3.21 Let X1, . . . , Xn be independent Exponential(1) random variables. (i) Prove the identity...

Question:

3.21 Let X1, . . . , Xn be independent Exponential(1) random variables.

(i) Prove the identity 1

n i=1 Xi

= 1 n

−

n i=1 Xi − n n2

+ (

n i=1 Xi − n)2 n3

− (

n i=1 Xi − n)3 n3

n i=1 Xi

(ii) Show that

n i=1 X2

i n

i=1 Xi

= 1 n

n i=1 X2 i

− 1 n2

n i=1 X2 i

n i=1 Xi − n

+ 1 n3

n i=1 X2 i

n i=1 Xi − n

+ OP(n

−3/2).

(iii) What are the orders of the first three terms on the right side of the equation in (ii)?

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