Introduction To Robust Estimation And Hypothesis Testing 5th Edition Rand R. Wilcox - Solutions

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5. For the data in Exercise 1, use the R function reglev to comment on the advisability of using M regression with Schweppe weights.
4. Let T be any regression estimator that is affine equivariant. Let A be any non-singular square matrix. Argue that the predicted y values, yˆi, remain unchanged when xi is replaced by xiA.
3. Using the data in Exercise 1, show that the estimate of the slope given by βˆch is 0.057.In contrast, the OLS estimate is 0.0045, and βˆm = 0.0042. Comment on the difference among the three estimates
2. Discuss the relative merits of βˆch.
1. The average LSAT scores (x) for the 1973 entering classes of 15 American law schools and the corresponding grade point averages (y) are as follows:x: 576 635 558 578 666 580 555 661 651 605 653 575 545 572 594 y: 3.39 3.30 2.81 3.03 3.44 3.07 3.00 3.43 3.36 3.13 3.12 2.74 2.76 2.88 2.96 Using
14. Let X be a standard normal random variable, and suppose Y is a contaminated normal with probability density function given by Eq. (1.1). Let Q = ρX + 1 − ρ2Y , −1 ≤ρ ≤ 1. Verify that the correlation between X and Q is
13. If in the definition of the biweight midcovariance, the median is replaced by the biweight measure of location, the biweight midcovariance is equal to zero under independence.Describe some negative consequences of replacing the median with the biweight measure of location.
12. The definition of the percentage bend correlation coefficient, ρpb, involves a measure of scale, ωx, that is estimated with ωˆ = W(m), where Wi = |Xi − Mx| and m = [(1 − β)n], where 0 ≤ β ≤ 0.5. Note that this measure of scale is defined even when 0.5 0. Argue that the finite
11. For the data in the file read.dat, test for independence using the data in columns 4 and 5 and β = 0.1.
10. For the data used in the last two exercises, test the hypothesis of independence using the function indt. Why might indt find an association not detected by any of the correlations covered in this chapter?
9. Examine the variables in the last exercise using the R function mscor.
8. Use the data in the file read.dat and test for independence using the data in columns 2, 3, and 10 and the R function pball. Try β = 0.1, 0.3, and 0.5. Comment on any discrepancies.
7. The method for detecting outliers, described in Section 6.4.3, could be modified by replacing the MVE estimator with the Winsorized mean and covariance matrix. Discuss how this would be done and its relative merits.
6. Repeat the previous problem using the data for group 2.
5. Using the group 1 alcohol data in Section 8.6.2, compute the MVE estimate of correlation, and compare the results to the biweight midcorrelation, the percentage bend correlation using β = 0.1, 0.2, 0.3, 0.4, and 0.5, the Winsorized correlation using γ = 0.1 and 0.2, and the skipped correlation.
4. Use the function cov.mve(m,cor=TRUE) to compute the MVE correlation for the star data in Fig. 9.2. Compare the results to the Winsorized, percentage bend, skipped, and biweight correlations, as well as the M-estimate of correlation returned by the R function relfun.
3. Demonstrate that heteroscedasticity affects the probability of a Type I error when testing the hypothesis of a zero correlation based on any type M correlation and non-bootstrap method covered in this chapter.
2. Repeat Exercise 1 with Spearman’s rho, the percentage bend correlation, and the Winsorized correlation.
1. Generate 20 observations from a standard normal distribution, and store them in the R variable ep. Repeat this, and store the values in x. Compute y=x+ep, and compute Kendall’s tau. Generally, what happens if two pairs of points are added at (2.1, −2.4)?Does this have a large impact on tau?
6. Analyze the data for murderers in Section 6.3.5 using the methods in Sections 8.6.1 and 8.6.4.
5. Repeat Exercises 3 and 4 using the data for the murderers in Table 6.1.
4. Repeat Exercise 3 using the rank-based method in Section 8.5. How do the results compare to using a measure of location?
3. Analyze the data for the control group mentioned in Section 6.12.1 using the methods in Sections 8.1 and 8.2. The data are stored in the file schiz2.data, which can be accessed as described in Section 1.10. Compare and contrast the results.
2. For the data used in Exercise 1, compute confidence intervals for all pairs of trimmed means using the R function pairdepb.
1. Section 8.6.2 reports data on hangover symptoms. For group 2, use the R function rmanova to compare the trimmed means corresponding to times 1, 2, and 3.
