In Problem 14 we collected a balanced RSS of 96 subjects from the NHANES III population using a set size of k = 6 and arm circumference as the auxiliary variable to perform the judgment rankings with the goal of making inferences about the BMI for the population. Since this auxiliary variable does not provide perfect rankings for the BMI values, there will be ranking errors within the sets. Let

for i = 1, ... , 6 and j = 1, ... , 6. Thus, p_ij , i ≠ j, is the probability that the ranking process in a given set incorrectly assigns the jth judgment rank to the BMI value that is actually the ith ordered BMI value in the set and p_ii , i = 1, ... , 6, is the probability that the ranking process correctly identifies the ith ordered BMI value in a set of size 6. Use the known BMI values for all 576 subjects involved in the ranking process leading to this RSS of size 96 to obtain sample estimates of the 36 probabilities p_ij , i = 1, ... , 6 and j = 1, ... , 6. Discuss the effectiveness of using arm circumference as the auxiliary ranking variable for BMI.