Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

394 Maximum Likelihood Methods

EXERCISES

6.4.1.A survey is taken of the citizens in a city as to whether or not they sup-

port the zoning plan that the city council is considering. The responses are: Yes,

No, Indifferent, and Otherwise. Letp 1 ,p 2 ,p 3 ,andp 4 denote the respective true

probabilities of these responses. The results of the survey are:

Yes No Indifferent Otherwise

60 45 70 25

(a)Obtain the mles ofpi,i=1,...,4.

(b)Obtain 95% confidence intervals, (4.2.7), forpi,i=1,...,4.

6.4.2.LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ymbe independent random samples from
N(θ 1 ,θ 3 )andN(θ 2 ,θ 4 ) distributions, respectively.

(a)If Ω⊂R^3 is defined by

Ω={(θ 1 ,θ 2 ,θ 3 ):−∞<θi<∞,i=1,2; 0<θ 3 =θ 4 <∞},

find the mles ofθ 1 ,θ 2 ,andθ 3.

(b)If Ω⊂R^2 is defined by

Ω={(θ 1 ,θ 3 ):−∞<θ 1 =θ 2 <∞;0<θ 3 =θ 4 <∞},

find the mles ofθ 1 andθ 3.

6.4.3.LetX 1 ,X 2 ,...,Xnbe iid, each with the distribution having pdff(x;θ 1 ,θ 2 )=
(1/θ 2 )e−(x−θ^1 )/θ^2 ,θ 1 ≤x<∞,−∞<θ 2 <∞, zero elsewhere. Find the maximum
likelihood estimators ofθ 1 andθ 2.

6.4.4.ThePareto distributionis a frequently used model in the study of incomes
and has the distribution function

F(x;θ 1 ,θ 2 )=

{

1 −(θ 1 /x)θ^2 θ 1 ≤x

0elsewhere,

whereθ 1 >0andθ 2 >0. IfX 1 ,X 2 ,...,Xnis a random sample from this distri-
bution, find the maximum likelihood estimators ofθ 1 andθ 2 .(Hint:This exercise
deals with a nonregular case.)

6.4.5. LetY 1 <Y 2 <···<Ynbe the order statistics of a random sample of
sizenfrom the uniform distribution of the continuous type over the closed interval
[θ−ρ, θ+ρ]. Find the maximum likelihood estimators forθandρ.Arethesetwo
unbiased estimators?

6.4.6.LetX 1 ,X 2 ,...,Xnbe a random sample fromN(μ, σ^2 ).