Questions and Answers of Nonparametric Statistical Inference

6.10 (a) Show that the distribution of the precedence statistic P(i) under the null hypothesis FX ¼ FY, given in Problem 2.28(c), can be expressed as P(P(i) ¼ jjH0) ¼n m þ n mj n 1 i 1 m
6.11 For the control median test statistic V, use Problem 2.28, or otherwise, to show that when FX ¼ FY, E(V) ¼m 2and var(V) ¼2r þ m þ 2 4m(2r þ 3)[Hint: Use the fact that E(X) ¼ EYE(XjY) and
6.12 Show that when m, n!1such that m=(m þ n) ! l, 0 < l < 1, the null distribution of the precedence statistic P(i) given in Problem 6.10 tends to the negative binomial distribution with parameters
6.13 In some applications the quantity jp ¼ FX(kp), where kp is the pth quantile of FY, is of interest. Let limn!1 (m=n) ¼ l, where l is a fixed quantity, and let {rn} be a sequence of positive
6.14 A sample of three girls and five boys are given instructions on how to complete a certain task. Then they are asked to perform the task over and over until they complete it correctly. The
6.15 A researcher is interested in learning if a new drug is better than a placebo in treating a certain disease. Because of the nature of the disease, only a limited number of patients can be found.
7.1 One of the simplest linear rank statistics is defined as WN ¼XN i¼1 iZi This is the Wilcoxon rank-sum statistic to be discussed in the next chapter. Use Theorem 7.3.2 to evaluate the mean and
7.2 Express the two-sample median-test statistic U defined in Section 6.4 in the form of a linear rank statistic and use Theorem 7.3.2 to find its mean and variance. Hint: For the appropriate
7.3 Prove the three properties stated in Theorem 7.3.7.
7.4 For m¼n¼2, derive the probability mass function of TN, the sum of the X ranks, under H0. Determine whether this distribution is symmetric and if so, identify the point of symmetry. Calculate

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