12. For the schizophrenia data in Section 7.8.4, compare the groups with t1way and pbadepth.
11. Generate data for a 2-by-3 design, and use the function pbad2way. Note the contrast coefficients for interactions. If you again use pbad2way, but with conall=FALSE, what will happen to these contrast coefficients? Describe the relative merits of using conall=TRUE.
10. Snedecor and Cochran (1967) report weight gains for rats randomly assigned to one of four diets that varied in the amount and source of protein. The data are stored in the file Snedecor_dat.txt, which can be retrieved as described in Section 1.10.Verify the results based on the R function
9. For the data in the previous two exercises, perform all pairwise comparisons using the Harrell–Davis estimate of the median.
8. For the data in the previous exercise, compare the groups using both the Rust–Fligner and the Brunner–Dette–Munk method.
7. Suppose three different drugs are being considered for treating some disorder, and it is desired to check for side effects related to liver damage. Further suppose that the following data are collected on 28 participants.The values under the columns headed by ID indicate which of the three drugs
6. Using the data from the previous two exercises, compare the 20% trimmed means of the experimental group to the control, taking into account grade. Also test for no interactions using lincon and linconb. Is there reason to suspect that the confidence interval returned by linconbt will be longer
5. Using the data in the previous exercise, use the function lincon to compare the experimental group to the control group, taking into account grade and the two tracking abilities. (Again, tracking abilities 2 and 3 are combined.) Comment on whether the results support the conclusion that the
4. Some psychologists have suggested that teachers’ expectancies influence intellectual functioning. The file VIQ.dat contains pretest verbal IQ scores for students in grades 1 and 2 who were assigned to one of three ability tracks. (The data are from Elashoff and Snow, 1970, and originally
3. From well-known results on the random effects model (e.g., Graybill, 1976; Jeyaratnam and Othman, 1985), it follows that Use these results to derive an alternative estimate of ρWI.
2. If data are generated from exponential distributions, what problems would you expect in terms of probability coverage when computing confidence intervals? What problems with power might arise?
1. Describe how M-measures of location might be compared in a two-way design with a percentile bootstrap method. What practical problem might arise when using the bootstrap and sample sizes are small?
18. This exercise deals with the likelihood of a correct decision when using classification method C3 in Section 6.16. Imagine that training data contain 100 individuals from the first of two groups and 200 from the other. Further imagine that the test data consist of 10 individuals from the first
16. Argue that when testing Eq. (6.27), this provides a metric-free method for comparing groups based on scatter.17. The goal is to run simulations to compare the mean squared error and bias of two estimators designed for functional data. Here is some R code for generating data according to a
15. For the EEG data, compare the two groups with the method in Section 6.11.
14. For the EEG data, compare the two groups with the method in Section 6.10.
13. For the EEG data, compare the two groups with the method in Section 6.9.
12. For the EEG data used in Exercise 1, compare the two groups with the method in Section 6.7.
11. For the cork boring data mentioned in Section 5.9.10, imagine that the goal is to compare the north, east, and south sides to the west side. How might this be done with the software in Section 6.6.1? Perform the analysis and comment on the results. (The data are stored in the file corkall.dat;
10. The file read.dat contains data from a reading study conducted by L. Doi. Columns 4 and 5 contain measures of digit naming speed and letter naming speed. Use both the relplot and the MVE method to identify any outliers. Compare the results and comment on any discrepancies.
9. The MVE method of detecting outliers, described in Section 6.4.3, could be modified by replacing the MVE estimator of location with the Winsorized mean, and replacing the covariances with the Winsorized covariances described in Section 5.9.3. Discuss how this would be done and its relative
8. The average LSAT scores (X) for the 1973 entering classes of 15 American law schools and the corresponding grade point averages (Y ) are as follows.X: 576 635 558 578 666 580 555 661 651 605 653 575 545 572 594 Y : 3.39 3.30 2.81 3.03 3.44 3.07 3.00 3.43 3.36 3.13 3.12 2.74 2.76 2.88 2.96 Use a
7. Give a general description of a situation where for n = 20, the minimum depth among all points is 3/20.
6. Suppose that for each row of an n-by-p matrix, its depth is computed relative to all n points in the matrix. What are the possible values that the depths might be?
5. Repeat the last two exercises, but now use the cork data mentioned in Section 6.7.3.
4. Repeat the last exercise using the data for group 2.
3. For the data used in the last two exercises, check for outliers among the first group using the methods in Section 6.4. Comment on why the number of outliers found differs among the methods.
