Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.

在中小样本量场景下，多余参数（nuisance parameter）往往会干扰感兴趣参数的基于似然推断的质量。本文针对形状-尺度族（shape-scale family）中形状参数的基于似然推断问题，提出一种最大尺度不变变换，以规避多余尺度参数带来的影响。该变换可适用于完全样本或II型截尾样本（type-II censored samples）。针对常用的威布尔分布（Weibull distribution）、伽马分布（Gamma distribution）与广义指数分布（Generalized exponential distribution），对所提估计量开展基于仿真的性能评估，结果显示：在完全样本与II型截尾样本情境下，针对形状参数的各类基于似然推断的性能均得到显著提升。仿真研究表明，针对威布尔分布与伽马分布，经典最大似然估计量（maximum likelihood estimator, MLE）与基于变换的MLE的偏差（bias）之间存在线性关系。借此线性关系，我们可为这两类分布提出形状参数的近似无偏估计量。此外，本文还探讨了尺度参数的联合估计问题。