1. Suppose that X ~Unif (0, 30) and we draw a random sample Xi,..., Xn Find...
1. Suppose that X Unif(0, 30) and we draw a random sample X1,..., Xn Find the MME and compute its relative efficiency to 6, = 2X1-3X2. 2. In class, I showed the below picture. Here, I have changed the vertical axis from variance to SD. In this new picture, how can we visualize the MSE? How does this way of seeing the MSE help us decide which of two (possibly biased) estimators is more efficient? SD 04 Bias (B) 0...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...
4. Let Xi,..., Xn be a random sample with density 303 for 0 < θ < x NOTE: We have previously found that θMLE-X(1) and that FX(1) (x)-1-(!)3m (a) Using the probability integral transform method, find a pivot for 0 based on the MLE. (b) Use the pivot found in (a) to get an ezact 100(1-a)% C.1. for θ (c) Find an approximate 100(1-a)% C.1. for θ based on our result for the MLE. (d) Suppose that we get n...
Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X...
1. Suppose that xi,..., xn are a random sample having probability density function f(x; δ)-¡0 otherwise (a) Determine the method of moments estimator of δ based on the first moment. (b) Determine the MLE of δ.
2. Let Xi,... ,Xn be a random sample from a distribution with p.d.f for 0 < x < θ f(x; 0) - 0 elsewhere . (a) Find an estimator for θ using the method of moments. (b) Find the variance of your estimator in (a).
1. Let Xi,..., Xn be a random sample from a distribution with p.d.f. f(x:0)-829-1 , 0 < x < 1. where θ > 0. (a) Find a sufficient statistic Y for θ. (b) Show that the maximum likelihood estimator θ is a function of Y. (c) Determine the Rao-Cramér lower bound for the variance of unbiased estimators 12) Of θ
3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2 < oo. Find cov(X,-x,x) for i 1,..,n. 3. [6 pts] Let Xi, . . . , Xn be a random sample from a distribution with variance σ2
Problem 8.2 Suppose that Xi, X,.., Xn is a random sample of size n is to be taken from a population with pdf 2 In>X (In2) x We are interested in determining the approximate distribution of the sample geometric mean given by [x. If we let Y-In X, then we can re-express the geometric mean as a) Determine the mean of Y. Hint, if u = In x, then du = 1/x dx. b) Determine the variance of Y. c)...
xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+ ), where θ e (-00, Exercise 7.5: Suppose X1, X2, . .. , sufficient for θ. a) Show that the smallest and largest of Xi, ..., Xn are jointliy (b) If p@-constant, θ e (-00, oo), is the prior distribution of θ, find its posterior distribution xercise 7.5: Suppose Xi, X2, ..., Xn are a random sample from the u distribution U(9-2 ,0+...