2. Repeat the last exercise using the data for group 2. These are the measures in the last four columns.
1. For the EEG data mentioned in Section 6.3.5, compute the MVE, MCD, OP, and the Donoho–Gasko 0.2 trimmed mean for group 1. This corresponds to the first four measures, which are based on murderers.
16. Section 5.9.15 described a method for comparing the variances of two dependent variables. It was noted that when distributions differ in shape, the method can fail to control the Type I error probability. Execute the following R commands to get a sense of why:
15. Using R, generate 30 observations from a standard normal distribution and store the values in x. Generate 20 observations from a chi-squared distribution with one degree of freedom, and store them in z. Compute y=4(z-1), so, x and y contain data sampled from distributions having identical
14. Let D = X − Y , let θD be the population median associated with D, and let θX and θY be the population medians associated with X and Y , respectively. Verify that under general conditions, θD = θX − θY .
13. The file tumor.dat contains data on the number of days to occurrence of a mammary tumor in 48 rats injected with a carcinogen and subsequently randomized to receive either the treatment or the control. The data were collected by Gail et al. (1980) and represent only a portion of the results
12. Continuing the last exercise, examine a boxplot of the data. What would you expect to happen if the 0.95 confidence interval is computed using a bootstrap-t method? Verify your answer using the R function yuenbt.
11. The file pyge.dat (see Section 1.8) contains pretest reasoning IQ scores for students in grades 1 and 2 who were assigned to one of three ability tracks. (The data are from Elashoff and Snow, 1970, and originally collected by R. Rosenthal.) The file pygc.dat contains data for a control group.
10. Section 5.9.6 used some hypothetical data to illustrate the R function yuend with 20%trimming. Use the function to compare the means. Verify that the estimated standard error of the difference between the sample means is smaller than the standard error of the difference between the 20% trimmed
9. The example at the end of Section 5.3.3 examined some data from an experiment on the effects of drinking alcohol. Another portion of the study consisted of measuring the effects of alcohol over 3 days of drinking. The scores for the control group, for the first 2 days of drinking, are 4, 47, 35,
8. Compute a confidence interval for p using Salk’s data.
7. Apply the Yuen–Welch method using Salk’s data, where the amount of trimming is 0, 0.05, 0.1, and 0.2. Compare the estimated standard errors of the difference between the trimmed means.
6. Verify that if X and Y are independent, the third moment about the mean of X − Y isμx[3] − μy[3].
5. Compare the deciles only, using the Harrell–Davis estimator and Salk’s data.
4. Consider two independent groups having identical distributions. Suppose four observations are randomly sampled from the first and three from the second. Determine P(D = 1) and P(D = 0.75), where D is given by Eq. (5.4). Verify your results with the R function kssig.
3. Summarize the relative merits of using the weighted versus unweighted Kolmogorov–Smirnov test. Also discuss the merits of the Kolmogorov–Smirnov test relative to comparing measures of location.
2. For the ozone data stored in the file rat_data.txt, compare the two groups using the weighted Kolmogorov–Smirnov test. Plot the shift function and its 0.95 confidence band.Compare the results with the unweighted test.
1. For Salk’s data stored in the file salk_dat.txt, compare the two groups using the weighted Kolmogorov–Smirnov test. Plot the shift function and its 0.95 confidence band. Compare the results with the unweighted test.
12. Generate 20 observations from a g-and-h distribution with g = h = 0.5. (This can be done with the R function ghdist, written for this book.) Examine a boxplot of the data.Repeat this 10 times. Comment on the strategy of examining a boxplot to determine whether the confidence interval for the
11. For the LSAT data in Table 4.3, compute a 0.95 bootstrap-t confidence interval for mean using the R function trimcibt with plotit=T. Note that a boxplot finds no outliers. Comment on the plot created by trimcibt in terms of achieving accurate probability coverage when using Student’s t. What
10. Verify Eq. (4.5) using the decision rule about whether to reject H0 described in Section 4.4.3.
9. Discuss the relative merits of using the R function sint versus qmjci and hdci.
8. For the exponential distribution, would the sample median be expected to have a relatively high or low standard error? Compare your answer to the estimated standard error obtained with data generated from the exponential distribution.
7. Do the skewness and kurtosis of the exponential distribution suggest that the bootstrapt method will provide a more accurate confidence interval for μt versus the confidence interval given by Eq. (4.3)?
6. If the exponential distribution has variance μ[2] = σ2, then μ[3] = 2σ3 and μ[4] = 9σ4.Determine the skewness and kurtosis. What does this suggest about getting an accurate confidence interval for the mean?
