Question

Let X1, · · · , Xn be independent identically distributed random variables with probability density function f(x) = 1 2σ exp − |x| σ , −[infinity] < x < [infinity] where σ > 0 is some unknown parameter. This is known as the Laplace distribution or double exponential distribution.

a. Find the moment estimator of σ.
b. Find the maximum likelihood estimator of σ.

Answer 1

σˆ =

sPn

i=1 X2

i

2n

Explanation:

To obtain, the maximum likelihood ratio, we use the following method.

l(σ) = Xn

i=1 "

− log 2 − log σ −

|Xi

|

σ

#

Let the derivative with respect to θ be zero:

l

0

(σ) = Xn

i=1 "

−

1

σ

+

|Xi

|

σ

2

#

= −

n

σ

+

Pn

i=1 |Xi

|

σ

2

= 0

and this gives us the MLE for σ as

σˆ =

Pn

i=1 |Xi

|

n

Again this is different from the method of moment estimation which is

σˆ =

sPn

i=1 X2

i

2n

As our answer