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To facilitate robust inference, we employ three distinct estimation methodologies, including maximum likelihood, maximum product of spacings, and Bayesian approaches. Within the Bayesian framework, estimators of  are derived under both the squared error loss function and the LEL function, assuming independent gamma priors for the unknown parameters. Furthermore, approximate confidence intervals for the index  are constructed using conventional frequentist methods and are compared with the Bayesian HPD intervals. To comprehensively evaluate the efficacy of the proposed estimation procedures, a comprehensive Monte Carlo simulation is carried out. The accuracy of the point estimators is measured by their mean squared errors and biases, whereas the reliability of the interval estimates is judged by their average widths along with coverage probabilities. The practical applicability of the proposed index and estimation methodologies is further illustrated through a real data example concerning the failure times of transmission components in Caterpillar\u00a0tractors.<\/jats:p>","DOI":"10.1002\/qre.70107","type":"journal-article","created":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T13:28:06Z","timestamp":1761658086000},"page":"793-814","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Inference of Cpyk$\\mathcal {C}_{pyk}$ for Progressively Censored Log\u2013Log Distributed Samples"],"prefix":"10.1002","volume":"42","author":[{"given":"Tanmay","family":"Kayal","sequence":"first","affiliation":[{"name":"School of Mathematical and Statistical Sciences Indian Institute of Technology Mandi Mandi India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8389-5257","authenticated-orcid":false,"given":"Amulya Kumar","family":"Mahto","sequence":"additional","affiliation":[{"name":"Mehta Family School of Data Science and Artificial Intelligence Indian Institute of Technology Guwahati Guwahati India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sanku","family":"Dey","sequence":"additional","affiliation":[{"name":"Department of Statistics St. Anthony's College Shillong India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aman","family":"Gupta","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences Indian Institute of Technology Mandi Mandi India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,10,28]]},"reference":[{"key":"e_1_2_10_2_1","volume-title":"Juran's Quality Control Handbook","author":"Juran J. 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