In Exercise 7.8 you showed that p n b 2 2 ! d N(0,V ) as n !1 for some V . Let b

In Exercise 7.8 you showed that p

n

¡

b¾

2 ¡¾2¢

¡!

d N(0,V ) as n !1 for some V . Let b V be an estimator of V .

(a) Using this result construct a t-statistic for H0 : ¾2 Æ 1 against H1 : ¾2 6Æ 1.

(b) Using the DeltaMethod find the asymptotic distribution of p

n (b¾

¡¾).

(c) Use the previous result to construct a t-statistic for H0 : ¾ Æ 1 against H1 : ¾ 6Æ 1.

(d) Are the null hypotheses in

(a) and

(c) the same or are they different? Are the tests in

(a) and

(c) the same or are they different? If they are different, describe a context in which the two tests would give contradictory results.