{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T13:13:26Z","timestamp":1790082806691,"version":"4.0.1"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,12]],"date-time":"2015-06-12T00:00:00Z","timestamp":1434067200000},"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>Stress-strength reliability problems arise frequently in applied statistics and related fields. Often they involve two independent and possibly small samples of measurements on strength and breakdown pressures (stress). The goal of the researcher is to use the measurements to obtain inference on reliability, which is the probability that stress will exceed strength. This paper addresses the case where reliability is expressed in terms of an integral which has no closed form solution and where the number of observed values on stress and strength is small. We find that the Lagrange approach to estimating constrained likelihood, necessary for inference, often performs poorly. We introduce a penalized likelihood method and it appears to always work well. We use third order likelihood methods to partially offset the issue of small samples. The proposed method is applied to draw inferences on reliability in stress-strength problems with independent exponentiated exponential distributions. Simulation studies are carried out to assess the accuracy of the proposed method and to compare it with some standard asymptotic methods.<\/jats:p>","DOI":"10.3390\/e17064040","type":"journal-article","created":{"date-parts":[[2015,6,12]],"date-time":"2015-06-12T10:56:00Z","timestamp":1434106560000},"page":"4040-4063","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Penalized Likelihood Approach to Parameter Estimation with Integral Reliability Constraints"],"prefix":"10.3390","volume":"17","author":[{"given":"Barry","family":"Smith","sequence":"first","affiliation":[{"name":"Department of Economics, York University, Toronto ON M3J 1P3, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Steven","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto ON M3J 1P3, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Augustine","family":"Wong","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto ON M3J 1P3, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaofeng","family":"Zhou","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto ON M3J 1P3, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,12]]},"reference":[{"key":"ref_1","unstructured":"Bertsekas, D.P. 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