10.7. Simple Linear Model 62510.6.3.LetF(x) be a distribution function of a distribution of the continuous type
that is symmetric about its medianθ.WewishtotestH 0 :θ= 0 againstH 1 :θ>0.
Use the fact that the 2nvalues,Xiand−Xi,i=1, 2 ,...,n, after ordering, are
complete sufficient statistics forF, provided thatH 0 is true.
(a)As in Exercise 10.5.15, determine the one-sample signed-rank test statistics
corresponding to the two-sample score functionsφ 1 (u),φ 2 (u), andφ 3 (u)de-
fined in the last exercise. Use the asymptotic test statistics. Note that these
score functions are odd about^12 ; hence, their top halves serve as score func-
tions for signed-rank statistics.(b)We are assuming symmetric distributions in this problem; hence, we use only
Q 2 as our score selector. IfQ 2 ≥7, then selectφ 2 (u); if 2<Q 2 <7, then
selectφ 1 (u); and finally, ifQ 2 ≤2, then selectφ 3 (u). Construct this adaptive
distribution-free test.(c)Use your adaptive procedure on Darwin’sZea maysdata; see Example 10.3.1.
Obtain thep-value.10.7SimpleLinearModel...........................
In this section, we consider the simple linear model and briefly develop the rank-
based procedures for it.
Suppose the responsesY 1 ,Y 2 ,...,Ynfollow the modelYi=α+β(xi−x)+εi,i=1, 2 ,...,n, (10.7.1)whereε 1 ,ε 2 ,...,εnare iid with continuous cdfF(x)andpdff(x). In this model, the
variablesx 1 ,x 2 ,...,xnare considered fixed. Oftenxis referred to as apredictor
ofY. Also, the centering, usingx, is for convenience (without loss of generality)
and we do not use it in the examples of this section. The parameterβis the slope
parameter, which is the expected change inY (provided expectations exist) when
xincreases by one unit. A natural null hypothesis is
H 0 :β=0versusH 1 : β =0. (10.7.2)UnderH 0 , the distribution ofY is free ofx.
In Chapter 3 of Hettmansperger and McKean (2011), rank-based procedures
for linear models are presented from a geometric point of view; see also Exercises
10.9.11–10.9.12 of Section 10.9. Here, it is easier to present a development which
parallels the preceding sections. Hence we introduce a rank test ofH 0 and then
invert the test to estimateβ. Before doing this, though, we present an example that
shows that the two-sample location problem of Section 10.4 is a regression problem.
Example 10.7.1.As in Section 10.4, letX 1 ,X 2 ,...,Xn 1 be a random sample from
a distribution with a continuous cdfF(x−α), whereαis a location parameter.
LetY 1 ,Y 2 ,...,Yn 2 be a random sample with cdfF(x−α−Δ). Hence Δ is the
shift between the cdfs ofXiandYj. Redefine the observations asZi=Xi,for