{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T06:47:24Z","timestamp":1774594044237,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T00:00:00Z","timestamp":1663632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Mongkut\u2019s University of Technology North Bangkok, Thailand","award":["KMUTNB-66-BASIC-04"],"award-info":[{"award-number":["KMUTNB-66-BASIC-04"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The notion of length-biased distribution can be used to develop adequate models. Length-biased distribution was known as a special case of weighted distribution. In this work, a new class of length-biased distribution, namely the two-sided length-biased inverse Gaussian distribution (TS-LBIG), was introduced. The physical phenomenon of this scenario was described in a case of cracks developing from two sides. Since the probability density function of the original TS-LBIG distribution cannot be written in a closed-form expression, its generalization form was further introduced. Important properties such as the moment-generating function and survival function cannot be provided. We offered a different approach to solving this problem. Some distributional properties were investigated. The parameters were estimated by the method of the moment. Monte Carlo simulation studies were carried out to appraise the performance of the suggested estimators using bias, variance, and mean square error. An application of a real dataset was presented for illustration. The results showed that the suggested estimators performed better than the original study. The proposed distribution provided a more appropriate model than other candidate distributions for fitting based on Akaike information criterion.<\/jats:p>","DOI":"10.3390\/sym14101965","type":"journal-article","created":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:08:09Z","timestamp":1663718889000},"page":"1965","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Generalization of Two-Sided Length Biased Inverse Gaussian Distributions and Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0210-3623","authenticated-orcid":false,"given":"Teerawat","family":"Simmachan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 10120, Thailand"},{"name":"Thammasat University Research Unit in Data Learning, Thammasat University, Pathum Thani 12120, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6082-4779","authenticated-orcid":false,"given":"Wikanda","family":"Phaphan","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"465","DOI":"10.18187\/pjsor.v11i4.1008","article-title":"Length-biased Weighted Maxwell distribution","volume":"11","author":"Modi","year":"2015","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/s00477-008-0215-9","article-title":"A length-biased version of the Birnbaum-Saunders distribution with application in water quality","volume":"23","author":"Leiva","year":"2009","journal-title":"Stoch. Env. Res. Risk Assess."},{"key":"ref_3","first-page":"465","article-title":"On Some Length-Biased Weighted Weibull Distribution","volume":"2","author":"Das","year":"2011","journal-title":"Adv. Appl. Sci. Res."},{"key":"ref_4","first-page":"21","article-title":"A Generalization of Length-biased Nakagami Distribution","volume":"17","author":"Abdullahi","year":"2022","journal-title":"Int. J. Math. Comput. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"462","DOI":"10.1198\/004017008000000442","article-title":"Length bias in the measurements of carbon nanotubes","volume":"50","author":"Kvam","year":"2008","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"701","DOI":"10.1088\/0253-6102\/57\/4\/26","article-title":"Multi-type directed scale-free percolation","volume":"57","author":"Shang","year":"2012","journal-title":"Commun. Theor. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/24.46490","article-title":"Characterization of inverse-Gaussian and gamma distributions through their length-biased distributions","volume":"38","author":"Khattree","year":"1989","journal-title":"IEEE Trans. Reliab."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1080\/00949659208811366","article-title":"A comparison of various estimators of the mean of an inverse Gaussian distribution","volume":"40","author":"Akman","year":"1992","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/0378-3758(94)00148-O","article-title":"On the reliability studies of a weighted inverse Gaussian model","volume":"48","author":"Gupta","year":"1995","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_10","first-page":"1581","article-title":"On the convoluted gamma to length-biased inverse Gaussian distribution and application in financial modeling","volume":"24","author":"Naik","year":"2021","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_11","first-page":"107","article-title":"Parameter estimation for re-parameterized length-biased inverse Gaussian distribution","volume":"17","author":"Budsaba","year":"2022","journal-title":"IJMCS"},{"key":"ref_12","first-page":"77","article-title":"On the mixture of the inverse Gaussian distribution with its complementary reciprocal","volume":"18","author":"Jorgensen","year":"1991","journal-title":"Scand. J. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s11009-008-9112-4","article-title":"Two new mixture models related to the inverse Gaussian distribution","volume":"12","author":"Kotz","year":"2010","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2695","DOI":"10.1080\/02664763.2011.567251","article-title":"Weighted inverse Gaussian\u2014A versatile lifetime model","volume":"38","author":"Gupta","year":"2011","journal-title":"J. Appl. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"319","DOI":"10.2307\/3212003","article-title":"A new family of life distribution","volume":"6","author":"Birnbaum","year":"1969","journal-title":"J. Appl. Probab."},{"key":"ref_16","first-page":"213","article-title":"Parametric Estimation for the Birnbaum-Saunders Lifetime Distribution Based on New Parametrization","volume":"6","author":"Ahmed","year":"2008","journal-title":"Thail. Stat. Thail."},{"key":"ref_17","first-page":"195","article-title":"Some properties of the three-parameter Crack distribution","volume":"9","author":"Bowonrattanaset","year":"2011","journal-title":"Thail. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"758","DOI":"10.3923\/jas.2014.758.766","article-title":"A new two-parameter crack distribution","volume":"14","author":"Saengthong","year":"2014","journal-title":"J. Appl. Sci."},{"key":"ref_19","unstructured":"Lisawadi, S. (2009). Parameter Estimation for the Two-Sided BS and IG Distributions: And Laws of Large Numbers for Arrays under a Condition of Uniform Integrability, VDM Verlag."},{"key":"ref_20","first-page":"2826","article-title":"On two-sided length biased inverse Gaussian distribution","volume":"45","author":"Simmachan","year":"2018","journal-title":"Chiang Mai J. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1080\/00401706.1977.10489586","article-title":"The Inverse Gaussian distribution as a lifetime model","volume":"19","author":"Chhikara","year":"1977","journal-title":"Technometrics"},{"key":"ref_22","unstructured":"Pichetverachai, P. (2015). Bootstrap Confidence Intervals for a Population Mean of Crack Distribution. [Ph.D. Thesis, Thammasat University]."},{"key":"ref_23","first-page":"1455","article-title":"R package for the two-parameters crack distribution","volume":"16","author":"Phaphan","year":"2021","journal-title":"IJMCS"},{"key":"ref_24","unstructured":"(2022, August 01). R Core Team. Available online: http\/\/www.R-project.org."},{"key":"ref_25","unstructured":"(2021, March 01). Backblaze Hard Drive Data and Stats. Available online: https:\/\/www.backblaze.com\/b2\/hard-drive-test-data.html."},{"key":"ref_26","first-page":"460","article-title":"On the new weight parameter of the mixture Pareto distribution and its application to real data","volume":"14","author":"Chananet","year":"2021","journal-title":"Appl. Sci. Eng. Prog."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1965\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:35:53Z","timestamp":1760142953000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/1965"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,20]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14101965"],"URL":"https:\/\/doi.org\/10.3390\/sym14101965","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,20]]}}}