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The goal is to estimate these distributions under the sole assumption that$$Q_x$$<\/jats:tex-math>Q<\/mml:mi>x<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>is isotonic inx<\/jats:italic>with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution$$\\mathcal {L}(X,Y)$$<\/jats:tex-math>L<\/mml:mi>(<\/mml:mo>X<\/mml:mi>,<\/mml:mo>Y<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>under the sole assumption that it is totally positive of order two. An algorithm is developed which estimates the unknown family of distributions$$(Q_x)_x$$<\/jats:tex-math>(<\/mml:mo>Q<\/mml:mi>x<\/mml:mi><\/mml:msub>)<\/mml:mo><\/mml:mrow>x<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.<\/jats:p>","DOI":"10.1007\/s11222-023-10370-9","type":"journal-article","created":{"date-parts":[[2023,12,31]],"date-time":"2023-12-31T11:02:01Z","timestamp":1704020521000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Estimation of a likelihood ratio ordered family of distributions"],"prefix":"10.1007","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8270-3724","authenticated-orcid":false,"given":"Alexandre","family":"M\u00f6sching","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0172-9285","authenticated-orcid":false,"given":"Lutz","family":"D\u00fcmbgen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,12,31]]},"reference":[{"key":"10370_CR1","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1017\/S0266466614000401","volume":"31","author":"BK Beare","year":"2015","unstructured":"Beare, B.K., Moon, J.-M.: Nonparametric tests of density ratio ordering. 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