570 Nonparametric and Robust Statistics
Remark 10.1.1(Natural Nonparametric Estimators). Functionals induce non-
parametric estimators naturally. LetX 1 ,X 2 ,...,Xndenote a random sample from
some distribution with cdfF(x)andletT(F) be a functional. Letx 1 ,x 2 ,...,xnbe
a realization of this sample. Recall that the empirical distribution function of the
sample is given by
F̂n(x)=n−^1 [#{xi≤x}], −∞<x<∞. (10.1.1)
Hence,Fnis a discrete cdf that puts mass (probability) 1/nat eachxi. Because
F̂n(x)isacdf,T(F̂n) is well defined. Furthermore,T(F̂n) depends only on the
sample; hence, it is a statistic. We callT(F̂n)theinduced estimatorofT(F).
For example, ifT(F) is the mean of the distribution, then it is easy to see that
T(F̂n)=x; see Exercise 10.1.3.
For another example, consider the median. Note thatFˆnis a discrete cdf; hence,
we use the general definition of a median of a distribution that is given in Definition
1.7.2 of Chapter 1. Let ˆθdenote the usual sample median which is defined in
expression (4.4.4); that is,θˆ=x((n+1)/2)ifnis odd whileˆθ=[x(n/2)+x((n/2)+1)]/ 2
ifnis even. To show thatθˆsatisfies Definition 1.7.2, note that: