Statistical Inference for the Inverted Scale Family under General Progressive Type-II Censoring

• Main Authors: Jing Gao, Kehan Bai, Wenhao Gui
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Symmetry
• Subjects: Monte Carlo simulation; inverted scale family; interval estimation; maximum likelihood estimation; Bayesian estimation; approximate maximum likelihood estimator
• Online Access: https://www.mdpi.com/2073-8994/12/5/731

Two estimation problems are studied based on the general progressively censored samples, and the distributions from the inverted scale family (ISF) are considered as prospective life distributions. One is the exact interval estimation for the unknown parameter <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula>, which is achieved by constructing the pivotal quantity. Through Monte Carlo simulations, the average <inline-formula> <math display="inline"> <semantics> <mrow> <mn>90</mn> <mo>%</mo> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>95</mn> <mo>%</mo> </mrow> </semantics> </math> </inline-formula> confidence intervals are obtained, and the validity of the above interval estimation is illustrated with a numerical example. The other is the estimation of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>R</mi> <mo>=</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Y</mi> <mo><</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> in the case of ISF. The maximum likelihood estimator (MLE) as well as approximate maximum likelihood estimator (AMLE) is obtained, together with the corresponding R-symmetric asymptotic confidence intervals. With Bootstrap methods, we also propose two R-asymmetric confidence intervals, which have a good performance for small samples. Furthermore, assuming the scale parameters follow independent gamma priors, the Bayesian estimator as well as the HPD credible interval of <i>R</i> is thus acquired. Finally, we make an evaluation on the effectiveness of the proposed estimations through Monte Carlo simulations and provide an illustrative example of two real datasets.