Inference on overlapping coefficient in two exponential populations based on adaptive type-II progressive hybrid censoring

This article considers a life test scheme called the adaptive type-II progressive hybrid censoring scheme introduced by Ng et al. (Naval Res Logist 5(8):687-698, 2009). Based on this type of censoring, we draw inferences about the three well-known measures of overlap, namely Matusita's measure (rho\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rho $$\end{document}), Morisita's measure (lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}), and Weitzman's (Delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document}) for two exponential populations with different means. The asymptotic bias and variance of the overlap measure estimators are derived. Monte Carlo evaluations are employed in cases with small sample sizes, where computing the precision or bias of these estimators becomes challenging due to the lack of closed-form expressions for their variances and exact sampling distributions. Confidence intervals for those measures are also constructed via the bootstrap method and Taylor expansion approximation. To emphasize the practical relevance of our proposed estimators, we illustrate their application using a real data set from head and neck cancer research.