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 shouldnot 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 forthis problem.
c) Is Yn unbiased? Explain.
d) Show that Yn is consistent asymptotically normal and identifythe asymptotic normal variance.
e) Variance-stabilize your result in (d) or show there is noneed 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 shouldnot!).
