4.4. Order Statistics 261lower quantiles ofZmust be spread out as are the higher quantiles; i.e., symmetric
distributions may be more appropriate. The plots in Panels B and C are more
linear than that of Panel D, but they still contain some curvature. Of the two,
Panel C appears to be more linear. Actually, the data were generated from a
Laplace distribution, so one would expect that Panel C would be the most linear of
the three plots.
Many computer packages have commands to obtain the population quantiles
used in this example. The R functionqqplotc4s2.r, at the site listed in Chapter
1, obtains the normal, Laplace, and exponential quantiles used for Figure 4.4.1 and
the plot. The call isqqplotc4s2(x)where the R vectorxcontains the data.Theq−qplot using normal quantiles is often called anormalq−qplot. If the
data are in the R vectorx, the plot is obtained by the callqqnorm(x).
4.4.2 Confidence Intervals for Quantiles
LetXbe a continuous random variable with cdfF(x). For 0<p<1, define the
100 pth distribution percentile to beξp,whereF(ξp)=p. For a sample of sizenon
X,letY 1 <Y 2 <···<Ynbe the order statistics. Letk=[(n+1)p]. Then the
100 pth sample percentileYkis a point estimate ofξp.
We now derive adistribution freeconfidence interval forξp, meaning it is a
confidence interval forξpwhich is free of any assumptions aboutF(x) other than
it is of the continuous type. Leti<[(n+1)p]<j, and consider the order statistics
Yi<Yjand the eventYi<ξp<Yj.Fortheith order statisticYito be less than
ξp,itmustbetruethatatleastiof theXvalues are less thanξp.Moreover,for
thejth order statistic to be greater thanξp, fewer thanjof theXvalues are less
thanξp. To put this in the context of a binomial distribution, the probability of
success isP(X<ξp)=F(ξp)=p. Further, the eventYi<ξp<Yjis equivalent to
obtaining betweeni(inclusive) andj(exclusive) successes innindependent trials.
Thus, taking probabilities, we have
P(Yi<ξp<Yj)=∑j−^1w=i(
n
w)
pw(1−p)n−w. (4.4.8)When particular values ofn,i,andjare specified, this probability can be computed.
By this procedure, suppose that it has been found thatγ=P(Yi<ξp<Yj). Then
the probability isγthat the random interval (Yi,Yj) includes the quantile of order
p. If the experimental values ofYiandYjare, respectively,yiandyj,theinterval
(yi,yj) serves as a 100γ% confidence interval forξp, the quantile of orderp.Weuse
this in the next example to find a confidence interval for the median.
Example 4.4.7(Confidence Interval for the Median).LetXbe a continuous ran-
dom variable with cdfF(x). Letξ 1 / 2 denote the median ofF(x); i.e.,ξ 1 / 2 solves
F(ξ 1 / 2 )=1/2. SupposeX 1 ,X 2 ,...,Xnis a random sample from the distribution
ofXwith corresponding order statisticsY 1 <Y 2 <···<Yn. As before, letQ 2
denote the sample median, which is a point estimator ofξ 1 / 2. Selectα,sothat
0 <α<1. Takecα/ 2 to be theα/2th quantile of a binomialb(n, 1 /2) distribution;