Statistical inference for the bathtub-shaped distribution using balanced and unbalanced sampling techniques

In order to reduce errors and enhance precision while estimating the unknown parameters of the distributions, it is crucial to choose a representative sample. The common estimation methods that estimate the parameters associated with the bathtub-shaped distribution include maximum likelihood (ML), maximum product of spacings estimation (MPSE), and Cramér-von Mises estimation (CME) methods. However, four modifications are used with the sample selection technique. They are simple random sampling (SRS), ranked set sampling (RSS), maximum ranked set sampling (MaxRSS), and double ranked set sampling (DBRSS), which is due to small sample sizes. Based on the estimation methods such as ML, MPSE, and CME, the ranked set sampling techniques do not have simple functions to manage them. The MaxRSS matrix has variable dimensions but requires fewer observations than RSS. DBRSS requires a greater number of observations than MaxRSS and RSS. According to simulation studies, the RSS, MaxRSS, and DBRSS estimators were more effective than the SRS estimator for different sample sizes. Additionally, MaxRSS was discovered to be the most efficient RSS-based technique. Other techniques, however, proved more effective than RSS for high mean squared errors. The CM method estimated the true values of the parameters more accurately and with smaller biases than ML and MPSE. The MPSE method was also found to have significant biases and to be less accurate in estimating the values of the parameters when compared to the other estimate methods. Finally, two datasets demonstrated how the bathtub-shaped distribution could be feasible based on different sampling techniques.