and 0 < π < 1 (Z ∼ B(M, π)). We recall the following formulas

Final examen in Asymptotic Statistics Preparation 1 hour

Manuscript notes and handout of the lectures are allowed

1 Cheap Bin

Let Z be a random variable with binomial distribution of parameters M ∈ N

and 0 < π < 1 (Z ∼ B(M, π)). We recall the following formulas

E (Z ) = M π,

E (Z

) = M π(1 + (M − 1)π),

E (Z

) = M π(1 − 3π + 3M π + 2π

− 3M π

+ M

π

),

E (Z

) = M π(1 − 7π + 7M π + 12π

− 18M π

+ 6M

π

− 6π

+ 11M π

− 6M

π

+ M

π

).

For 0 < π

< 1, let X

, . . . , X

be an i.i.d. sample with common law B(1, √

π

). To estimate the parameter π

, we consider the maximum likelihood estimator b π.

1. Show that b π = X

. Here, as usual, X

denotes the empirical mean built on the sample X

, . . . , X

. Compute the two first moments of b π. Compute its mean square error :

R

(π

) := E [( π b − π

)

].

2. Modify b π to build an unbiased estimator b

b π. Compute R

bb π

(π

).

3. Show that √

n( b π − π

) and √

n(b b π − π

) converge both in distribution towards the same law. What is the limit law ?

4. What estimator should we use ? Why ? 5. Show that the statistical model is LAN. Is b

b π optimal ?

1