{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T07:14:59Z","timestamp":1762067699882,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T00:00:00Z","timestamp":1664236800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We consider the problem of estimating tail probabilities of random sums of scale mixture of phase-type distributions\u2014a class of distributions corresponding to random variables which can be represented as a product of a non-negative but otherwise arbitrary random variable with a phase-type random variable. Our motivation arises from applications in risk, queueing problems for estimating ruin probabilities, and waiting time distributions, respectively. Mixtures of distributions are flexible models and can be exploited in modelling non-life insurance loss amounts. Classical rare-event simulation algorithms cannot be implemented in this setting because these methods typically rely on the availability of the cumulative distribution function or the moment generating function, but these are difficult to compute or are not even available for the class of scale mixture of phase-type distributions. The contributions of this paper are that we address these issues by proposing alternative simulation methods for estimating tail probabilities of random sums of scale mixture of phase-type distributions which combine importance sampling and conditional Monte Carlo methods, showing the efficiency of the proposed estimators for a wide class of scaling distributions, and validating the empirical performance of the suggested methods via numerical experimentation.<\/jats:p>","DOI":"10.3390\/a15100350","type":"journal-article","created":{"date-parts":[[2022,9,27]],"date-time":"2022-09-27T21:23:27Z","timestamp":1664313807000},"page":"350","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Estimating Tail Probabilities of Random Sums of Phase-Type Scale Mixture Random Variables"],"prefix":"10.3390","volume":"15","author":[{"given":"Hui","family":"Yao","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia"}]},{"given":"Thomas","family":"Taimre","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Asmussen, S., and Albrecher, H. 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(TOMACS)"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1214\/aop\/1176996756","article-title":"Mixtures of distributions, moment inequalities and measures of exponentiality and normality","volume":"2","author":"Keilson","year":"1974","journal-title":"Ann. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1080\/03461238.2013.865257","article-title":"Calculation of Ruin Probabilities for a Dense Class of Heavy Tailed Distributions","volume":"2015","author":"Bladt","year":"2015","journal-title":"Scand. Actuar. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1017\/S1748499517000136","article-title":"Asymptotic tail behaviour of phase-type scale mixture distributions","volume":"12","author":"Xie","year":"2018","journal-title":"Ann. Actuar. Sci."},{"key":"ref_12","unstructured":"Neuts, M. (1975). Probability Distributions of Phase Type, Department of Mathematics, University of Louvain."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/s11857-009-0088-0","article-title":"On the Efficiency of the Asmussen\u2013Kroese-estimator and its Application to Stop-loss Transforms","volume":"30","author":"Hartinger","year":"2009","journal-title":"Bl\u00e4tter der DGVFM"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1214\/aos\/1176343541","article-title":"Importance Sampling in the Monte Carlo Study of Sequential Tests","volume":"4","author":"Siegmund","year":"1976","journal-title":"Ann. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kroese, D., Taimre, T., and Botev, Z. (2011). Handbook of Monte Carlo Methods, John Wiley and Sons. 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Probab."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/350\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:40:14Z","timestamp":1760143214000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/15\/10\/350"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,27]]},"references-count":18,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["a15100350"],"URL":"https:\/\/doi.org\/10.3390\/a15100350","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2022,9,27]]}}}