Introduction To Robust Estimation And Hypothesis Testing 2nd Edition Rand R. Wilcox - Solutions

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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=F, what will happen to these contrast coefficients?Describe the relative merits of using conall=T.
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 results were:Beef Cereal Low High Low High 90 73 107 98 76 102 95 75 90 118 97 56 64 104 80 111 86 81 98 95 51 107 74 88 72 100 74 82 90 87 67 77
9. For the data in Exercises 7 and 8, perform all pairwise comparisons using the Harrell–Davis estimate of the median.
8. For the data in Exercise 7, compare the groups using both the Rust–Fligner and Brunner–Dette–Munk methods.
7. Suppose three different drugs are being considered for treating some disorder and that it is desired to check for side effects related to
6. Using the data from Exercises 4 and 5, 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 linconb will be longer than the
5. With the data in Exercise 4, 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 experimental and
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, from Elashoff and Snow, 1970, were originally
3. From well-known results on the random effects model (e.g., Graybill, 1976; Jeyaratnam and Othman, 1985), it follows that?
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?
17. For the data in Table 6.6, compare the groups using the method in Section 6.7?
16. Argue that when testing Eq. (6.27), this provides a metric-free method for comparing groups based on scatter.
15. For the data in Table 6.1, compare the two groups with the method in Section 6.11.
14. For the data in Table 6.1, compare the two groups with the method in Section 6.10.
13. For the data in Table 6.1, compare the two groups with the method in Section 6.9.
12. For the data in Table 6.1, compare the two groups with the method in Section 6.7.
11. For the cork boring data in Table 6.5, 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;see Chapter 1.)
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 letternaming 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:
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 Exercises 3 and 4, but now use the data in Table 6.3.
4. Repeat Exercise 3 using the data for group 2.
3. For the data in Table 6.1, 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 Exercise 1 using the data for group 2.
1. For the EEG data in Table 6.1, compute the MVE, MCD, OP, and Donoho–Gasko .2 trimmed mean for group 1.
17. Using S-PLUS (or R), generate 30 observations from a standard normal distribution, and store the values in x. Generate 20 observations from a chi-squared distribution with 1 degree of freedom, and store them in z. Compute y=4(z-1), so x and y contain data sampled from distributions having
16. 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.
15. 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, collected by Gail et al. (1980), represent only a portion of the results they
14. Continuing the last exercise, examine a boxplot of the data. What would you expect to happen if the .95 confidence interval is computed using a bootstrap-t method? Verify your answer using the S-PLUS (or R)function yuenbt.
13. 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 were originally collected by R. Rosenthal.) The file pygc.dat contain data for a control
12. Section 5.9.4 used some hypothetical data to illustrate the S-PLUS 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%
11. 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,
10. Compute a confidence interval for p using the data in Table 5.1.
9. Comment on the relative merits of testing H0: p = 1/2 with Mee’s method versus comparing two independent groups with the Kolmogorov–Smirnov test.
8. Describe a situation where testing H0: p = 1/2 with Mee’s method can have lower power than the Yuen–Welch procedure.
7. Apply the Yuen–Welch method to the data in Table 5.1. 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, then the third moment about the mean of X − Y is µx[3] − µy[3].
5. Compare the deciles only, using the Harrell–Davis estimator and the data in Table 5.1.
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 = .75), where D is given by Eq. (5.4). Verify your results with the S-PLUS 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. Compare the two groups of data in Table 5.3 using the weighted Kolmogorov–Smirnov test. Plot the shift function and its .95 confidence band. Compare the results with the unweighted test.
1. Compare the two groups of data in Table 5.1 using the weighted Kolmogorov–Smirnov test. Plot the shift function and its .95 confidence band. Compare the results with the unweighted test.
12. Generate 20 observations from a g-and-h distribution with g = h = .5.(This can be done with the R or S-PLUS 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
11. For the LSAT data in Table 4.3, compute a .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 or S-PLUS 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 bootstrap-t 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. If Z is generated from a uniform distribution on the unit interval, then X = ln(1/(1 − Z)) has an exponential distribution. Use R or S-PLUS to estimate the probability of getting at least one outlier, based on a boxplot, when sampling from this distribution. Discuss the implications for
4. Use the R or S-PLUS functions qmjci, hdci, and sint to compute a .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 .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 bootstrap-t method.
2. Compute a .95 confidence interval for the mean, 10% mean, and 20%mean using the data in Table 3.1. 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. Verify that Eq. (3.29) reduces to s2/n if no observations are flagged as being unusually large or small by?
12. Argue that if is taken to be the biweight, it approximates the optimal choice for under normality when observations are not too extreme.
11. Set Xi = i, i = 1, ... , 20, and compute the Harrell–Davis estimate of the median. Repeat this, but with X20 equal to 1000 and then 100,000. When X20 = 100,000, would you expect xˆ.5 or the Harrell–Davis estimator to have the smaller standard error? Verify your answer.
10. 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.
9. 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
8. Use results on Winsorized expected values in Chapter 2 to show that Xw is a Winsorized unbiased estimate of µw.
