Type II Half Logistic Family of Distributions with Applications

In this paper, the estimation of the stress–strength reliability is taken into account when the stress and strength variables have unit Gompertz distributions with a similar scale parameter. The consideration of the unit Gompertz distribution in this context is because of its intriguing symmetric and asymmetric properties that can accommodate various histogram proportional-type data shapes. As the main contribution, the reliability estimate is determined via seven frequentist techniques using the ranked set sampling (RSS) and simple random sampling (SRS). The proposed methods are the maximum likelihood, least squares, weighted least squares, maximum product spacing, Cramér–von Mises, Anderson–Darling, and right tail Anderson–Darling methods. We perform a simulation work to evaluate the effectiveness of the recommended RSS-based estimates by using accuracy metrics. We draw the conclusion that the reliability estimates in the maximum product spacing approach have the lowest value compared to other approaches. In addition, we note that the RSS-based estimates are superior to those obtained by a comparable SRS approach. Additional results are obtained using two genuine data sets that reflect the survival periods of head and neck cancer patients.

Unit distributions are typically used in probability theory and statistics to illustrate useful quantities with values between zero and one. In this paper, we investigated an appropriate transformation to propose the unit-exponentiated half-logistic distribution (UEHLD), which is also beneficial for modelling data on the unit interval. This distribution’s mathematical features are supplied, including moments, probability-weighted moments, incomplete moments, various entropy measures, and stress–strength reliability. Using well-known estimation techniques such as the maximum likelihood, maximum product of spacing, and Bayesian inference, the estimators of the parameters relevant to the proposed distribution were determined. A comprehensive simulation analysis is provided to examine the performance of parameter estimation approaches on finite samples. The proposed distribution was realistically applied to data on economic growth and data on the tensile strength of polyester fibers to provide an explanation. Furthermore, the analysis of COVID-19 data from Britain as a medical statistical dataset is provided. The experimental results demonstrate that the suggested UEHLD yields a better comparison with some new unit distributions, as well as other unbounded distributions.

The unit exponentiated half logistic power series (UEHLPS), a family of compound distributions with bounded support, is introduced in this study. This family is produced by compounding the unit exponentiated half logistic and power series distributions. In the UEHLPS class, some interesting compound distributions can be found. We find formulas for the moments, density and distribution functions, limiting behavior, and other UEHLPS properties. Five well-known estimating approaches are used to estimate the parameters of one sub-model, and a simulation study is created. The simulated results show that the maximum product of spacing estimates had lower accuracy measure values than the other estimates. Ultimately, three real data sets from various scientific areas are used to analyze the performance of the new class.

The unit–power Burr X distribution (UPBXD), a bounded version of the power Burr X distribution, is presented. The UPBXD is produced through the inverse exponential transformation of the power Burr X distribution, which is also beneficial for modelling data on the unit interval. Comprehensive analysis of its key characteristics is performed, including shape analysis of the primary functions, analytical expression for moments, quantile function, incomplete moments, stochastic ordering, and stress–strength reliability. Rényi, Havrda and Charvat, and d-generalized entropies, which are measures of uncertainty, are also obtained. The model’s parameters are estimated using a Bayesian estimation approach via symmetric and asymmetric loss functions. The Bayesian credible intervals are constructed based on the marginal posterior distribution. Monte Carlo simulation research is intended to test the accuracy of various estimators based on certain measures, in accordance with the complex forms of Bayesian estimators. Finally, we show that the new distribution is more appropriate than certain other competing models, according to their application for COVID-19 in Saudi Arabia and the United Kingdom.