2. Let X ~ Pois (λ) λ > 0 a. Show explicitly that this family is...

2. Let X ~ Pois (λ) λ > 0

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.

R1 - parameter space does not contain its own endpoints.

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

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

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

b. Find the maximum likelihood estimator of λ, 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 (λ) where I is Fisher’s Information.

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

Expert Solution

milcah answered 2 years ago

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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...

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