(4) Suppose that {X;}-1 iid random variables from a Binomial distribution Bin(m, p). Using your answer...
Suppose that X1,X2,. X are iid random variables with pdf ,220 (a) Find the maximum likelihood estimate of the parameter a (b) Find the Fisher Information of X1,X2,.. ., Xn and use it to estimate a 95% confidence interval on the MLE of a (c) Explain how the central limit theorem relates to (b).
Suppose that Xi, X2Xn are iid random variables with pdf eter a. (b) Find the Fisher Information of X1,X2, ,Xn and use it to estimate a 95% confidence interval on the MLE of a. (c) Explain how the central limit theorem relates to (b).
Suppose that {X}}=1 are iid random variables uniformly distributed random variables with density fr A f(x; 0) = S (0 – 10)- € (10,0) 0 otherwise (i) Derive the MLE of e. (ii) Obtain the asymptotic sampling properties of 0. Is the distribution of the MLE asymptotically normal?
3. Suppose that ai . ,,an are a random sample from a N( ,02) distribution. Recall that the MLE in this case is [a, σ]T = [x, V (n-1)s2/n]T and the information matrix is Consider the data s2-4.84 with n 16 (a) Use the delta-method to obtain an approximate 95% confidence interval for log(o) (b) Obtain an approximate 95% confidence interval for σ2 using the confidence interval from (a). Compare to the exact interval, [2.21,15.77], and approximate interval [0.43, 10.50...
please answer 3). Assume two random variables X and Y both follow the binomial distribution Bin(n, p). Does X+Y also follow a binomial distribution? Why?
Question 3: Let X1,..., X.be iid Poisson (2) random variables. a. Find the maximum likelihood estimate for X. b. Obtain the Fisher expected information. c. Obtain the observed information evaluated at the maximum likelihood estimate. d. For large n, obtain a 95% confidence interval for based on the Central Limit Theorem. e. Repeat part (a), but use the Wald method. f. Repeat part (d), but use the Score method. 8. Repeat part (a), but use the likelihood ratio method.
- Suppose that the binomial distribution parameter a is to be estimated by P = X/n, where X is the number of successes in n independent trials, i.e. P is the sample proportion of successes. i. Write down the endpoints of an approximate 100(1 – a)% confidence interval for at, stating any necessary conditions which should be satisfied for such an approximate confidence interval to be used. You should also state the approximate sampling distribution of P = X/n. ii....
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ), derive the randomized UMP level α test for l, P-Po p reassigned oE versus H p Po where p, is a number between 0 and 1 8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ),...
S. 14 points] Suppose random variables X, x.. X, are iid (independent, identical distributed) with density function (with 0>0 and x20) (a) Find the MLE (maximal likelihood estimate) for the parameter 9 (b) Explain (without mathematical proof) how will you check whether your estimate ,f(x) = 3ar'e® in (a) is an unbiased estimate or not?
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...