Exercise 13 (#5.23). Let (X1, ..., Xn) be a random sample from a distri- bution F and let 7(x) = P(8; = 1|X; = x), where d; = 1 if X; is observed and d; = 0 if X; is missing. Assume that 0 < T = [ T(x)dF(x) < 1. (i) Let F1(x) = P(X; < x|ô¡ = 1). Show that F and F1 are the same if and only if T(x) = T. (ii) Let F be the empirical distribution putting mass r-1 to each observed Xi, where r is the number of observed X;'s. Show that F(x) is unbiased and consistent for F1(x), x E R. (iii) When T(x) = T, show that F(x) in part (ii) is unbiased and consistent for F(x), x E R. When 7(x) is not constant, show that F(x) is biased and inconsistent for F(x) for some x E R. %3D
Exercise 13 (#5.23). Let (X1, ..., Xn) be a random sample from a distri- bution F and let 7(x) = P(8; = 1|X; = x), where d; = 1 if X; is observed and d; = 0 if X; is missing. Assume that 0 < T = [ T(x)dF(x) < 1. (i) Let F1(x) = P(X; < x|ô¡ = 1). Show that F and F1 are the same if and only if T(x) = T. (ii) Let F be the empirical distribution putting mass r-1 to each observed Xi, where r is the number of observed X;'s. Show that F(x) is unbiased and consistent for F1(x), x E R. (iii) When T(x) = T, show that F(x) in part (ii) is unbiased and consistent for F(x), x E R. When 7(x) is not constant, show that F(x) is biased and inconsistent for F(x) for some x E R. %3D
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Exercise 13 (#5.23). Let (X1, ..., Xn) be a random sample from a distri-
bution F and let 1(x) = P(8; = 1|X; = x), where d; = 1 if X; is observed
and d; = 0 if X; is missing. Assume that 0 < T = [ ¤(x)dF(x) < 1.
(i) Let F1(x) = P(X¡ < x|8; = 1). Show that F and F1 are the same if and
only if T(x) = T.
(ii) Let F be the empirical distribution putting mass r
Xi, where r is the number of observed X;'s. Show that F(x) is unbiased
and consistent for F1(x), x E R.
(iii) When 7(x) = 7, show that F(x) in part (ii) is unbiased and consistent
for F(x), x E R. When 1(x) is not constant, show that F(x) is biased and
inconsistent for F(x) for some x E R.
-1
to each observed
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