7. Use results on Winsorized expected values in Chapter 2 to show that if the error term in Eq. (3.4) is ignored, X t is a Winsorized unbiased estimate of µt.
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?
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 2, 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 2. What do the results suggest? Estimate the standard errors
3. For the data in Exercise 2, 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?
2. In the study by Dana (1990) on self-awareness, described in this chapter(in connection with Table 3.2), a second group of subjects yielded the observations 59 106 174 207 219 237 313 365 458 497 515 529 557 615 625 645 973 1065 3215.Compute the sample median, the Harrell–Davis estimate of the
1. Included among the R and S-PLUS functions written for this book is the function ghdist(n,g=0,h=0), which 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,.5) will generate 30 observations from a symmetric,
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
1. Comment on the relative merits of using a linear model versus a smoother in the context of ANCOVA.
12. Generate data from a bivariate normal distribution with the R command x=rmul(200).Then enter the R command y=x[,1]+x[,2]+x[,1]*x[,2]+rnorm(200), and examine the plot returned by the R command gamplot(x,y,scale=TRUE). Compare this to the plot returned by the R command gamplotINT(x,y,scale=TRUE).
11. Generate 25 pairs of observations from a bivariate normal distribution having correlation 0, and store them in x. (The R function rmul, written for this book, can be used.) Generate 25 more observations, and store them in y. Create a smooth using rplot using scale=T, and compare it to the
10. Generate 25 observations from a standard normal distribution, and store the results in the R variable x. Generate 25 more observations, and store them in y. Use rungen to plot a smooth based on the Harrell–Davis estimator of the median. Also create a smooth with the argument scat=F. Comment
9. For the reading data in the upper right panel of Fig. 11.5, recreate the smooth. If you wanted to find a parametric regression equation, what might be tried? Examine how well your suggestions perform.
8. The data in the lower left panel of Fig. 11.5 are stored in the file agegesell.dat. Remove the two pairs of points having the largest x value, and create a running interval smoother using the data that remain.
7. For the reading data in the file read.dat, use the R function rplot to investigate the shape of the regression surface when predicting the 20% trimmed mean of WWISST2 (the data in column 8) with RAN1T1 and RAN2T1 (the data in columns 4 and 5).
6. For the reading data in file read.dat, let x be the data in column 2 (TAAST1), and suppose it is desired to predict y, the data in column 8 (WWISST2). Speculate on whether there are situations where it would be beneficial to use x2 to predict y taking into account the value stored in column 3
5. The example at the end of Section 11.5.5 are based on the data stored in the file A3B3C_dat.txt, which can be downloaded as described in Section 1.10. Verify the results in Section 11.5.5 when leverage points are removed.
4. Use the function winreg to estimate the slope and intercept of the star data using 20%Winsorization. (The data are stored in the file star.dat. See Section 1.8 on how to obtain the data.)
3. For the data in Exercise 1, test H0: β1 = 0 with the functions regci and regtest. Comment on the results.
2. Section 8.6.2 reports data on the effects of consuming alcohol on three different occasions. Using the data for group 1, suppose it is desired to predict the response at time 1 using the responses at times 2 and 3. Test H0: β1 = β2 = 0 using the R function regtest and βˆm.
1. For the data in Exercise 1 of Chapter 10, the 0.95 confidence interval for the slope, based on the least squares regression line, is (0.0022, 0.0062). Using R, the 0.95 confidence interval for the slope returned by lsfitci is (0.003, 0.006). The 0.95 confidence interval returned by the R
15. Graphically illustrate the difference between a regression outlier and a good leverage point. That is, plot some points for which y = β1x + β0 and then add some points that represent regression outliers and good leverage points.
14. For the data in Exercise 13, identify any leverage points using the hat matrix. Next, identify leverage points with the function reglev. How do the results compare?
13. For the data used in Exercise 11, RAN1T1 and RAN2T1 (stored in columns 4 and 5)are measures of digit naming speed and letter naming speed. Use M regression with Schweppe weights to estimate the regression parameters when predicting WWISST2.Use the function elimna, described in Chapter 1, to
12. For the data used in Exercise 11, compute the hat matrix and identify any leverage points. Also check for leverage points with the R function reglev. How do the results compare?
11. The file read.dat contains reading data collected by Doi. Of interest is predicting WWISST2, a word identification score (stored in column 8), using TAAST1, a measure of phonological awareness stored in column 2, and SBT1 (stored in column 3), another measure of phonological awareness. Compare
10. For the data in Exercise 6, verify that the 0.95 confidence interval for the regression parameters, using the R function regci with M regression and Schweppe weights, are(−0.2357, 0.3761) and (−0.0231, 1.2454). Also verify that if regci is used with OLS, the confidence intervals are
9. Referring to Exercise 6, how do the results compare to the results obtained with the R function reglev?
8. For the data used in the previous exercise, compute 0.95 confidence intervals for the parameters using OLS as well as M regression with Schweppe weights.
7. The example in Section 6.6.1 reports the results of drinking alcohol for two groups of subjects measured at three different times. Using the group 1 data, compute an OLS estimate of the regression parameters for predicting the time 1 data using the data based on times 2 and 3. Compare the
6. Compute the hat matrix for the data in Exercise 1. Which x values are identified as leverage points? Relate the result to the previous exercise.

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Introduction To Robust Estimation And Hypothesis Testing 2nd Edition Rand R. Wilcox - Solutions

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