Bayesian and non-Bayesian estimation of the bivariate inverse Weibull distribution parameters using ranked set sampling with concomitant variable

Estimating the bivariate distribution parameter is crucial for modeling paired variable dependencies, but highly variable or resource-intensive data may not respond well to traditional simple random sampling (SRS). In order to maximize efficiency, Ranked Set Sampling (RSS) ranks a subset of observations based on a concurrent variable, hence selecting just a subset for measurement. This study use both Bayesian and non-Bayesian estimation techniques to estimate the parameters of the Bivariate Inverse Weibull (BIW) distribution under RSS and SRS. According to the Marshall-Olkin approach, dependencies are captured by the BIW model using the parameters. We compute the probability functions for RSS and SRS because the ranking technique and dependence structure are intricate. Based on SRS and RSS, Bayesian estimators are explicitly derived by applying conjugate gamma priors for model parameters under squared error loss, whereas Maximum Likelihood Estimation (MLE) solutions are derived numerically via the Newton-Raphson technique because of the likelihood equations' nonlinearity. Mean Squared Error (MSE), Bias, and Efficiency (EFF), simulations conducted with four different parameter settings that showed that RSS routinely performs better than SRS. In particular, under RSS, Bayesian estimation frequently produces lower MSE and bias than MLE. Nevertheless, prior decisions have an impact on Bayesian performance, particularly when the parameters are tiny, Simulations with 10,000 Monte Carlo replications across four parameter sets show that RSS consistently outperforms SRS, with MSE reduced by up to 50% and EFF exceeding 10 for large samples. Bayesian estimation with conjugate gamma priors yields lower MSE than MLE, particularly under RSS, though prior selection is critical for small parameters. We recommend RSS with Bayesian methods for applications in reliability and lifespan analysis, as demonstrated on a real dataset of 243 men's body fat and chest circumference.