The MLE of the uniform distribution with right-censored data

We carry out parametric inferences to a breast cancer data set which is right censored using the uniform distribution U(a, b). Under right censoring, it is rare that one can find the explicit solution to the maximum likelihood estimator (MLE) under the parametric set-up, except for the exponential distribution Exp(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Exp(\theta )$$\end{document}. We show that the MLE of a has a closed form solution, whereas the MLE of b has a closed form solution in some sense. We further propose a diagnostic plotting method and test for U(a, b). The asymptotic properties of the MLE are also investigated. It turns out that this breast cancer data set fits both U(a, b) and Exp(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Exp(\theta )$$\end{document}. Moreover, U(a, b) leads to more useful and reasonable inferences than those using the product-limit estimator or using the MLE of Exp(θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Exp(\theta )$$\end{document}.