Composite Tests under Corrupted Data
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 , … , n, we observe X i = Z i + δ V i, with an unknown parame...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-01-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | http://www.mdpi.com/1099-4300/21/1/63 |
| _version_ | 1798024317258170368 |
|---|---|
| author | Michel Broniatowski Jana Jurečková Ashok Kumar Moses Emilie Miranda |
| author_facet | Michel Broniatowski Jana Jurečková Ashok Kumar Moses Emilie Miranda |
| author_sort | Michel Broniatowski |
| collection | DOAJ |
| description | 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 , … , n, we observe X i = Z i + δ V i, with an unknown parameter δ 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 δ. 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 δ δ and g δ δ 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. |
| first_indexed | 2024-04-11T18:00:30Z |
| format | Article |
| id | doaj.art-0e7f5aaaa96c47a78c98ead60f8bffae |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T18:00:30Z |
| publishDate | 2019-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-0e7f5aaaa96c47a78c98ead60f8bffae2022-12-22T04:10:32ZengMDPI AGEntropy1099-43002019-01-012116310.3390/e21010063e21010063Composite Tests under Corrupted DataMichel Broniatowski0Jana Jurečková1Ashok Kumar Moses2Emilie Miranda3Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, 75005 Paris, FranceInstitute of Information Theory and Automation, The Czech Academy of Sciences, 18208 Prague, Czech RepublicDepartment of ECE, Indian Institute of Technology, Palakkad 560012, IndiaLaboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, 75005 Paris, FranceThis 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 , … , n, we observe X i = Z i + δ V i, with an unknown parameter δ 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 δ. 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 δ δ and g δ δ 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.http://www.mdpi.com/1099-4300/21/1/63composite hypothesescorrupted dataleast-favorable hypothesesNeyman Pearson testdivergence based testingChernoff Stein lemma |
| spellingShingle | Michel Broniatowski Jana Jurečková Ashok Kumar Moses Emilie Miranda Composite Tests under Corrupted Data Entropy composite hypotheses corrupted data least-favorable hypotheses Neyman Pearson test divergence based testing Chernoff Stein lemma |
| title | Composite Tests under Corrupted Data |
| title_full | Composite Tests under Corrupted Data |
| title_fullStr | Composite Tests under Corrupted Data |
| title_full_unstemmed | Composite Tests under Corrupted Data |
| title_short | Composite Tests under Corrupted Data |
| title_sort | composite tests under corrupted data |
| topic | composite hypotheses corrupted data least-favorable hypotheses Neyman Pearson test divergence based testing Chernoff Stein lemma |
| url | http://www.mdpi.com/1099-4300/21/1/63 |
| work_keys_str_mv | AT michelbroniatowski compositetestsundercorrupteddata AT janajureckova compositetestsundercorrupteddata AT ashokkumarmoses compositetestsundercorrupteddata AT emiliemiranda compositetestsundercorrupteddata |