A weighted sample mean takes the form X n 1n Pni 1wi Xi for some

A weighted sample mean takes the form X

¤

n Æ 1n Pni

Æ1wi Xi for some non-negative constants wi satisfying 1n Pni

Æ1wi Æ 1. Assume Xi is i.i.d.

(a) Show that X

¤

n is unbiased for ¹ Æ E[X] .

(b) Calculate var h

X

¤

n i

.

(c) Show that a sufficient condition for X

¤

n ¡!

p

¹ is that n¡2Pni

Æ1w2 i

!0.

(d) Show that a sufficient condition for the condition in part 3 is maxi·n wi !0 as n!1.