Question

2. Let X ~ Geometric (p) where 0 < p <1

a. Show explicitly that this family is “very regular,” that is, that R0,R1,R2,R3,R4 hold.

R 0 - different parameter values have different functions.

R 1 - parameter space does not contain its own endpoints.

R 2. - the set of points x where f (x, p) is not zero and should not depend on p.

R 3. One derivative can be found with respect to p.

R 4. Two derivatives can be found with respect to p.

b. Find the maximum likelihood estimator of p, call it Yn for this problem.

c. Is Yn unbiased? Explain.

d. Show that Yn is consistent asymptotically normal and identify the asymptotic normal variance.

e. Variance-stabilize your result in (d) or show there is no need to do so.

f. Compute I (p) where I is Fisher’s Information.

g. Compute the efficiency of Yn for p (or show that you should not!).

Answer 1

It is given that X ~ Geometric (p) where 0 < p <1

the defination of geometric distribution states;

"a random variable X is said to have geometric distribution if it assumes only non-negative values and its probability mass function is given by;

P(X=x)= q^xp ; x=0,1,2,3......... ; 0<p<1 ; q=1-p "

R 0 - according to its probability mass function, different values of p will give a different function.

R 1 - parameter space does not contain its own endpoints.

i.e.= for p=0;   P(X=x)= (1-0)^x.0= 0

for p=1; P(X=x)= (1-1)^x.1= 0

first derivative with respect to p=

  

second derivative with respect to p=

MLE of p=

f(X)= q^xp

liklihood function can be written as

by taking log both the sides

by differentiating with respect to p

by putting equal to zero

this is MLE of p.

again differentiating with respect to p

by taking negative expectation of it, it is equal to Fisher's information=