374 Maximum Likelihood MethodsCompare this with the variance of (n+1)Yn/n,whereYnis the largest observation
of a random sample of sizenfrom this distribution. Comment.6.2.3.Given the pdff(x;θ)=1
π[1 + (x−θ)^2 ]
, −∞<x<∞, −∞<θ<∞,show that the Rao–Cram ́er lower bound is 2/n,wherenis the size of a random sam-
ple from this Cauchy distribution. What is the asymptotic distribution of√
n(̂θ−θ)
if̂θis the mle ofθ?
6.2.4.Consider Example 6.2.2, where we discussed the location model.
(a)Write the location model wheneihas the logistic pdf given in expression
(4.4.11).(b)Using expression (6.2.8), show that the informationI(θ)=1/3forthemodel
in part (a). Hint: In the integral of expression (6.2.8), use the substitution
u=(1+e−z)−^1 .Thendu=f(z)dz,wheref(z) is the pdf (4.4.11).6.2.5. Using the same location model as in part (a) of Exercise 6.2.4, obtain the
ARE of the sample median to mle of the model.
Hint:The mle ofθfor this model is discussed in Example 6.2.7. Furthermore, as
shown in Theorem 10.2.3 of Chapter 10,Q 2 is asymptotically normal with asymp-
totic meanθand asymptotic variance 1/(4f^2 (0)n).
6.2.6. Consider a location model (Example 6.2.2) when the error pdf is the con-
taminated normal (3.4.17) with as the proportion of contamination and withσ^2 c
as the variance of the contaminated part. Show that the ARE of the sample median
to the sample mean is given by
e(Q 2 ,X)=2[1 + (σ^2 c−1)][1− +( /σc)]^2
π. (6.2.34)
Use the hint in Exercise 6.2.5 for the median.
(a)Ifσ^2 c= 9, use (6.2.34) to fill in the following table:0 0.05 0.10 0.15
e(Q 2 ,X)(b)Notice from the table that the sample median becomes the “better” estimator
when increases from 0.10 to 0.15. Determine the value for where this occurs
[this involves a third-degree polynomial in , so one way of obtaining the root
is to use the Newton algorithm discussed around expression (6.2.32)].6.2.7.Recall Exercise 6.1.1 whereX 1 ,X 2 ,...,Xnis a random sample onXthat
has a Γ(α=4,β=θ) distribution, 0<θ<∞.