Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.1. Sampling and Statistics 233

Histogram of Poisson Variates

Number of events

0123456

024681

0

Figure 4.1.2:Histogram of the Poisson variates of Example 4.1.6.

estimate off(x)atagivenx:

f̂(x)=#{x−h<Xi<x+h}

2 hn

. (4.1.13)

To write this more formally, consider the indicator statistic

Ii(x)=

{

1 x−h<Xi<x+h

0otherwise,

i=1,...,n.

Then a nonparametric estimator off(x)is

f̂(x)=^1

2 hn

∑n

i=1

Ii(x). (4.1.14)

Since the sample items are identically distributed,

E[f̂(x)] =

1

2 hn

nf(ξ)2h=f(ξ)→f(x),

ash→0. Hencef̂(x) is approximately an unbiased estimator of the densityf(x).
In density estimation terminology, the indicator functionIiis called arectangular
kernel withbandwidth 2 h. See Sheather and Jones (1991) and Chapter 6 of
Lehmann (1999) for discussions of density estimation. The R functiondensity
provides a density estimator with several options. For the examples in the text, we
use the default option as in Example 4.1.7.