Free Trial

Safari Books Online is a digital library providing on-demand subscription access to thousands of learning resources.

Share this Page URL
Help

4.6.2 Alternative Method for the Median > 4.6.2 Alternative Method for the Medi... - Pg. 130

130 Introduction to Robust Estimation and Hypothesis Testing 0.7 0.6 0.5 Rel. freq. Rel. freq. 0.4 0.3 0.2 0.1 0.0 * 3.0 * 3.5 4.0 Median 4.5 5.0 * 0.6 * 0.8 * 0.4 * 0.2 * 4.8 5.0 0.0 4.0 4.2 * 4.4 4.6 Median Figure 4.5: When tied values can occur, the sample median might not be asymptotically normal. The left panel shows the sampling distribution of the median with n = 20. The right panel is the sampling distribution with n = 100. 4.6.2 Alternative Method for the Median When the goal is to compute a confidence interval for the population median, the following method can be used even when there are tied values. Suppose W is a binomial random variable with probability of success p = .5 and n trials. For any integer k between 0 and [n/2], let k = P(k W n - k), the probability that the number of successes, W , is between k and n - k, inclusive. Then a distribution-free k confidence interval for the median is (X (k) , X (n-k+1) ). That is, the probability coverage is exactly k under random sampling (e.g., Hettmansperger & McKean, 1998; also see Yohai & Zamar, 2004). This is just a special case of the first method described in the previous section. Because the binomial distribution is discrete, it is not possible, in general, to choose k so that the probability coverage is exactly equal to 1 - . For example, if n = 10, 0.891 and 0.978 confidence intervals can be computed, but not a 0.95 confidence interval as is often desired. However, linear interpolation can be used along the lines suggested by Hettmansperger and Sheather (1986) so that the probability coverage is approximately 1 - . First determine k such that k+1 < 1 - < k . Next, compute I = www.elsevierdirect.com k - 1 - k - k+1