Solve all 4 parts: (i)(ii)(iii)(iv), exact answer or downvote. bution F and let (x)=P(i=1Xi=x), where i=1 if Xi is observed and i=0 if Xi is missing. Assume that 0< =(x)dF(x)<1. (i) Let F1(x)=P(Xixi=1). Show that F and F1 are the same if and only if (x). (ii) Let F be the empirical distribution putting mass r1 to each observed Xi, where r is the number of observed Xi 's. Show that F^(x) is unbiased and consistent for F1(x),xR. (iii) When (x), show that F ^(x) in part (ii) is umbiased and consistent for F(x),xR. When (x) is not constant, show that F^(x) is biased and inconsistent for F(x) for some xR.(,2)=(2)n/2(2)n/2exp{221i=1n(Xi)2}. For fixed 2,(,2)(X, 2), since i=1n(Xi)2i=1n(XiX)2, where X is the sample mean. Hence the maximum does not depend on 2 and the profile likelihood function is (X,2)=(2)n/2(2)n/2exp{221i=1n(XiX)2}. By the result in the previous exereise, the profile MLE of 2 is the same as the MLE of 2. This can also be shown by directly verifying that (X,2) is maximized at ^2=n1i=1n(XiX)2. For fixed ,(,2) is maximized at 2()=n1i =1n(Xi)2. Then the profile likelihood function is (,2())=(2)n/2en/2[i=1n(Xi)2n]n/2.

1. Solve all 4 parts: (i)(ii)(iii)(iv), exact answer or downvote.
bution F and let (x)=P(i=1Xi=x), where i=1 if Xi is observed and i=0 if Xi is missing. Assume that 0<
=(x)dF(x)<1. (i) Let F1(x)=P(Xixi=1). Show that F and F1 are the same if and only if (x). (ii) Let F
be the empirical distribution putting mass r1 to each observed Xi, where r is the number of
observed Xi 's. Show that F^(x) is unbiased and consistent for F1(x),xR. (iii) When (x), show that F
^(x) in part (ii) is umbiased and consistent for F(x),xR. When (x) is not constant, show that F^(x) is
biased and inconsistent for F(x) for some xR.(,2)=(2)n/2(2)n/2exp{221i=1n(Xi)2}. For fixed 2,(,2)(X,
2), since i=1n(Xi)2i=1n(XiX)2, where X is the sample mean. Hence the maximum does not depend
on 2 and the profile likelihood function is (X,2)=(2)n/2(2)n/2exp{221i=1n(XiX)2}. By the result in the
previous exereise, the profile MLE of 2 is the same as the MLE of 2. This can also be shown by
directly verifying that (X,2) is maximized at ^2=n1i=1n(XiX)2. For fixed ,(,2) is maximized at 2()=n1i
=1n(Xi)2. Then the profile likelihood function is (,2())=(2)n/2en/2[i=1n(Xi)2n]n/2