Consider the continuous probability distribution f ( x ). Suppose that is an unknown location parameter and that the density may be written as

Consider the continuous probability distribution f ( x ). Suppose that θ is an unknown location parameter and that the density may be written as f ( x − θ )

for −∞ < θ < ∞ . Let x1 , x2 , . . . , x n be a random sample of size n from the density.

a. Show that the maximum - likelihood estimator of θ is the solution to

ψ θ xi i

n

( ) − =

=

∑1 0

that maximizes the logarithm of the likelihood function In L fx i n ( ) μ θ =∑ − =1 ( ) i ln , where ψ ( x ) = ρ′ ( x ) and ρ ( x ) = − ln f ( x ).

b. If f ( x ) is a nonmal distribution, fi nd ρ ( x ), ψ ( x ) and the corresponding maximum - likelihood estimator of θ .

c. If f ( x ) = (2 σ ) − 1 e−

|

x

|

/σ

(the double - exponential distribution), fi nd ρ ( x ) and

ψ ( x ). Show that the maximum - likelihood estimator of θ is the sample median. Compare Ibis estimator with the estimator found in part

b. Does the sample median seem to be a reasonable estimator in this case?

d. If f ( x ) = [ π (1 + x2

)] − 1

(the Cauchy distribution), fi nd ρ ( x ) and ψ ( x ). How would you solve ∑ − i= ( ) n 1ψ θ xi in this case?