Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

232 Some Elementary Statistical Inferences

Medium Fair Dark Red Black

Haircolor

0.0

0.1

0.2

0.3

0.4

Bar Chart of Haircolor of Scottish Schoolchildren

Figure 4.1.1:Bar chart of the Scottish hair color data discussed in Example 4.1.5.

The nonparametric estimate of the pmf is

j 0 1 2 3 4 5 ≥ 6

̂p(j) 0.067 0.367 0.233 0.167 0.067 0.067 0.033

The histogram for this data set is given in Figure 4.1.2. Note that counts are used

for the vertical axis. If the R vectorxcontains the 30 data points, then the following

R code computes this histogram:

brs=seq(-.5,6.5,1);hist(x,breaks=brs,xlab="Number of events",ylab="")

The Distribution ofXIs Continuous
For this section, assume that the random sampleX 1 ,...,Xnis from a continuous
random variableXwith continuous pdff(t). We first sketch an estimate for this
pdf at a specified value ofx. Then we use this estimate to develop a histogram
estimate of the pdf. For an arbitrary but fixed pointxand a givenh>0, consider
the interval (x−h, x+h). By the mean value theorem for integrals, we have for
someξ,|x−ξ|<h,that

P(x−h<X<x+h)=

∫x+h

x−h

f(t)dt=f(ξ)2h≈f(x)2h.

The nonparametric estimate of the leftside is the proportion of the sample items
that fall in the interval (x−h, x+h). This suggests the following nonparametric