{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:46:55Z","timestamp":1760233615288,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T00:00:00Z","timestamp":1611792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP20K11702"],"award-info":[{"award-number":["JP20K11702"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper presents a difference-type lower bound for the Bayes risk as a difference-type extension of the Borovkov\u2013Sakhanenko bound. The resulting bound asymptotically improves the Bobrovsky\u2013Mayor\u2013Wolf\u2013Zakai bound which is difference-type extension of the Van Trees bound. Some examples are also given.<\/jats:p>","DOI":"10.3390\/e23020161","type":"journal-article","created":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T11:54:53Z","timestamp":1611834893000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Improvement of Bobrovsky\u2013Mayor\u2013Wolf\u2013Zakai Bound"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8664-1172","authenticated-orcid":false,"given":"Ken-ichi","family":"Koike","sequence":"first","affiliation":[{"name":"College of Commerce, Nihon University, Tokyo 157-8570, Japan"}]},{"given":"Shintaro","family":"Hashimoto","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hiroshima University, Hiroshima 739-8521, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,28]]},"reference":[{"key":"ref_1","unstructured":"Van Trees, H.L. (1968). Detection, Estimation, and Modulation Theory, Part I, Wiley."},{"key":"ref_2","first-page":"185","article-title":"On estimates of the expected quadratic risk (in Russian)","volume":"1","author":"Borovkov","year":"1980","journal-title":"Probab. Math. Statist."},{"key":"ref_3","unstructured":"Borovkov, A.A. (1998). Mathematical Statistics, Gordon and Breach."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2307\/3318681","article-title":"Applications of the van Trees inequality: A Bayesian Cram\u00e9r-Rao bound","volume":"1","author":"Gill","year":"1995","journal-title":"Bernoulli"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2185","DOI":"10.1080\/03610920600854496","article-title":"An integral Bhattacharyya type bound for the Bayes risk","volume":"35","author":"Koike","year":"2006","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5213","DOI":"10.1080\/03610926.2013.810265","article-title":"Bhattacharyya type information inequality for the Bayes risk","volume":"44","author":"Hashimoto","year":"2015","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1109\/18.2647","article-title":"A general class of lower bounds in parameter estimation","volume":"34","author":"Weinstein","year":"1988","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5334","DOI":"10.1109\/TSP.2008.927075","article-title":"A fresh look at the Bayesian bounds of the Weiss-Weinstein family","volume":"56","author":"Renaux","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5064","DOI":"10.1109\/TIT.2010.2059890","article-title":"General classes of performance lower bounds for parameter estimation\u2014Part II: Bayesian bounds","volume":"56","author":"Todros","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1109\/TIT.1969.1054301","article-title":"Some lower bounds on signal parameter estimation","volume":"15","author":"Ziv","year":"1969","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1109\/18.556118","article-title":"Extended Ziv-Zakai lower bound for vector parameter estimation","volume":"43","author":"Bell","year":"1997","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2152","DOI":"10.1109\/TSP.2012.2187523","article-title":"A general class of outage error probability lower bounds in Bayesian parameter estimation","volume":"60","author":"Routtenberg","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Van Trees, H.L., and Bell, K.L. (2007). Bayesian Bounds for Parameter Estimation and Nonlinear Filtering\/Tracking, Wiley and IEEE Press.","DOI":"10.1109\/9780470544198"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/tpms\/945","article-title":"On asymptotic Borovkov-Sakhanenko inequality with unbounded parameter set","volume":"90","author":"Veretennikov","year":"2015","journal-title":"Theory Probab. Math. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Koike, K. (2020). Asymptotic comparison of some Bayesian information bounds. Commun. Stat. Theory Methods, 49.","DOI":"10.1080\/03610926.2020.1752722"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Koike, K. (2019). Attainments of the Bayesian information bounds. Commun. Stat. Theory Methods, 48.","DOI":"10.1080\/03610926.2019.1676445"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1214\/aos\/1176350602","article-title":"Some classes of grobal Cram\u00e9r-Rao bounds","volume":"15","author":"Bobrovsky","year":"1987","journal-title":"Ann. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"761","DOI":"10.2307\/2529262","article-title":"A generalization of the probit and logit models for dose response curves","volume":"32","author":"Prentice","year":"1976","journal-title":"Biometrics"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Small, C.G. (2010). 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