Estimating the Reliability of A Lomax Distribution Based on Double Ranked Set Sampling (DRSS) (With Application)

Abstract

Sometimes it is difficult to obtain the data and information required for research for reasons that may be related to time and effort. In addition, in many applied issues, obtaining real measurements of the variable of interest may be very expensive. To solve these problems, the sampling method is used, which ensures that the researcher achieves research objectives with the least time, effort, and cost. One of the important sampling methods in this aspect is the Ranked Set Sampling (RSS) method. From this standpoint, the main goal of the research was to find the probability function of the Lomax distribution by estimating the scale and shape parameters, in addition to estimating the reliability function of the distribution. All of this will be done based on the Double Ranked Set Sampling (DRSS), using both estimation methods, which are the Maximum likelihood Estimation (MLE) and the Maximum Product Spacing  (MPS). For the purpose of comparing the two estimation methods of the reliability function, the Monte-Carlo simulation method was used to generate data for the random variable that follows the Lomax distribution with the use of the statistical criterion represented by the mean square error (MSE). The research found that the (MLE) gives the best estimate of the reliability function of the Lomax distribution when using the (DRSS) method with different sample sizes and number of cycles. This has been proven through experimental results, as the larger the sample size, the better the estimation results, as the (MLE) method achieved a percentage of preference (72.22%). To demonstrate the importance of the topic from an applied and realistic perspective, real data was used on bladder cancer patients, in which the random variable represented the disease remission times in months (beginning of complete recovery), recorded from the time the patients entered the hospital until they left it. It was tested that the data follows a Lomax distribution using goodness of fit tests (Anderson-Darling, Kolmogorov-Smirnov and Chi-square). Using the (DRSS)  method, a sample size of 96 was drawn from the total sample through four groups, and the process was repeated 24 times. Using the (MLE) method as the best way to estimate the reliability function, it was observed that the reliability of recovery from bladder cancer decreases with increasing patient stay times in the hospital when using Treatment and recovery protocol.