buja (1986). A numerical experiment illustrates the principles and performance of the novel test. View Full-Text
Keywords: measurement errors; robust testing; two-sample test; misspecified hypothesis and alternative; 2-alternating capacities measurement errors; robust testing; two-sample test; misspecified hypothesis and alternative; 2-alternating capacities
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Broniatowski, M.; Jurečková, J.; Kalina, J. Likelihood Ratio Testing under Measurement Errors. Entropy 2018, 20, 966.

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