This study develops and evaluates a suite of Bayesian shrinkage estimators for the shape parameter ($\alpha$) of the two-parameter Power distribution under a complete sampling scheme. The primary objective is to identify a robust estimation strategy that improves upon classical and standard Bayesian methods, particularly in scenarios with limited data. The methodological framework incorporates three distinct prior distributions (non-informative Uniform and Jeffreys; informative Exponential), five symmetric and asymmetric loss functions (Squared Error Loss Function (SELF), Weighted Squared Error Loss Function (WSELF), Kullback-Leibler Loss Function (KLF), Modified Quadratic Squared Error Loss Function (M/Q SELF), and Precautionary Loss Function (PLF)), and four shrinkage strategies (one with a constant factor and three modified Thompson-type adaptive estimators). Performance is rigorously assessed via a comprehensive Monte Carlo simulation study, comparing the proposed estimators against standard Maximum Likelihood and Bayes estimators in terms of absolute bias and mean squared error (MSE). The results consistently demonstrate the superiority of Bayesian methods over the Maximum Likelihood Estimator (MLE), especially for small sample sizes. Crucially, a modified shrinkage estimator based on the work of Mehta \& Srinivasan (1971) exhibits the lowest bias and MSE across nearly all considered scenarios, establishing its exceptional robustness and efficiency. The principal implication of this study is that the proposed adaptive shrinkage estimator offers a significant improvement in estimation accuracy for the Power distribution's shape parameter, presenting a valuable tool for practitioners in reliability engineering and lifetime data analysis.

This paper addresses the estimation of the stress-strength reliability parameter, defined as R = P(X < Y ), within a model where the stress variable X follows a Geometric distribution and the strength variable Y follows an Exponential distribution. The analysis is conducted for two prevalent data scenarios in reliability studies: complete and right-censored sampling. We derive and compare the performance of several estimation methodologies. The classical Maximum Likelihood Estimator (MLE) serves as a baseline. Its performance is contrasted with modern robust techniques, including three distinct shrinkage estimators based on a constant weight factor, a modified Thompson-type factor, and the Mehta and Srinivasan formulation. A comprehensive Monte Carlo simulation study is designed to evaluate these estimators under various conditions of sample size, true reliability, and censoring proportions. The performance is assessed based on Bias and Mean Squared Error (MSE). The results consistently demonstrate that shrinkage estimators offer substantial improvements in MSE over the traditional MLE, particularly in small to moderate sample sizes and under heavy censoring, highlighting the practical benefits of regularization in this mixed discrete-continuous reliability context.

Entropy measures are crucial in measuring uncertainty and unpredictability in complex systems and have found applications in various fields, including reliability theory, economic statistics, cybersecurity, pattern recognition, and artificial intelligence. This paper explores the estimation of Shannon and Past entropy of the Geometric distribution with Maximum Penalised Likelihood Estimation (MPLE), a method that aims to reduce overfitting and increase robustness in small or sparse samples. In contrast to traditional Maximum Likelihood Estimation (MLE), MPLE employs penalty functions to balance the accuracy and complexity of the model. There are two penalty structures considered: a Quadratic (ridge-type) penalty and a Lasso-type penalty. The results of the proposed estimators are compared under a complete sampling scheme, a right-censored sampling scheme, and a Type I censored sampling scheme, using extensive Monte Carlo simulations of 10,000 iterations with different sample sizes (n = 25, 50, 75) and parameter values (0.15, 0.5, 0.9). The criteria applied in evaluation are bias, mean square error (MSE) and relative efficiency (RE). The findings repeatedly indicate that the Quadratic Penalty 1 (QP1) estimator is the best with the lowest bias, least MSE and maximum RE in all cases. It is essential to note that QP1 can sustain a RE of above 80% with moderate sample sizes (n = 50) and mid-range parameter values. The results are helpful in entropy applications, especially in the design of cryptographic systems, the analysis of random number generators, and reliability analysis. The suggested MPLE framework enhances the accuracy of estimates and is robust, thereby contributing to improved predictive performance and the development of safe systems across a wide range of disciplines.

The classical Pareto distribution is a foundational model for the heavy-tailed phenomena, particularly in reliability analysis, where the Total Time on Test (TTT) transform serves as a key measure. However, conventional estimation techniques for the TTT transform, such as those based on Maximum Likelihood, are sample-sensitive and biased, especially in small samples, which are common challenges with heavy-tailed data. To address these limitations, the present study proposes a robust estimation framework based on Quasi-Bayesian methodology. The estimators are derived under both Gamma and Jeffrey’s priors, and with symmetric Squared Error and asymmetric loss functions to accommodate diverse inferential objectives. To further enhance estimator stability and mitigate overfitting, the shrinkagebased technique and penalization via L1 (Lasso-type) and L2 (Quadratic) are introduced. The performance of these novel estimators is systematically evaluated against classical methods and Thompson-type shrinkage estimators via a comprehensive Monte Carlo simulation study. The results demonstrate the superior finite-sample properties, in terms of bias and Mean Squared Error(MSE), of the proposed regularized and shrinkage estimators, particularly in small-sample settings. An application to a real-world insurance claims dataset illustrates the practical implications and utility of our proposed methodologies for risk assessment.