{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:30:36Z","timestamp":1760239836194,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T00:00:00Z","timestamp":1547424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper focuses on test procedures under corrupted data. We assume that the observations     Z i     are mismeasured, due to the presence of measurement errors. Thus, instead of     Z i     for     i = 1 , \u2026 , n, we observe      X i  =  Z i  +  \u03b4   V i, with an unknown parameter    \u03b4    and an unobservable random variable     V i. It is assumed that the random variables     Z i     are i.i.d., as are the     X i     and the     V i. The test procedure aims at deciding between two simple hyptheses pertaining to the density of the variable     Z i, namely     f 0     and     g 0. In this setting, the density of the     V i     is supposed to be known. The procedure which we propose aggregates likelihood ratios for a collection of values of    \u03b4. A new definition of least-favorable hypotheses for the aggregate family of tests is presented, and a relation with the Kullback-Leibler divergence between the sets       f \u03b4   \u03b4     and       g \u03b4   \u03b4     is presented. Finite-sample lower bounds for the power of these tests are presented, both through analytical inequalities and through simulation under the least-favorable hypotheses. Since no optimality holds for the aggregation of likelihood ratio tests, a similar procedure is proposed, replacing the individual likelihood ratio by some divergence based test statistics. It is shown and discussed that the resulting aggregated test may perform better than the aggregate likelihood ratio procedure.<\/jats:p>","DOI":"10.3390\/e21010063","type":"journal-article","created":{"date-parts":[[2019,1,14]],"date-time":"2019-01-14T12:20:07Z","timestamp":1547468407000},"page":"63","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Composite Tests under Corrupted Data"],"prefix":"10.3390","volume":"21","author":[{"given":"Michel","family":"Broniatowski","sequence":"first","affiliation":[{"name":"Laboratoire de Probabilit\u00e9s, Statistique et Mod\u00e9lisation, Sorbonne Universit\u00e9, 75005 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jana","family":"Jure\u010dkov\u00e1","sequence":"additional","affiliation":[{"name":"Institute of Information Theory and Automation, The Czech Academy of Sciences, 18208 Prague, Czech Republic"},{"name":"Faculty of Mathematics and Physics, Charles University, 18207 Prague, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5721-6705","authenticated-orcid":false,"given":"Ashok Kumar","family":"Moses","sequence":"additional","affiliation":[{"name":"Department of ECE, Indian Institute of Technology, Palakkad 560012, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Emilie","family":"Miranda","sequence":"additional","affiliation":[{"name":"Laboratoire de Probabilit\u00e9s, Statistique et Mod\u00e9lisation, Sorbonne Universit\u00e9, 75005 Paris, France"},{"name":"Safran Aircraft Engines, 77550 Moissy-Cramayel, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Broniatowski, M., Jure\u010dkov\u00e1, J., and Kalina, J. (2018). Likelihood ratio testing under measurement errors. Entropy, 20.","DOI":"10.3390\/e20120966"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Guo, D. (July, January 28). Relative entropy and score function: New information-estimation relationships through arbitrary additive perturbation. Proceedings of the IEEE International Symposium on Information Theory (ISIT 2009), Seoul, Korea.","DOI":"10.1109\/ISIT.2009.5205652"},{"key":"ref_3","first-page":"251","article-title":"Minimax tests and the Neyman-Pearson lemma for capacities","volume":"2","author":"Huber","year":"1973","journal-title":"Ann. Stat."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2034","DOI":"10.1016\/j.jmva.2006.03.007","article-title":"Interpreting Kullback-Leibler divergence with the Neyman-Pearson lemma","volume":"97","author":"Eguchi","year":"2006","journal-title":"J. Multivar. Anal."},{"key":"ref_5","unstructured":"Narayanan, K.R., and Srinivasa, A.R. (arXiv, 2007). On the thermodynamic temperature of a general distribution, arXiv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1214\/aoms\/1177705894","article-title":"Stochastic comparison of tests","volume":"31","author":"Bahadur","year":"1960","journal-title":"Ann. Math. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bahadur, R.R. (1971). Some Limit Theorems in Statistics, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611970630"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/BF00532119","article-title":"Vitesses maximales de d\u00e9croissance des erreurs et tests optimaux associ\u00e9s","volume":"55","year":"1981","journal-title":"Z. Wahrsch. Verw. Gebiete"},{"key":"ref_9","first-page":"385","article-title":"On asymptotically optimal tests","volume":"5","year":"1987","journal-title":"Ann. Stat."},{"key":"ref_10","unstructured":"Liese, F., and Vajda, I. (1987). Convex Statistical Distances, Teubner."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of BG statistics","volume":"52","author":"Tsallis","year":"1987","journal-title":"J. Stat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.1017\/S0305004100042225","article-title":"A class of infinitely divisible random variables","volume":"63","author":"Goldie","year":"1967","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Basu, A., Shioya, H., and Park, C. (2011). Statistical Inference: The Minimum Distance Approach, CRC Press.","DOI":"10.1201\/b10956"},{"key":"ref_14","unstructured":"Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory, John Wiley & Sons."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1646","DOI":"10.1214\/aoms\/1177696808","article-title":"Bounds for the power of likelihood ratio tests and their asymptotic properties","volume":"41","author":"Krafft","year":"1970","journal-title":"Ann. Math. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aoms\/1177728347","article-title":"Large-sample theory: Parametric case","volume":"27","author":"Chernoff","year":"1956","journal-title":"Ann. Math. 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