12. Let Xi, ,X, be lID N(?,78), where both parameters ? and ? are positive. a)...
Suppose that Xi, X2, ..., Xn is an iid sample from the distribution with density where θ > 0. (a) Find the maximum likelihood estimator (MLE) of θ (b) Give the form of the likelihood ratio test for Ho : θ-Bo versus H1: θ > θο. (c) Show that there is an appropriate statistic T - T(X) that has monotone likelihood ratio. (d) Derive the uniformly most powerful (UMP) level α test for versusS You must give an explicit expression...
, Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus , Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
Thank you so much! 5. We have two independent samples of n observations X1,Xy,.. . ,x, and Y. Ya, . … We want to test the hypothesis Ho : μ,-μυ versus the alternative Hi : μ, μν. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ (b) Then plug this estimate into the log likelihood along with the MLE μ'.. and 1,-j) to calculate the LRT statistic. (e) Is this likelihood ratio test...
2. Let Xi, , Х, be a random sample gamma(a, β). In parts (a-(d) assume a is known. 30 points a. Consider testing H. : β--βο. Derive Wald statistic for testing H, using the MLE of B both in the numerator and denominator of the statistic. b. Derive a test statistic for testing H, using the asymptotic distribution of the MLE of β. What is the relation between the two statistics in parts (a) and (b)? c. Derive the Score...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...
5. we have two independent samples of n observations X1,X2, ,x, and Yi, ½, ,y, We want to test the hypothesis Ho M Hy versus the alternative Hi: Hr y (a) First, assume that the null hypothesis Ho is true and find the MLE for μ Ha-μυ (b) Then plug this estimate into the log likelihood along with the MLE's μ--x and μυ-D to calculate the LRT statistic (c) Is this likelihood ratio test equivalent to the test that rejects...
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a laboratory specimen from the ith individual, with μί representing the amount of some antigen in the specimen; the measuring instrument is inaccurate, with normally distributed errors with constant variability. Let Z, X/V2. (a) Consider the estimate σ2- (b) Show that the...
1. We have n observations xi that are і. i. d. from a Normal distribution with mean-0 and unknown variance. We want to test using a Generalized Likelihood Ratio Test. Calculate the test statistic T for the GLRT. You can assume that the MLE for the variance is 2 i-1
1, we have n observations xi that are i. i. d. from a Normal distribution with mean= 0 and unknown variance. We want to test using a Generalized Likelihood Ratio Test. Calculate the test statistic T for the GLRT. You can assume that the MLE for the variance is Tt 2 -1