260 Some Elementary Statistical Inferences110
100
x^90
80Panel A70
60110
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90Sample quantiles80Panel B70
60- 1.5 –1.0 –0.5 0.0
Normal quantiles
0.5 1.0 1.5110
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90Sample quantiles80Panel C70
60- 2 – 1 0 1
Laplace quantiles
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90Sample quantiles80Panel D70
60
0.0 0.5 1.0 1.5
Exponential quantiles2.0 2.5Figure 4.4.1:Boxplot and quantile plots for the data of Example 4.4.4.from which we have the linear relationship
ξX,p=bξZ,p+a. (4.4.7)Thus, ifXhasacdfoftheformofF((x−a)/b), then the quantiles ofXare
linearly related to the quantiles ofZ. Of course, in practice, we do not know the
quantiles ofX, but we can estimate them. LetX 1 ,...,Xnbe a random sample from
the distribution ofXand letY 1 <···<Ynbe the order statistics. Fork=1,...,n,
letpk=k/(n+1). ThenYkis an estimator ofξX,pk. Denote the corresponding
quantiles of the cdfF(z)byξZ,pk=F−^1 (pk). Letykdenote the realized value of
Yk. The plot ofykversusξZ,pkis called aq−qplot, as it plots one set of quantiles
from the sample against another set from the theoretical cdfF(z). Basedonthe
above discussion, the linearity of such a plot indicates that the cdf ofXis of the
formF((x−a)/b).
Example 4.4.6(Example 4.4.5, Continued).Panels B, C, and D of Figure 4.4.1
containq−qplots of the data of Example 4.4.4 for three different distributions.
The quantiles of a standard normal random variable are used for the plot in Panel
B. Hence, as described above, this is the plot ofyk versus Φ−^1 (k/(n+ 1)), for
k=1, 2 ,...,n. For Panel C, the population quantiles of the standardLaplace
distribution are used; that is, the density ofZisf(z)=(1/2)e−|z|,−∞<z<∞.
For Panel D, the quantiles were generated from an exponential distribution with
densityf(z)=e−z, 0 <z<∞,zero elsewhere. The generation of these quantiles
is discussed in Exercise 4.4.1.
The plot farthest from linearity is that of Panel D. Note that this plot gives
an indication of a more correct distribution. For the points to lie on a line, the