Consider the median test statistic described in Example 11.14, which is a linear rank statistic with \(a(i)=\delta\left\{i ;\left\{\frac{1}{2}(m+n+1), \ldots, m+night\}ight\}\) and \(c(i)=\delta\{i ;\{m+1, \ldots, n+m\}\}\) for all \(i=1, \ldots, n+m\).

a. Under the null hypothesis that the shift parameter \(\theta\) is zero, find the mean and variance of the median test statistic.

b. Determine if there are conditions under which the distribution of the median test statistic under the null hypothesis is symmetric.

c. Prove that the regression constants satisfy Noether's condition.

d. Define \(\alpha(t)\) such that \(a(i)=\alpha\left[(m+n+1)^{-1} iight]\) for all \(i=1, \ldots, m+n\) and show that \(\alpha\) is a square integrable function.

e. Prove that the linear rank statistic

\[D=\sum_{i=1}^{n} a(i, n) c(i, n)\]

converges weakly to a \(\mathrm{N}(0,1)\) distribution when it has been properly standardized.

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Example 11.14. Once again consider the two-sample setup studied in Ex- ample 11.2, except in this case we will use the median test statistic proposed by Mood (1950) and Westenberg (1948). For this test we compute the me- dian of the combined sample X,..., Xm. Y..... Yn and then compute the number of values in the sample Y,..., Yn that exceed the median. Note that under the null hypothesis that = 0, where the combined sample all comes from the sample distribution, we would expect that half of these would be above the median. If 0 0 then we would expect either greater than, or fewer than, of these values to exceed the median. Therefore, counting the number of values that exceed the combined median provides a reasonable test statis- tic for the null hypothesis that = 0. This test statistic can be written in the form of a linear rank statistic with c(i) = 6{i; {m +1,...,n+m}} and a(i) = {i: {(n+m+1),...,n+m}}. The form of the score function is derived from the fact that if the rank of a value from the combined sample exceeds (n+m+1) then the corresponding value exceeds the median. This score function is called the median score function.