5. The R function rexp generates data from an exponential distribution. Use R to estimate the probability of getting at least one outlier, based on a boxplot, when sampling from this distribution. Discuss the implications for computing a confidence interval for μ.
4. Use the R functions qmjci, hdci, and sint to compute a 0.95 confidence interval for the median based on the LSAT data in Table 4.3. Comment on how these confidence intervals compare to one another.
3. Compute a 0.95 confidence interval for the mean, 10% mean, and 20% mean using the lifetime data listed in the example of Section 4.6.2. Use both Eq. (4.3) and the bootstrapt method.
2. Compute a 0.95 confidence interval for the mean, 10% mean, and 20% mean using the data in Table 3.1 of Chapter 3. Examine a boxplot of the data and comment on the accuracy of the confidence interval for the mean. Use both Eq. (4.3) and the bootstrap-t method.
1. Describe situations where the confidence interval for the mean might be too long or too short. Contrast this with confidence intervals for the 20% trimmed mean and μm.
13. Using R, generate n = 20 observations from the mixed normal distribution in Fig. 1.1.This can be done with the R function cnorm stored in Rallfun. Compute the mean, 20%trimmed mean, median, and one-step M-estimate, and repeat this 10,000 times. Compare the results using a boxplot.
12. Using R, generate n = 20 observations from a standard normal distribution, compute the mean, 20% trimmed mean, median, and one-step M-estimate, and repeat this 10,000 times. Compare the results using a boxplot.
11. Verify that Eq. (3.29) reduces to s2/n if no observations are flagged as being unusually large or small by .
10. Argue that if is taken to be the biweight, it approximates the optimal choice for under normality when observations are not too extreme.
9. Set Xi = i, i = 1, ..., 20, and compute the Harrell–Davis estimate of the median. Repeat this, but with X20 equal to 1,000 and then 100,000. When X20 = 100,000, would you expect xˆ0.5 or the Harrell–Davis estimator to have the smaller standard error? Verify your answer.
8. Repeat the previous exercise, only this time, compute the biweight midvariance, the 20%Winsorized variance, and the percentage bend midvariance. Comment on the resistance of these three measures of scale.
7. Set Xi = i, i = 1, ..., 20, and compute the 20% trimmed mean and the M-estimate of location based on Huber’s . Next set X20 = 200, and compute both estimates of location. Replace X19 with 200, and again, estimate the measures of location. Keep doing this until the upper half of the data is
6. Cushny and Peebles (1904) conducted a study on the effects of optical isomers of hyoscyamine hydrobromide in producing sleep. For one of the drugs, the additional hours of sleep for 10 patients were 0.7, −1.6, −0.2, −1.2, −0.1, 3.4, 3.7, 0.8, 0, and 2.
5. Comment on the strategy of applying the boxplot to the data in Exercise 2, removing any outliers, computing the sample mean for the data that remain, and then estimating the standard error of this sample mean based on the sample variance of the data that remain.
4. For the data in Exercise 1, estimate the deciles using the Harrell–Davis estimator. Do the same for the data in Table 3.2. Plot the difference between the deciles as a function of the estimated deciles for the data in Exercise 1. What do the results suggest? Estimate the standard errors
3. For the data in Exercise 1, compute MADN, the biweight midvariance, and the percentage bend midvariance. Compare the results to those obtained for the data in Table 3.2.What do the results suggest about which group is more dispersed?
1. Included among the R functions written for this book is the function ghdist(n,g=0,h=0).It generates n observations from a so-called g-and-h distribution, which is described in more detail in Chapter 4. The command ghdist(30,0,0.5) will generate 30 observations from a symmetric, heavy-tailed
7. Repeat the previous exercise, but now use method TAP. Compare the plot returned by the R function ancdet to the plot created by ancJN and comment on the results. (Gender is indicated by the variable BK_SEX in column 268.)
6. Repeat the first example in Section 12.2.6, only use the measure of meaningful activities stored in column 253 of the Well Elderly data. (The variable label is MAPAGLOB.)
5. For the variables used in Exercise 2, use method UB and TAP. Explain any differences in the results.
4. Comment generally on why method UB and TAP can give different results. Why will they tend to differ in terms of power?
3. Using the same variables as in the last exercise, perform the analysis using the R function ancova.
2. Repeat the first example in Section 12.2.6, only use the measure of meaningful activities stored in column 214 of the Well Elderly data. (The variable label is MAPAFREQ_SUM and gender is indicated by the variable BK_SEX in column 268. The data are stored on the author’s web page in the files

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