10.2. Sample Median and the Sign Test 58510.2.2.Show that the test given by (10.2.6) has asymptotically levelα;thatis,
show that underH 0 ,
S(θ 0 )−(n/2)
√
n/ 2→DZ,whereZhas aN(0,1) distribution.10.2.3.Letθdenote the median of a random variableX. Consider testingH 0 : θ=0versusH 1 :θ> 0.Suppose we have a sample of sizen= 25.
(a)LetS(0) denote the sign test statistic. Determine the level of the test: reject
H 0 ifS(0)≥16.(b)Determine the power of the test in part (a) ifXhasN(0. 5 ,1) distribution.(c)AssumingXhas finite meanμ=θ, consider the asymptotic test of rejecting
H 0 ifX/(σ/√
n)≥k. Assuming thatσ= 1, determinekso the asymptotic
test has the same level as the test in part (a). Then determine the power of
this test for the situation in part (b).10.2.4.To appreciate the importance of setting the location functional, consider
the length of rivers data set, as taken from Tukey (1977). This data set con-
tains the lengths of 141 American rivers in miles and it can be found in the file
lengthriver.rda.
(a)Suppose the location functional is the median. Obtain the estimate and a
95% confidence interval for it. Use the confidence interval discussed in Section
10.2.3. Interpret it in terms of the data. Use the R functiononesampsgn.R
for computation.(b)Suppose the location functional is the mean. Obtain the estimate and the
95%t-confidence interval for it. Interpret it in terms of the data.(c)Obtain the boxplot of the data and sketch the estimates and confidence inter-
vals on it. Discuss.10.2.5.Recall the definition of a scale functional given in Exercise 10.1.4. Show
that the parameterτSdefined in Theorem 10.2.2 is a scale functional.
10.2.6.Show that the sample mean solves Equation (10.2.32).10.2.7.Derive the approximation (10.2.37).10.2.8.Show that the power function of the sign test is nonincreasing for the
hypotheses
H 0 :θ=θ 0 versusH 1 : θ<θ 0. (10.2.38)