Fisher Information and Tukey’s Linear Sensitivity Measure Based on Ordered Ranked Set Samples

Abstract(1995) derived the Fisher information and discussed the maximum likelihood estimation (MLE) of the parameters of a location-scale family $$ F\left( {\tfrac{{x - \mu }} {\sigma }} \right) $$ based on the ranked set sample (RSS). She found that a RSS provided more information about both μ and σ than a simple random sample (SRS) of the same size. We also focus here on the location-scale family. We use the idea of order statistics from independent and nonidentical random variables (INID) to propose an ordered ranked set sample (ORSS) and develop the Fisher information and the maximum likelihood estimation based on such an ORSS. We use logistic, normal, and one-parameter exponential distributions as examples and conclude that in all these three cases, the ORSS does not provide as much Fisher information as the RSS, and consequently the MLEs based on the ORSS (MLE-ORSS) are not as efficient as the MLEs based on the RSS (MLE-RSS). In addition to the MLEs, we are also interested in best linear unbiased estimators (BLUE). For this purpose, we apply another measure of information, viz., Tukey’s linear sensitivity. Tukey (1965) proposed linear sensitivity to measure information contained in an ordered sample. We use logistic, normal, one- and two-parameter exponential, two-parameter uniform, and right triangular distributions as examples and show that in all these cases except the one-parameter the RSS, and consequently the BLUEs based on the ORSS (BLUE-ORSS) are more efficient than the BLUEs based on the RSS (BLUE-RSS). In the case of one-parameter exponential, the ORSS has only slightly less information than the RSS with the relative efficiency being very close to 1.