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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.

Relax

Respuesta :

Answer:

σˆ =

sPn

i=1 X2

i

2n

Step-by-step 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