Berkeley UC Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 7 Fall 2007 Issued: Thursday, October 25 Reading: Keener: Chapter 10, 11. B & D: Chapter 5. Problem 7.1 Suppose that X1 , . . . , Xn are i.i.d. N (0, 2 ). (a) Show that (X ) =
C n n i=1 |Xi |
Due: Thursday, October 18
is a consistent estimator of if and only if C =
1 n n 2 i=1 Xi .
/2.
(b) Show that the MLE of is given by M LE (X ) = totic relative eciency of with c =
Determine the asymp-
/2 compared to the MLE M LE .
Problem 7.2 Suppose that X1 , . . . Xn are i.i.d from the normal location model N (, 1), and that we wish to estimate the critical or cuto value ga () = P[X1 a], where a R is some xed number. (a) Let denote the CDF of the standard normal distribution. Show that the estimator n (X ) = n (a Xn ) n1 d n(n ga ()) N (0, 2 (a )), where
is an unbiased estimator of ga (). Prove that is the PDF of the standard normal.
(b) If we had doubts about the assumption of normality, we might prefer to use the non1 parametric estimator n (X ) = n # {Xi a}, that simply counts the fraction of Xi that are less than or equal to a. Prove that d n(n ga ()) N (0, (a )[1 (a )]) . (c) Compute the ARE of and . If Xi truly are Gaussian, how much is lost by the non-parametric estimator ? Problem 7.3 Suppose that Xi , i = 1, . . . , n are i.i.d. samples from a normal location model N (, 1), and that we are interested in estimating the quantity 1/. In order to do so, we use the estimator 1 (X ) = 1/Xn where Xn = n n Xi is the sample mean. i=1 (a) Show that is asymptotically normalviz.: 1 d n 1/Xn 1/ N (0, 1/4 ).
(b) Show that the expectation E[1/Xn ] fails to exist for all n. Why does this not contradict the result of part (a)? Problem (Censored 7.4 Poisson) Suppose that Yi , i = 1, . . . , n are drawn i.i.d. from a Poi() distribution, but that we only observe the censored quantities Xi = 0 if Yi = 0 1 if Yi > 0.
(a) Under what conditions does the ML estimate M LE of based on (X1 , . . . , Xn ) exist? Compute the probability that the MLE exists as a function of and n. What happens as n +? (b) Dene the modied estimator (X ) = Prove that M LE (X ) 1 when the MLE exists otherwise.
d n( ) N (0, exp() 1).
Problem 7.5 Suppose that we have n i.i.d. samples {(Xi , Yi )}n from the bivariate normal distribution i=1 2 2 with zero means, variances X and Y and correlation coecient (1, 1).
2 2 (a) First supposing that X and Y are known constants, show that the MLE M LE satises
d n (M LE ) N
0,
(1 2 )2 1 + 2
.
2 2 Now suppose that X and Y , in addition to , are unknown parameters. In the following 2 2 sub-problems, we analyze the asymptotic behavior of the MLE (X , Y , ). (Note that 2 2 will not be the same, in general, as the MLE M LE from part (a) in which X and Y are assumed to be known.) 2 2 (b) Compute the 3 3 Fisher information matrix associated with = (X , Y , ).
(c) Verify that the inverse of the Fisher information matrix computed in (b) is given by 4 22 2 2X 22 X Y (1 2 )X 22 4 2 2Y (1 2 )Y [I ()]1 = 22 X Y 2 2 (1 2 )X (1 2 )Y (1 2 )2 (Hint: You do not need to try and compute this matrix inverse explicitly; simply compute the Fisher information matrix, and verify by matrix multiplication this form 2 2 of its inverse.) Use this result to derive asymptotic distributions for n(X X ), 2 2 n(Y Y ), and n( ). (d) Compute the asymptotic relative eciency (ARE) of M LE from (a) compared to . 2 2 What does this tell you about the value of knowing X and Y ? 2