Question

Estimators' Properties A researcher knows a random variable X has E[X] = and V[X] = o both finite. Here the researcher knowns the true value for ji (so they don't have to estimate it) but does not know the value for o? and wants to estimate it. a) Consider the following estimator: ö: 15X; -M)a a) Use the weak law of large numbers to construct the probability limit of o?. Is õ? a consistent estimator for o?? b) Is õ? an unbiased estimator for the population variance o?? Justify your answer.

Answer 1

solm 20%-M)? lino If 02 ga 2 follows WLLN then LCOM) and VC62) 70 When my w thon nit (819) = 42xi-pija) As onyn, 2 (x - ² ~ June F( 26X1-M)? thom in Lay as n you 0 ? F(69) n

= 1 (Mat) o n o => 0-2 and ( n = van (ona) voor (xi-mya) van 2 (xi-M)2) Y2 Lorvar vanil (xi-M) 2 11 ท or g2 an 20?->0 N2 N as nai so on scitisfies the property of WWLLN hence o 2 is consistent estimator for o? per f Theonem :- til P[1027213&]->0 iff ExYA E(227-72 and van (012) 10] Using say probability limit of 042 18 0°. > 0 this theonem We can

Б) * Et++) = ( 2 *i -9° ) = F( 2 кг + mr an151) — ЕСхі?) +ECH+) N nisi — 3|— 2 الحصص 2 ви | TE(xi ) Үля схі) + Etsiy1] * * - архи { o? + * + *- АН- (no*+ mn*) +- - criven: Elxi)=M V (x) = 02 12 | | т П о лп 0 Нет се an is unbiased estimator A raniance or population