(c) Frequentist Estimation and Hypothesis Testing: Large Sample
7 points possible (graded, results hidden)
Now, suppose that we have observations with . Recall .
Compute the maximum likelihood estimate (MLE).
(Enter numerical answers accurate up to at least 3 decimal places.)
Compute the method of moments estimate.
(Enter numerical answers accurate up to at least 3 decimal places.)
Use the plug-in method to construct a confidence interval for of asymptotic confidence level centered around . Use the variance obtained from the asymptotic variance formula for the MLE and plug in for . Enter the lower and upper bounds of (the realization of) the confidence interval below.
(Enter numerical answers accurate up to at least 3 decimal places.)
where
Next, we decide to test the hypothesis : . Perform Wald's test using the test statistic
where is the Fisher information evaluated at .
Compute the p-value of Wald's test on our observations and model.
(Enter a numerical answer accurate to at least 3 decimal places.)
Another test we could use to test the same hypothesis is the Likelihood Ratio Test (LRT). Compute the p-value of the likelihood ratio test on our observations and model.
(Enter a numerical answer accurate to at least 3 decimal places.)
Suppose that we want our test to have asymptotic level . Decide whether Wald's test and/or the Likelihood Ratio Test would reject the null hypothesis .
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Frequentist Estimation and Hypothesis Testing: Large Sample (Ignore answers attempted, they are wrong)
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ABayes Is the confidence Region R(a,b) symmetric around the Bayesian estimatorA , as defined earlier? Why or Why not? (a)Yes, because by construction, confidence intervals and hence confidence regions are symmetric around any consistent of the parameter (b)Yes, because our posterior distribution is symmetric and we chose a and b such that (-o0, a) and (b,oo) have an equal 5% Probability. (c)No, because our posterior distribution is not symmetric (it is either skewed to the left or skewed to the...
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Thank you so much! 5. We have two independent samples of n observations X1,Xy,.. . ,x,...
Thank you so much!
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5. We have two independent samples of n observations X1,X2-…Xn and Yi,½, . …Ý, We want...
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4. Setup: Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d. draws from a Gaussian distribution with...
4.
Setup:
Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d.
draws from a Gaussian distribution with unknown mean μ and unknown
variance σ2.
Given Facts:
You are given the following:
15∑i=15Xi=0.90,15∑i=15X2i=1.31
Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
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2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...
ABayes Is the confidence Region R(a,b) symmetric around the Bayesian estimatorA , as defined earlier? Why or Why not? (a)Yes, because by construction, confidence intervals and hence confidence regions are symmetric around any consistent of the parameter (b)Yes, because our posterior distribution is symmetric and we chose a and b such that (-o0, a) and (b,oo) have an equal 5% Probability. (c)No, because our posterior distribution is not symmetric (it is either skewed to the left or skewed to the...
Consider a sample of i.i.d. random variables X1,..., X, and assume their common density is given by fo(a) = exp (3) 1(220), where is an unknown parameter Consider the following set of hypotheses: H:0=1 and H:0 1. You will perform the likelihood ratio test at significance level 7% for these hypotheses. Assume that the sample size is n=500. You observe that • The sample mean is 1 , X; = 0.86: • The (biased) sample variance is 15. (X- Xn)...
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...
5. We have two independent samples of n observations X1, X2, .. . , Xn and Yı, Y2,.. . , Yn We want to test the hvpothesis H 0 : μΧ-My versus the alternative H1 : μΧ * My (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ac-My (b) Then plug this estimate into the log likelihood along with the MLE's μχ-x and My-- to calculate the LRT statistic (c) Is this likelihood...
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...
5. We have two independent samples of n observations X1,X2-…Xn and Yi,½, . …Ý, We want to test the hypothesis H0 : μ®-ty versus the alternative H1 : μζ μυ. (a) First, assume that the null hypothesis H0 is true and find the MLE for μ- - y. (b) Then plug this estimate into the log likelihood along with the MLBs μτ-x and μ to calculate the LRT statistic. (c) Is this likelihood ratio test equivalent to the test that...
5. We have two independent samples of n observations X1, X2,... , Xn and Yi, Y2,..., Y, We want to test the hypothesis Ho : μ®-,ty versus the alternative H, : μ*-t ,ty. (a) First, assume that the null hypothesis Ho is true and find the MLE for μ-Ae-μΥ. (b) Then plug this estimate into the log likelihood along with the MLE's μΧ-x and My to calculate the LRT statistic. (c) Is this likelihood ratio test equivalent to the test...
4.
Setup:
Suppose you have observations X1,X2,X3,X4,X5 which are i.i.d.
draws from a Gaussian distribution with unknown mean μ and unknown
variance σ2.
Given Facts:
You are given the following:
15∑i=15Xi=0.90,15∑i=15X2i=1.31
Bookmark this page Setup: Suppose you have observations X1, X2, X3, X4, X5 which are i.i.d. draws from a Gaussian distribution with unknown mean u and unknown variance o? Given Facts: You are given the following: x=030, =1:1 Choose a test 1 point possible (graded, results hidden) To test...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
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