Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

644 Nonparametric and Robust Statistics

Example 10.9.2(Asymptotic Distribution of the Sample Median).The influence
function for the sample medianθ̂L 1 is given by (10.9.20). SinceE[sgn^2 (X−θ)] = 1,
by expression (10.9.25) the asymptotic distribution of the sample median is

√

n[̂θ−θ]→DN

(

0 ,[2fX(θ]−^2

)

,

whereθis the median of the pdffX(t). This agrees with the result given in Section
10.2.

Breakdown Point of an Estimator

The influence function of an estimator measures the sensitivity of an estimator to
a single outlier, sometimes called thelocal sensitivityof the estimator. We next
discuss a measure ofglobal sensitivityof an estimator. That is, what proportion of
outliers can an estimator tolerate without completely breaking down?
To be precise, letx′=(x 1 ,x 2 ,...,xn) be a realization of a sample. Suppose we
corruptmpoints of this sample by replacingx 1 ,...,xmbyx∗ 1 ,...,x∗m,wherethese
points are large outliers. Letxm=(x∗ 1 ,...,x∗m,xm+1,...,xn) denote the corrupted
sample. Define the bias of the estimator upon corruptingmdata points to be

bias(m,xn,̂θ)=sup|θ̂(xm)−̂θ(xn)|, (10.9.27)

where the sup is taken over all possible corrupted samplesxm. If this bias is infinite,
we say that the estimator hasbroken down. The smallest proportion of corruption
an estimator can tolerate until its breakdown is called itsfinite sample breakdown
point. More precisely, if

∗n=minm{m/n:bias(m,xn,̂θ)=∞}, (10.9.28)

then ∗nis called thefinite sample breakdown pointof̂θ. If the limit

∗n→ ∗ (10.9.29)

exists, we call ∗thebreakdown pointof̂θ.
To determine the breakdown point of the sample mean, suppose we corrupt
one data point, say, without loss of generality, the first data point. The corrupted
sample is thenx′=(x∗ 1 ,x 2 ,...,xn). Denote the sample mean of the corrupted
sample byx∗. Then it is easy to see that

x∗−x=

1

n

(x∗ 1 −x 1 ).

Hence bias(1,xn,x) is a linear function ofx∗ 1 and can be made as large (in absolute
value) as desired by takingx∗ 1 large (in absolute value). Therefore, the finite sample
breakdown of the sample mean is 1/n. Because this goes to 0 asn→∞,the
breakdown point of the sample mean is 0.