Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

6.1. Maximum Likelihood Estimation 361

is a mle ofθ. In particular, (4Y 1 +2Yn+1)/ 6 ,(Y 1 +Yn)/2, and (2Y 1 +4Yn−1)/ 6

are three such statistics. Thus, uniqueness is not, in general, a property of mles.

6.1.4.SupposeX 1 ,...,Xnare iid with pdff(x;θ)=2x/θ^2 , 0 <x≤θ, zero
elsewhere. Note this is a nonregular case. Find:

(a)The mleθˆforθ.

(b)The constantcso thatE(cˆθ)=θ.

(c)The mle for the median of the distribution. Show that it is a consistent

estimator.

6.1.5.Consider the pdf in Exercise 6.1.4.

(a)Using Theorem 4.8.1, show how to generate observations from this pdf.

(b)The following data were generated from this pdf. Find the mles ofθand the

median.

1.2 7.7 4.3 4.1 7.1 6.3 5.3 6.3 5.3 2.8

3.8 7.0 4.5 5.0 6.3 6.7 5.0 7.4 7.5 7.5

6.1.6.SupposeX 1 ,X 2 ,...,Xnare iid with pdff(x;θ)=(1/θ)e−x/θ, 0 <x<∞,
zero elsewhere. Find the mle ofP(X≤2) and show that it is consistent.

6.1.7.Let the table

x 01 2 345

Frequency 610141361

represent a summary of a sample of size 50 from a binomial distribution having
n=5. FindthemleofP(X≥3). For the data in the table, using the R function
pbinomdetermine the realization of the mle.

6.1.8.LetX 1 ,X 2 ,X 3 ,X 4 ,X 5 be a random sample from a Cauchy distribution with
medianθ,thatis,withpdf

f(x;θ)=

1

π

1

1+(x−θ)^2

, −∞<x<∞,

where−∞<θ<∞. Supposex 1 =− 1. 94 ,x 2 =0. 59 ,x 3 =− 5. 98 ,x 4 =− 0. 08 ,
andx 5 =− 0 .77.

(a)Show that the mle can be obtained by minimizing

∑^5

i=1

log[1 + (xi−θ)^2 ].