Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

630 Nonparametric and Robust Statistics

Example 10.7.3(Distance of Punts).Rasmussen (1992), page 562, presents a data

set concerning distance of punts along with several predictors. The actual response

is the average distance in feet of 10 punts for each of 13 punters. As a predictor, we

consider the average hang-time in seconds (the time the punted football is in the

air). The data are in the filepunter.rda. Based on the plot (see Exercise 10.7.1),

the simple linear model seems reasonable as an initial fit. Next is the code and

partial results of the Wilcoxon fit:

fit <- rfit(distance~hangtime); summary(fit)

Estimate Std. Error t.value p.value

(Intercept) -18.180 51.201 -0.3551 0.729254

hangtime 41.010 12.882 3.1834 0.008708 **

The second line of the summary table gives the Wilcoxon estimate of the slope

(41.01) and the standard error of the estimate (12.89). Hence, we predict that the

football travels an additional 41 feet for each additional second of hang-time. An

approximate 95% confidence interval for the true slope, using thet-critical with 11

degrees of freedom is (12. 66 , 69 .36). So with approximate confidence of 95% the

slope differs from 0.

EXERCISES

10.7.1.Consider the data on football punts in Example 10.7.3.

(a)Obtain the scatterplot of distance versus hang-time and overlay the Wilcoxon

fit.

(b)As a second predictor consider overall strength of the kicker which is in the

variablestrength. Obtain the scatterplot of distance versus strength and

overlay the Wilcoxon fit. What is the meaning of the slope parameter for this

predictor. Answer using a 95% confidence interval for the slope.

10.7.2.Establish expression (10.7.10). To do this, note first that the expression is

the same as

Eβ

[n

∑

i=1

(xi−x)aφ(R(Yi))

]

=E 0

[n

∑

i=1

(xi−x)aφ(R(Yi+xiβ))

]

.

Show that the cdfs ofYi(underβ)andYi+(xi−x)β(under 0) are the same.

10.7.3.Suppose we have a two-sample model given by (10.7.3). Assuming Wilcoxon
scores, show that the test statistic (10.7.4) is equivalent to the Wilcoxon test statistic
found in expression (10.4.5).

10.7.4.Show that the null variance of the test statisticTφis the value given in
(10.7.5).

10.7.5.Show that the translation property (10.7.10) implies that the power curve

for either one-sided test based on the test statisticTφofH 0 :β= 0 is monotone.