2. Suppose that a random variable Z has an unknown distribution with a mean of u...

  

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2. Suppose that a random variable Z has an unknown distribution with a mean of u and a variance of o. We take a sample of n i.i.d observations (2, 22,..., 22), calculate Zn-2 (the sample mean based on the first n-2 observations) and then construct defined by = (Zn-2) + (n-1 + n). a) Show that is an unbiased estimator of and then provide an intuitive explana- tion of why is not consistent for u. b) Find Var(), compare it with Var(Zn-2), and then discuss the efficiency of . Does attain the Cramer Rao lower bound? Briefly explain. Now let Z, Z2,... be a sequence of random variables, and suppose that for n = 1, 2, the probability distribution of Zn is given by Zn 0 n +5 Pr(Zn = Zn) 1 T c) Find an expression for E(Zn) and then briefly discuss why lim E(Zn) does not n-x exist. d) Briefly discuss why plim Zn =0. An intuitive argument is acceptable here. 2. Suppose that a random variable Z has an unknown distribution with a mean of u and a variance of o. We take a sample of n i.i.d observations (2, 22,..., 22), calculate Zn-2 (the sample mean based on the first n-2 observations) and then construct defined by = (Zn-2) + (n-1 + n). a) Show that is an unbiased estimator of and then provide an intuitive explana- tion of why is not consistent for u. b) Find Var(), compare it with Var(Zn-2), and then discuss the efficiency of . Does attain the Cramer Rao lower bound? Briefly explain. Now let Z, Z2,... be a sequence of random variables, and suppose that for n = 1, 2, the probability distribution of Zn is given by Zn 0 n +5 Pr(Zn = Zn) 1 T c) Find an expression for E(Zn) and then briefly discuss why lim E(Zn) does not n-x exist. d) Briefly discuss why plim Zn =0. An intuitive argument is acceptable here.

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a To show that is an unbiased estimator of we need to demonstrate that the expected value of equals the true population mean Lets calculate the expect... View the full answer