Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.9. Robust Concepts 645

Example 10.9.3(Breakdown Value of the Sample Median). Next consider the
sample median. Letxn=(x 1 ,x 2 ,...,xn) be a realization of a random sample. If
thesamplesizeisn=2k, then it is easy to see that in a corrupted samplexnwhen
x(k)tends to−∞, the median also tends to−∞. Hence the breakdown value of the
sample median isk/n, which tends to 0.5. By a similar argument, when the sample
size isn=2k+ 1, the breakdown value is (k+1)/nand it also tends to 0.5asthe
sample size increases. Hence we say that the sample median is a 50% breakdown
estimate. For a location model, 50% breakdown is the highest possible breakdown
point for an estimate. Thus the median achieves the highest possible breakdown
point.

In Exercise 10.9.10, the reader is asked to show that the Hodges–Lehmann esti-
mate has the breakdown point of 0.29.

10.9.2 LinearModel...........................

In Sections 9.6 and 10.7, respectively, we presented the least squares (LS) procedure

and a rank-based (Wilcoxon) procedure for fitting simple linear models. In this

section, we briefly compare these procedures in terms of their robustness properties.

Recall that the simple linear model is given by

Yi=α+βxci+εi,i=1, 2 ,...,n, (10.9.30)

whereε 1 ,ε 2 ,...,εnare continuous random variable that are iid. In this model, we
have centered the regression variables; that is,xci=xi−x,wherex 1 ,x 2 ,...,xnare
considered fixed. The parameter of interest in this section is the slope parameter
β, the expected change (provided expectations exist) when the regression variable
increases by one unit. The centering of thexs allows us to consider the slope
parameter by itself. The results we present are invariant to the intercept parameter
α. Estimates ofαare discussed at the end of this section. With this in mind, define
the random variableeito beεi+α. Then we can write the model as

Yi=βxci+ei,i=1, 2 ,...,n, (10.9.31)

wheree 1 ,e 2 ,...,enare iid with continuous cdfF(x)andpdff(x). We often refer
to the support ofY as theY-space. Likewise, we refer to the range ofXas the
X-space.TheX-spaceis often referred to as thefactor space.

Least Squares and Wilcoxon Procedures

The first procedure isleast squares(LS). The estimating equation forβis given by
expression (9.6.4) of Chapter 9. Using the fact that

∑

ixci= 0, this equation can

be reexpressed as

∑n

i=1

(Yi−xciβ)xci=0. (10.9.32)

This is the estimating equation (EE) for the LS estimator ofβ, which we use in
this section. It is often called thenormal equation. It is easy to see that the LS