{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,11]],"date-time":"2026-01-11T00:32:54Z","timestamp":1768091574518,"version":"3.49.0"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T00:00:00Z","timestamp":1615939200000},"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>In 2015, Lad, Sanfilippo and Agr\u00f2 proposed an alternative measure of uncertainty dual to the entropy known as extropy. This paper provides some results on a dispersion measure of extropy of random variables which is called varextropy and studies several properties of this concept. Especially, the varextropy measure of residual and past lifetimes, order statistics, record values and proportional hazard rate models are discussed. Moreover, the conditional varextropy is considered and some properties of this measure are studied. Finally, a new stochastic comparison method, named varextropy ordering, is introduced and some of its properties are presented.<\/jats:p>","DOI":"10.3390\/e23030356","type":"journal-article","created":{"date-parts":[[2021,3,17]],"date-time":"2021-03-17T11:48:22Z","timestamp":1615981702000},"page":"356","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Results on Varextropy Measure of Random Variables"],"prefix":"10.3390","volume":"23","author":[{"given":"Nastaran Marzban","family":"Vaselabadi","sequence":"first","affiliation":[{"name":"Department of Statistics, Persian Gulf University, Bushehr 7516913817, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4025-6791","authenticated-orcid":false,"given":"Saeid","family":"Tahmasebi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Persian Gulf University, Bushehr 7516913817, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9957-6963","authenticated-orcid":false,"given":"Mohammad Reza","family":"Kazemi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, Fasa University, Fasa 7461686131, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3569-4052","authenticated-orcid":false,"given":"Francesco","family":"Buono","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica e Applicazioni \u201cRenato Caccioppoli\u201d, Universit\u00e0 degli Studi di Napoli Federico II, I-80126 Naples, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,17]]},"reference":[{"key":"ref_1","unstructured":"Clausius, R., and Hirst, T.A. (1867). The Mechanical Theory of Heat: With Its Application to the Steam Engine and to the Physical Properties of Bodies, J. Van Voorst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"AT&T Bell Labs. Tech. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0378-3758(00)00169-5","article-title":"R\u00e9nyi information, log likelihood and an intrinsic distribution measure","volume":"93","author":"Song","year":"2001","journal-title":"J. Statist. Plann. Inference"},{"key":"ref_4","unstructured":"Liu, J. (2007). Information Theoretic Content and Probability. [Ph.D. Thesis, University of Florida]."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Houdre, C., Mason, D., Reynaud-Bouret, P., and Rosinski, J. (2016). Optimal concentration of information content for logconcave densities. High Dimensional Probability VII. Progress in Probability, Springer.","DOI":"10.1007\/978-3-319-40519-3"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Di Crescenzo, A., and Paolillo, L. (2020). Analysis and applications of the residual varentropy of random lifetimes. Probab. Eng. Inf. Sci.","DOI":"10.1017\/S0269964820000133"},{"key":"ref_7","unstructured":"Maadani, S., Mohtashami Borzadaran, G.R., and Rezaei Roknabadi, A.H. (2020). Varentropy of order statistics and some stochastic comparisons. Commun. Stat. Theory Methods, 1\u201316."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1214\/14-STS430","article-title":"Extropy: Complementary dual of entropy","volume":"30","author":"Lad","year":"2015","journal-title":"Statist. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.spl.2016.09.016","article-title":"The extropy of order statistics and record values","volume":"120","author":"Qiu","year":"2017","journal-title":"Stat. Probabil. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.spl.2017.09.014","article-title":"The residual extropy of order statistics","volume":"133","author":"Qiu","year":"2018","journal-title":"Stat. Probabil. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1080\/10485252.2017.1404063","article-title":"Extropy estimators with applications in testing uniformity","volume":"30","author":"Qiu","year":"2018","journal-title":"J. Nonparametr. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Gilchrist, W. (2000). Statistical Modelling with Quantile Functions, Champman & Hall.","DOI":"10.1201\/9781420035919"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2193","DOI":"10.1109\/TIT.2018.2806965","article-title":"\u03a6-Entropic Measures of Correlation","volume":"64","author":"Beigi","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., Buono, F., and Longobardi, M. (2020). On weighted extropies. Commun. Stat. Theory Methods.","DOI":"10.1080\/03610926.2020.1860222"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Krishnan, A.S., Sunoj, S.M., and Nair, N.U. (2020). Some reliability properties of extropy for residual and past lifetime random variables. J. Korean Stat. Soc.","DOI":"10.1007\/s42952-019-00023-x"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kamari, O., and Buono, F. (2020). On extropy of past lifetime distribution. Ric. Mat.","DOI":"10.1007\/s11587-020-00488-7"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3525","DOI":"10.1016\/j.jspi.2007.03.029","article-title":"Proportional reversed hazard rate model and its applications","volume":"137","author":"Gupta","year":"2007","journal-title":"J. Statist. Plan. Inference"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5185","DOI":"10.1109\/TIT.2015.2452253","article-title":"Monotone Measures for Non-Local Correlations","volume":"61","author":"Beigi","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Shaked, M., and Shanthikumar, J.G. (2007). Stochastic Orders, Springer.","DOI":"10.1007\/978-0-387-34675-5"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1214\/aop\/1176994428","article-title":"A note on an inequality involving the normal distribution","volume":"9","author":"Chernoff","year":"1981","journal-title":"Ann. Probab."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1080\/03610928608829189","article-title":"On tail-ordering and comparison of failure rates","volume":"15","author":"Bagai","year":"1986","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.2307\/1427262","article-title":"Dispersive and Superadditive Ordering","volume":"18","author":"Ahmed","year":"1986","journal-title":"Adv. Appl. Probab."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/356\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:37:00Z","timestamp":1760161020000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,17]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030356"],"URL":"https:\/\/doi.org\/10.3390\/e23030356","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,17]]}}}