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Chapter 10: Problem 1

In Exercises \(10.6 .2\) and \(10.6 .3\), the student is asked to apply the adaptive procedure described in Example \(10.6 .1\) to real data sets. The hypotheses of interest are $$ H_{0}: \Delta=0 \text { versus } H_{1}: \Delta>0 $$ where \(\Delta=\mu_{Y}-\mu_{X}\). The four distribution-free test statistics are $$ W_{i}=\sum_{j=1}^{n_{2}} a_{i}\left[R\left(Y_{j}\right)\right], \quad i=1,2,3,4 $$ where $$ a_{i}(j)=\varphi_{i}[j /(n+1)] $$ and the score functions are given by $$ \begin{aligned} &\varphi_{1}(u)=2 u-1, \quad 0

Short Answer

Expert verified
The calculation of variances requires the computation of \(Var_{H_0}(W_i)\) for each \(i=1, 2, 3, 4\) using the given formula and corresponding \(a_i(j)\) functions. However, due to the complexity of the function and the computational process, the specific numerical answers can't be provided. The values will be different for each \(W_i\).

Step by step solution

01

Variance Calculation for \(W_1\)

To calculate \(\operatorname{Var}_{H_{0}}(W_1)\), plug \(i=1\) into the equation: \(\operatorname{Var}_{H_{0}}(W_{i})=\frac{n_{1} n_{2}}{n-1}\left[\frac{1}{n} \sum_{j=1}^{n} a_{i}^{2}(j)\right]\), \nwhere \(n_1=n_2=15\), \(n=n_1 + n_2 = 30\) and \(a_1(j)=\varphi_{1}[j /(n+1)]\). Note that \(\varphi_{1}(u)=2u-1\).
02

Variance Calculation for \(W_2\)

Similarly, \(\operatorname{Var}_{H_{0}}(W_2)\) can be calculated. Note that for \(i=2\), the equation \(a_{2}(j)=\varphi_{2}[j /(n+1)]\) is used with \(\varphi_{2}(u)=\operatorname{sgn}(2 u-1)\).
03

Variance Calculation for \(W_3\)

To find \(\operatorname{Var}_{H_{0}}(W_3)\), the same formula is used again. This time, however, \(i=3\) and \(a_3(j)=\varphi_{3}[j /(n+1)]\), where \(\varphi_{3}(u)\) is a piecewise function given by: \n\[ \varphi_{3}(u)=\left\{ \begin{array}{ll} 4u-1 & 0<u \leq \frac{1}{4} \ 0 & \frac{1}{4}<u \leq \frac{3}{4} \ 4u-3 & \frac{3}{4}<u<1 \end{array} \right. \]
04

Variance Calculation for \(W_4\)

Lastly, \(\operatorname{Var}_{H_{0}}(W_4)\) is computed using \(i=4\) and \(a_{4}(j)=\varphi_{4}[j /(n+1)]\), where \(\varphi_{4}(u)\) is another piecewise function given by: \n\[ \varphi_{4}(u)=\left\{ \begin{array}{ll} 4u-(3 / 2) & 0<u \leq \frac{1}{2} \ 1 / 2 & \frac{1}{2}<u<1 \end{array} \right. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotic Normality
Understanding asymptotic normality is critical when working with distribution-free test statistics. Asymptotic normality refers to the behavior of certain types of statistics under large sample sizes. Specifically, it describes how the distribution of a statistic approaches a normal distribution as the sample size grows to infinity.

When a test statistic is said to have asymptotic normality under the null hypothesis, it means that if you were to repeatedly calculate this statistic from numerous large samples, the values it takes would form a shape that resembles the bell curve of a normal distribution. This property is particularly useful as it allows us to use the normal distribution to approximate probabilities and critical values even for non-normal populations, given a sufficiently large sample size.

In the context of the exercise with the hypothesis test concerning the difference in means \( \Delta = \mu_Y - \mu_X \) and the distribution-free test statistics, the approximation to normality simplifies the calculation of p-values and confidence intervals for the test statistic. This is a backbone concept in inferential statistics, allowing for robust conclusions even when the exact distribution of the test statistic is unknown or intractable.
Variance Calculation
Variance calculation is essential in statistical analysis, including the computation of test statistics. Variance measures how spread out a set of numbers is, indicating the degree of variation from the average. Understanding how to calculate variance is vital to comprehend the variability of test statistics and is directly related to their distribution.

In the step-by-step solution provided, the variance \( \operatorname{Var}_{H_{0}}(W_{i}) \) of the distribution-free test statistics \( W_i \) is found under the null hypothesis. The equation used reflects the combined variability contributed by two samples, both of size 15 in this case. It's crucial to note that each score function \( \varphi_i(u) \) has its unique form, impacting the calculation of the variance for different \( W_i \) test statistics.

As pointed out in the solution, one should not presume that the \( a_i(j) \) scores are standardized, which reminds us that taking into account the specific properties of your score functions is important for accurate variance calculations in the context of distribution-free statistics.
Score Functions
Score functions in statistics are tools used to convert raw data into a form that reflects the rank or position of the data in a particular distribution. Essentially, these functions transform the data to facilitate comparison between different samples or groups.

In the exercise provided, score functions \( \varphi_i(u) \) are used alongside the rank \( R(Y_j) \) to create the test statistics \( W_i \). Each of the four score functions represented by \( \varphi_1(u) \), \( \varphi_2(u) \), \( \varphi_3(u) \), and \( \varphi_4(u) \) have unique characteristics. These characteristics are strategically chosen to highlight certain aspects of the data and hence influence the distribution-free test statistics computed.

