A new five‐parameter transmuted generalization of the Lomax distribution (TGL) is introduced in this study which is more flexible than current distributions and has become the latest distribution theory trend. Transmuted generalization of Lomax distribution is the name given to the new model. This model includes some previously unknown distributions. The proposed distribution′s structural features, closed forms for an rth moment and incomplete moments, quantile, and Rényi entropy, among other things, are deduced. Maximum likelihood estimate based on complete and Type‐II censored data is used to derive the new distribution′s parameter estimators. The percentile bootstrap and bootstrap‐t confidence intervals for unknown parameters are introduced. Monte Carlo simulation research is discussed in order to estimate the characteristics of the proposed distribution using point and interval estimation. Other competitive models are compared to a novel TGL. The utility of the new model is demonstrated using two COVID‐19 real‐world data sets from France and the United Kingdom.

In real-life experiments, collecting complete data is time-, finance-, and resources-consuming as stated by statisticians and analysts. Their goal was to compromise between the total time of testing, the number of units under scrutiny, and the expenditures paid through a censoring scheme. Comparing failure-censored schemes (Type-Ⅱ and Progressive Type-Ⅱ) to Time-censored schemes (Type-Ⅰ), it's worth noting that the former is time-consuming and is no more suitable to be applied in real-life situations. This is the reason why the Type-Ⅰ adaptive progressive hybrid censoring scheme has exceeded other failure-censored types; Time-censored types enable analysts to accomplish their trials and experiments in a shorter time and with higher efficiency. In this paper, the parameters of the inverse Weibull distribution are estimated under the Type-Ⅰ adaptive progressive hybrid censoring scheme (Type-Ⅰ APHCS) based on competing risks data. The model parameters are estimated using maximum likelihood estimation and Bayesian estimation methods. Further, we examine the asymptotic confidence intervals and bootstrap confidence intervals for the unknown model parameters. Monte Carlo simulations are carried out to compare the performance of the suggested estimation methods under Type-Ⅰ APHCS. Moreover, Markov Chain Monte Carlo by applying Metropolis-Hasting algorithm under the square error of loss function is used to compute Bayes estimates and related to the highest posterior density. Finally, two data sets are studied to illustrate the introduced methods of inference. Based on our results, we can conclude that the Bayesian estimation outperforms the maximum likelihood estimation for estimating the inverse Weibull parameters under Type-Ⅰ APHCS.

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters θ₁, θ₂, and β and the system of stress‐strength reliability R = P(Y < X) for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute‐log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér‐von Mises estimation (CME), and Anderson‐Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias‐corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.

This study discusses the Gompertz distribution’s statistical inference using unified hybrid censored data. Under various loss functions, the maximum likelihood and Bayesian approaches are explored for estimating the parameters for the Gompertz distribution. To compare the suggested methodologies, the Monte Carlo simulation is used, which also introduces the simulation research. By examining a genuine dataset, the applicability of the presented inference in practice is finally demonstrated.

Estimating entropy measures under progressive censoring poses a significant challenge in reliability and lifetime analysis. Despite the widespread use of the inverse Gaussian distribution in modeling skewed lifetime data, a comprehensive inferential framework for its Shannon and Rényi entropies under progressive Type-II censoring remains absent. This paper develops a unified Bayesian framework integrating maximum likelihood and Bayesian inference under squared error, general entropy, and LINEX loss functions, employing Lindley’s approximation, importance sampling, and Markov chain Monte Carlo methods. Two parameter configurations were examined to assess robustness under varying likelihood surface complexity, along with prior sensitivity analysis. Results demonstrate that Markov chain Monte Carlo and importance sampling are the only consistently reliable methods across all scenarios, whereas maximum likelihood suffered severe bias and collapse of Wald interval coverage under challenging settings, and Lindley’s approximation exhibited numerical instability at small samples. Shannon entropy proved substantially more sensitive to parameter variation than Rényi entropy, with the LINEX and general entropy loss functions showing superior performance. The study offers clear practical guidance, validated on real lifetime data.