I need the solutions for those statistical math problems.
1. Suppose that X1, …., Xn is a random sample from a distribution whose density for xi is a) Derive the probability density function(pdf) of the MLE of θ. b) Compute the MSE (mean squared error) for the MOM of θ. c) Find the limiting distribution of √n(X̅ - c) for an appropriate constant c. d. Let M = 1 n ∑ log Xi n i=1 . Find an appropriate expression for a and b such that (M -a) /b ~ N(0,1) approximately. (You do not need to obtain/evaluate any values of a and b.) 2. 3. 4. 5. 6. 7. See the Question 3(c) table. 8. Let X1, …., Xn be i.i.d from a distribution with probability density function (pdf) f (x: θ ) = θxθ−1, 0 < x="">< 1,="" θ=""> 0 a. Find the method of moments estimator of θ. b. Find the Fisher information of θ . 9. Let X1, …., Xn be a random sample from a beta (?, ?) distribution. For Y ~ Beta (?, ?) , f (y: ?, ? ) = Γ(?+?) Γ(?)Γ(?) y?-1(1-y)?-1, 0 < y="">< 1,=""> 0, ? > 0 E[Y] = ? ?+? , Var[Y]= ?? (?+?)2(?+?+1) Γ(?) = (?-1)!. a. Find the limiting distribution of √n(X̅ - c) for an appropriate constant c. b. Let M = M = 1 n ∑ log Xi n i=1 . Find an appropriate expression for a and b such that approximately. (You do need to obtain / evaluate any value any values of d and f). 10. Let Y1, …., Yn be a random sample from a normal distribution with mean 0 and variance ?2. f (?i: ? 2) = 1 √2?? exp(- ?i 2 2?2 ), -∞ < i="">< ∞,="" 2=""> 0 a. Construct a statistic whose sampling distribution is a t-distribution with 3 degrees of freedom. b. Find the rejection rule (critical region ) of the UPM (uniformly most powerful) test for H0: ? = 1 vs H1: ? > 1. c. If n = 20 and the significance level ? = 0.10 , determine explicitly the constant value in the rejection region using the table below. df ? 0.01 0.05 0.1 0.25 0.75 0.9 0.95 0.99 20 8.260 10.851 12.443 15.452 23.828 28.412 31.410 37.566 30 14.953 18.493 20.599 24.478 34.800 40.256 43.773 50.892 40 22.164 26.509 29.051 33.660 45.616 51.805 55.758 63.691 d. What is the power of the above UPM test if ?2 = 2.8847? i.e what is the probability of rejecting H0 when ? 2 = 2.8847? 11. Y1, …., Yn is a random sample from Poisson distribution with mean ?, f (y: ? ) = ?ye-? y! , y=0,1,2,… E(Y) = ? = Var(Y). a. Drive the score test statistic for testing H0: ? = ? 0 vs H1: ? ≠ ? 0 U( ? 0 ) = S2( ? 0: Y) I( ? 0: Y) , where S( ? 0: Y) = ? ? ? logL(? ∶ Y) b. Find a large sample (approximate) pivot for based on the sufficient statistic using the central limit theorem (CLT). c. Give the upper and lower limits for the approximate 95% confidence interval for using the pivot found in . d. In c), suppose that the confidence interval estimates computed from samples of size 100 were (1.65 , 3.24). Briefly explain the meaning of this obtained interval. 12. Suppose that X and Y are independent exponentially distributed random variables with E(X) =? and E(Y) = ? . The pdf of X is f (x: ? ) = 1 ? exp(- x ? ), x > 0 a. Find the MLEs of ? and ? , respectively, (There are called the unrestricted MLEs). b. Find the restricted MLEs under H0:? = ?. c. Show that the likelihood ratio test of H0:? = ? vs. H1:? ≠ ? rejects the null hypothesis whenever T(1-T)< k , where t=x/(x+y) and k is a constant. k="" ,="" where="" t="X/(X+Y)" and="" k="" is="" a="">