Chapter 10
Nonparametric and Robust
Statistics
10.1LocationModels
In this chapter, we present some nonparametric procedures for the simple location
problems. As we shall show, the test procedures associated with these methods
are distribution-free under null hypotheses. We also obtain point estimators and
confidence intervals associated with these tests. The distributions of the estimators
are not distribution-free; hence, we use the termrank-basedto refer collectively to
these procedures. The asymptotic relative efficiencies of these procedures are easily
obtained, thus facilitating comparisons among them and procedures that we have
discussed in earlier chapters. We also obtain estimators that are asymptotically
efficient; that is, they achieve asymptotically the Rao–Cram ́er bound.
Our purpose is not a rigorous development of these concepts, and at times we
simply sketch the theory. A rigorous treatment can be found in several advanced
texts, such as Randles and Wolfe (1979) or Hettmansperger and McKean (2011).
For an applied discussion using R, see Kloke and McKean (2014).
In this and the following section, we consider the one-sample problem. For the
most part, we consider continuous random variablesXwith cdf and pdfFX(x)
andfX(x), respectively. We assume thatfX(x)>0 on the support ofX;so,in
particular,FX(x) is strictly increasing on the support. In this and the succeeding
chapters, we want to identify classes of parameters. Think of a parameter as a
function of the cdf (or pdf) of a given random variable. For example, consider the
meanμofX. WecanwriteitasμX=T(FX)ifTis defined as
T(FX)=E(X).As another example, recall that the median of a random variableXis a parameter
ξsuch thatFX(ξ)=1/2; i.e.,ξ=FX−^1 (1/2). Hence, in this notation, we say that
the parameterξis defined by the functionT(FX)=FX−^1 (1/2). Note that theseTs
are functions of the cdfs (or pdfs). We shall call themfunctionals.
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