{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:59:39Z","timestamp":1766138379869,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,7]],"date-time":"2023-07-07T00:00:00Z","timestamp":1688688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007345","name":"King Mongkut\u2019s University of Technology North Bangkok, Thailand","doi-asserted-by":"publisher","award":["KMUTNB-66-BASIC-04"],"award-info":[{"award-number":["KMUTNB-66-BASIC-04"]}],"id":[{"id":"10.13039\/501100007345","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In recent decades, there have been numerous endeavors to develop a novel category of survival distributions possessing enhanced flexibility through the extension of existing distributions. This article constructs and validates the statistical properties of a novel survival distribution in order to obtain an alternative distribution that is suitable for analyzing survival data by presenting the novel mixture of the Fr\u00e9chet distribution along with statistical properties such as the probability density function (PDF), cumulative distribution function (CDF), rth ordinary moment, skewness, kurtosis, moment-generating function, mean, variance, mode, survival function, hazard function, and asymptotic behavior, as well as constructing the estimators of the unknown parameter by employing the expectation-maximization (EM) algorithm, and simulated annealing. Additionally, the performance of the proposed estimators was compared with bias, mean squared errors (MSE), and simulated variances, and given an illustrative example of the proposed distribution to the survival data set in order to show that the proposed distribution is appropriate for the right-skewed data. This will be extremely advantageous in survival analysis.<\/jats:p>","DOI":"10.3390\/sym15071380","type":"journal-article","created":{"date-parts":[[2023,7,10]],"date-time":"2023-07-10T00:45:37Z","timestamp":1688949937000},"page":"1380","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Properties and Maximum Likelihood Estimation of the Novel Mixture of Fr\u00e9chet Distribution"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6082-4779","authenticated-orcid":false,"given":"Wikanda","family":"Phaphan","sequence":"first","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"},{"name":"Research Group in Statistical Learning and Inference, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7280-3035","authenticated-orcid":false,"given":"Ibrahim","family":"Abdullahi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Yobe State University, Damaturu 500501, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-5447-3607","authenticated-orcid":false,"given":"Wirawan","family":"Puttamat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Chaiyaphum Rajabhat University, Chaiyaphum 36000, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.2307\/2529885","article-title":"The Use of Mixture Models for the Analysis of Survival Data with Long-Term Survivors","volume":"38","author":"Farewell","year":"1982","journal-title":"Biometrics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1002\/sim.3543","article-title":"A finite mixture survival model to characterize risk groups of neuroblastoma","volume":"28","author":"Hunsberger","year":"2009","journal-title":"Stat Med."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Joudaki, H., Hashemi, R., and Khazaei, S. (2022). Survival analysis using Dirichlet process mixture model with three-parameter Burr XII distribution as kernel. Commun. Stat. Simul. Comput., 1\u201319.","DOI":"10.1080\/03610918.2022.2076868"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Niyomdecha, A., and Srisuradetchai, P. (2023). Complementary Gamma Zero-Truncated Poisson Distribution and Its Application. Mathematics, 11.","DOI":"10.3390\/math11112584"},{"key":"ref_5","first-page":"731","article-title":"Some Properties of the New Mixture of Nakagami Distribution","volume":"20","author":"Abdullahi","year":"2022","journal-title":"Thail. Stat."},{"key":"ref_6","first-page":"191","article-title":"A New Mixture Pareto Distribution and Its Application","volume":"13","author":"Nanuwong","year":"2015","journal-title":"Thail. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1134\/S1995080220060037","article-title":"Discrete Generalized Odd Lindley\u2013Weibull Distribution with Applications","volume":"41","author":"Aryuyuen","year":"2020","journal-title":"Lobachevskii J. Math."},{"key":"ref_8","first-page":"1","article-title":"A four-parameter negative binomial-Lindley regression model to analyze factors influencing the number of cancer deaths using Bayesian inference","volume":"2023","author":"Tonggumnead","year":"2023","journal-title":"Commun. Math. Biol. Neurosci."},{"key":"ref_9","first-page":"93","article-title":"Sur la loi de probabilit\u00e9 de l\u2019\u00e9cart maximum","volume":"6","year":"1927","journal-title":"Ann. Soc. Polon. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1017\/S0305004100015681","article-title":"Limiting forms of the frequency distribution of the largest or smallest member of a sample","volume":"24","author":"Fisher","year":"1928","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gumbel, E.J. (1958). Statistics of Extremes, Columbia University Press.","DOI":"10.7312\/gumb92958"},{"key":"ref_12","first-page":"58","article-title":"Comparison of estimation methods for Fr\u00e9chet distribution with known shape","volume":"1","author":"Abbas","year":"2012","journal-title":"Casp. J. Appl. Sci. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"246","DOI":"10.14419\/ijbas.v4i3.4644","article-title":"Bayes approach to study shape parameter of Fr\u00e9chet distribution","volume":"4","author":"Nasir","year":"2015","journal-title":"Int. J. Basic. Appl. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"62","DOI":"10.9734\/BJMCS\/2016\/29231","article-title":"QE-Bayesian and E-Bayesian estimation of the Fr\u00e9chet model","volume":"19","author":"Reyad","year":"2016","journal-title":"BJMCS"},{"key":"ref_15","first-page":"185","article-title":"On five-parameter Lomax distribution: Properties and applications","volume":"1","author":"Mead","year":"2016","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kotz, S., and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications, Imperial College Press.","DOI":"10.1142\/p191"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.jhydrol.2008.02.011","article-title":"On the tails of extreme event distributions in hydrology","volume":"355","author":"Adlouni","year":"2008","journal-title":"J. Hydrol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"100771","DOI":"10.1016\/j.ejrh.2020.100771","article-title":"Probability distributions of daily rainfall extremes in Lazio and Sicily, Italy, and design rainfall inferences","volume":"23","author":"Moccia","year":"2021","journal-title":"J. Hydrol. Reg. Stud."},{"key":"ref_19","first-page":"549","article-title":"The Fr\u00e9chet distribution: Estimation and application\u2014An overview","volume":"23","author":"Ramos","year":"2020","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1080\/02664763.2016.1142945","article-title":"The Weibull Fr\u00e9chet distribution and its applications","volume":"43","author":"Afify","year":"2016","journal-title":"J. Appl. Stat."},{"key":"ref_21","first-page":"189","article-title":"The length-biased weighted Fr\u00e9chet distribution: Properties and estimation","volume":"3","author":"Hesham","year":"2017","journal-title":"Int. J. Appl. Math. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Acute Leukemia Group B, Freireich, E.J., Gehan, E.A., Frei, E., Schroeder, L.R., Wolman, I.J., Anbari, R., Burgert, E.O., Mills, S.D., and Pinkel, D. (1963). The Effect of 6-Mercaptopurine on the Duration of Steroid-Induced Remissions in Acute Leukemia: A Model for Evaluation of Other Potential Useful Therapy. Blood, 21, 699\u2013716.","DOI":"10.1182\/blood.V21.6.699.699"},{"key":"ref_23","first-page":"93","article-title":"On Interval Estimation of the Geometric Parameter in a Zero\u2013inflated Geometric Distribution","volume":"21","author":"Srisuradetchai","year":"2023","journal-title":"Thail. Stat."},{"key":"ref_24","first-page":"357","article-title":"On Interval Estimation of the Poisson Parameter in a Zero-inflated Poisson Distribution","volume":"20","author":"Srisuradetchai","year":"2022","journal-title":"Thail. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lee, E.T., and Wang, J.W. (2003). Statistical Methods for Survival Data Analysis, Wiley. 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