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For problems 2-7, consider X, X,, ..., X, to be independent random variables from a Normal (u,o) where both parameters are unknown. (As you might imagine, the solutions to this are available in almost any mathematical statistics text.) 2. Find maximum likelihood estimators of both parameters. Call these two estimators u and 0? (You may skip the step of showing that the critical values are a maximum, because that's somewhat harder for these two-dimensional problems than I intend for you to try to do now.) Recall the definition of Bias. Bias(U) = E(U)–0. An unbiased estimator of a parameter 0 is an estimator for which the bias is zero. 3. Is the MLE of u an unbiased estimator? Show your work. 4. Show that the MLE of o is not an unbiased estimator of o?. 5. Find a linear function of that MLE of o? which IS an unbiased estimator of o?. (If we call our MLE of the name 0 , this says: find some constants a and b so that E(a-ô +b)=o² ) We will call this estimator “theta-squared-tildle" ở. (We also know it as S.) (One can't do that for all similar problems, but for this problem you can. ) The estimator obtained here is one you are quite used to using. Do you recognize it? (We know it as S.) 6. What is the distribution of the estimator u ? (n–1)ð? 7. One can prove that (That proof is beyond the scope of this course. It is, however, the basis of the last half of the theoretical dist'ns on our statistical formulas document.)

For problems 2-7, consider X, X,, ..., X, to be independent random variables from a Normal (u,o) where both parameters are unknown. (As you might imagine, the solutions to this are available in almost any mathematical statistics text.) 2. Find maximum likelihood estimators of both parameters. Call these two estimators u and 0? (You may skip the step of showing that the critical values are a maximum, because that's somewhat harder for these two-dimensional problems than I intend for you to try to do now.) Recall the definition of Bias. Bias(U) = E(U)–0. An unbiased estimator of a parameter 0 is an estimator for which the bias is zero. 3. Is the MLE of u an unbiased estimator? Show your work. 4. Show that the MLE of o is not an unbiased estimator of o?. 5. Find a linear function of that MLE of o? which IS an unbiased estimator of o?. (If we call our MLE of the name 0 , this says: find some constants a and b so that E(a-ô +b)=o² ) We will call this estimator “theta-squared-tildle" ở. (We also know it as S.) (One can't do that for all similar problems, but for this problem you can. ) The estimator obtained here is one you are quite used to using. Do you recognize it? (We know it as S.) 6. What is the distribution of the estimator u ? (n–1)ð? 7. One can prove that (That proof is beyond the scope of this course. It is, however, the basis of the last half of the theoretical dist'ns on our statistical formulas document.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Consider  X₁,  X₂, . . . ,  Xn   to be independent random variables from a Normal(μ,σ ² ) where both parameters are unknown.  

For problems 2-7, consider X, X,, ..., X, to be independent random variables from a 19 Normal (u,0 ) where both parameters are unknown. (As you might imagine, the solutions to this are available in almost any mathematical statistics text.) 2. Find maximum likelihood estimators of both parameters. Call these two estimators u and 0? (You may skip the step of showing that the critical values are a maximum, because that's somewhat harder for these two-dimensional problems than I intend for you to try to do now.) Recall the definition of Bias. Bias(U)= E(U)-0. An unbiased estimator of a parameter 0 is an estimator for which the bias is zero. 3. Is the MLE of u an unbiased estimator? Show your work. 4. Show that the MLE of o is not an unbiased estimator of o? 5. Find a lincar function of that MLE of o which IS an unbiased estimator of o (If we call our MLE of the name 0 , this says: find some constants a and b so that E a-O +b)=o) We will call this estimator "theta-squared-tildle" ô² . (We also know it as S'.) (One can't do that for all similar problems, but for this problem you can. ) The estimator obtained here is one you are quite used to using. Do you recognize it? (We know it as S'.) 6. What is the distribution of the estimator û ? (n– 1)õ² 7. One can prove that (That proof is beyond the scope of this course. It is, ~ n-1 however, the basis of the last half of the theoretical dist'ns on our statistical formulas document.)
Transcribed Image Text:For problems 2-7, consider X, X,, ..., X, to be independent random variables from a 19 Normal (u,0 ) where both parameters are unknown. (As you might imagine, the solutions to this are available in almost any mathematical statistics text.) 2. Find maximum likelihood estimators of both parameters. Call these two estimators u and 0? (You may skip the step of showing that the critical values are a maximum, because that's somewhat harder for these two-dimensional problems than I intend for you to try to do now.) Recall the definition of Bias. Bias(U)= E(U)-0. An unbiased estimator of a parameter 0 is an estimator for which the bias is zero. 3. Is the MLE of u an unbiased estimator? Show your work. 4. Show that the MLE of o is not an unbiased estimator of o? 5. Find a lincar function of that MLE of o which IS an unbiased estimator of o (If we call our MLE of the name 0 , this says: find some constants a and b so that E a-O +b)=o) We will call this estimator "theta-squared-tildle" ô² . (We also know it as S'.) (One can't do that for all similar problems, but for this problem you can. ) The estimator obtained here is one you are quite used to using. Do you recognize it? (We know it as S'.) 6. What is the distribution of the estimator û ? (n– 1)õ² 7. One can prove that (That proof is beyond the scope of this course. It is, ~ n-1 however, the basis of the last half of the theoretical dist'ns on our statistical formulas document.)
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