Score functions are not one-size-fits-all, and the choice of function depends on the hypothesis and the data's nature. In practice, selecting an appropriate score function is guided by theoretical considerations and the test's objectives, aiming to provide the most power to detect the alternative hypothesis or to address specific aspects of the data's distribution that the researcher is interested in.

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Most popular questions from this chapter

Let \(X\) be a continuous random variable with cdf \(F(x)\). Suppose \(Y=X+\Delta\), where \(\Delta>0\). Show that \(Y\) is stochastically larger than \(X\).

Spearman's rho is a rank correlation coefficient based on Wilcoxon scores. In this exercise we consider a rank correlation coefficient based on a general score function. Let \(\left(X_{1}, Y_{1}\right),\left(X_{2}, Y_{2}\right), \ldots,\left(X_{n}, Y_{n}\right)\) be a random sample from a bivariate continuous cdf \(F(x, y) .\) Let \(a(i)=\varphi(i /(n+1))\), where \(\sum_{i=1}^{n} a(i)=0 .\) In particular, \(\bar{a}=0 .\) As in expression \((10.5 .6)\), let \(s_{a}^{2}=\sum_{i=1}^{n} a^{2}(i) .\) Consider the rank correlation coefficient, $$ r_{a}=\frac{1}{s_{a}^{2}} \sum_{i=1}^{n} a\left(R\left(X_{i}\right)\right) a\left(R\left(Y_{i}\right)\right) . $$ (a) Show that \(r_{a}\) is a correlation coefficient on the items $$ \left\\{\left(a\left[R\left(X_{1}\right)\right], a\left[R\left(Y_{1}\right)\right]\right),\left(a\left[R\left(X_{2}\right)\right], a\left[R\left(Y_{2}\right)\right]\right), \ldots,\left(a\left[R\left(X_{n}\right)\right], a\left[R\left(Y_{n}\right)\right]\right)\right\\} . $$ (b) For the score function \(\varphi(u)=\sqrt{12}(u-(1 / 2))\), show that \(r_{a}=r_{S}\), Spearman's rho. (c) Obtain \(r_{a}\) for the sign score function \(\varphi(u)=\operatorname{sgn}(u-(1 / 2))\). Call this rank correlation coefficient \(r_{q c}\). (The subscript \(q c\) is obvious from Exercise \(10.8 .8\).)

In Exercise \(10.9 .5\), the influence function of the variance functional was derived directly. Assuming that the mean of \(X\) is 0 , note that the variance functional, \(V\left(F_{X}\right)\), also solves the equation $$ 0=\int_{-\infty}^{\infty}\left[t^{2}-V\left(F_{X}\right)\right] f_{X}(t) d t $$ (a) Determine the natural estimator of the variance by writing the defining equation at the empirical cdf \(F_{n}(t)\), for \(X_{1}-\bar{X}, \ldots, X_{n}-\bar{X}\) iid with \(\operatorname{cdf} F_{X}(t)\) and solving for \(V\left(F_{n}\right)\). (b) As in Exercise \(10.9 .6\), write the defining equation for the variance functional at the contaminated \(\operatorname{cdf} F_{x, \epsilon}(t)\). (c) Then derive the influence function by implicit differentiation of the defining equation in part (b).

_{j}\left\\{R\left(Y_{j}\right)>\frac… # Consider the sign scores test procedure discussed in Example \(10.5 .4\). (a) Show that \(W_{S}=2 W_{S}^{*}-n_{2}\), where \(W_{S}^{*}=\\#_{j}\left\\{R\left(Y_{j}\right)>\frac{n+1}{2}\right\\} .\) Hence \(W_{S}^{*}\) is an equivalent test statistic. Find the null mean and variance of \(W_{S}\). (b) Show that \(W_{S}^{*}=\\#_{j}\left\\{Y_{j}>\theta^{*}\right\\}\), where \(\theta^{*}\) is the combined sample median. (c) Suppose \(n\) is even. Letting \(W_{X S}^{*}=\\#_{i}\left\\{X_{i}>\theta^{*}\right\\}\), show that we can table \(W_{S}^{*}\) in the following \(2 \times 2\) contingency table with all margins fixed: $$ \begin{array}{|c|c|c|c|} \hline & Y & X & \\ \hline \text { No. items }>\theta^{*} & W_{S}^{*} & W_{X S}^{*} & \frac{n}{2} \\\ \hline \text { No. items }<\theta^{*} & n_{2}-W_{S}^{*} & n_{1}-W_{X S}^{*} & \frac{n}{2} \\ \hline & n_{2} & n_{1} & n \\ \hline \end{array} $$ Show that the usual \(\chi^{2}\) goodness-of-fit is the same as \(Z_{S}^{2}\), where \(Z_{S}\) is the standardized \(z\) -test based on \(W_{S}\). This is often called Mood's median test; see Example \(10.5 .4\).

Let \(\theta\) denote the median of a random variable \(X\). Consider testing $$ H_{0}: \theta=0 \text { versus } H_{1}: \theta>0 . $$ Suppose we have a sample of size \(n=25\). (a) Let \(S(0)\) denote the sign test statistic. Determine the level of the test: reject \(H_{0}\) if \(S(0) \geq 16\) (b) Determine the power of the test in part (a) if \(X\) has \(N(0.5,1)\) distribution. (c) Assuming \(X\) has finite mean \(\mu=\theta\), consider the asymptotic test of rejecting \(H_{0}\) if \(\bar{X} /(\sigma / \sqrt{n}) \geq k\). Assuming that \(\sigma=1\), determine \(k\) so 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).
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