Bayes Estimator Lecture 15 Biostatistics 602 - Statistical - PowerPoint PPT Presentation

. . March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang March 12th, 2013 Hyun Min Kang Bayes Estimator Lecture 15 Biostatistics 602 - Statistical Inference . . Summary Conjugate Family . Bayes Estimator Bayesian Statistics Recap . . . . . . . . 1 / 26 . . . . . . . . . . . . . . . . . . . .

• When Cramer-Rao bound is attainable, can Cramer-Rao bound be • What is another way to find the best unbiased estimator? • Describe two strategies to obtain the best unbiased estimators for . . March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang , using complete sufficient statistics. ? restriction on ? If not, what is the used for find best unbiased estimator for any If not, in which cases? for any distribution? Last Lecture Summary . Conjugate Family Bayes Estimator Bayesian Statistics Recap . . . . . . . . 2 / 26 . . . . . . . . . . . . . . . . . . . . • Can Cramer-Rao bound be used to find the best unbiased estimator

• When Cramer-Rao bound is attainable, can Cramer-Rao bound be • What is another way to find the best unbiased estimator? • Describe two strategies to obtain the best unbiased estimators for . Last Lecture March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang , using complete sufficient statistics. ? restriction on ? If not, what is the used for find best unbiased estimator for any for any distribution? If not, in which cases? Summary . . Conjugate Family Bayes Estimator Bayesian Statistics Recap . . . . . . . . 2 / 26 . . . . . . . . . . . . . . . . . . . . • Can Cramer-Rao bound be used to find the best unbiased estimator

• What is another way to find the best unbiased estimator? • Describe two strategies to obtain the best unbiased estimators for . Conjugate Family March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang , using complete sufficient statistics. for any distribution? If not, in which cases? Last Lecture Summary . . Bayes Estimator Bayesian Statistics Recap . . . . . . . . 2 / 26 . . . . . . . . . . . . . . . . . . . . • Can Cramer-Rao bound be used to find the best unbiased estimator • When Cramer-Rao bound is attainable, can Cramer-Rao bound be used for find best unbiased estimator for any τ ( θ ) ? If not, what is the restriction on τ ( θ ) ?

• Describe two strategies to obtain the best unbiased estimators for . Conjugate Family March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang , using complete sufficient statistics. for any distribution? If not, in which cases? Last Lecture Summary . . 2 / 26 . . . . Bayesian Statistics . Recap . Bayes Estimator . . . . . . . . . . . . . . . . . . . . . . • Can Cramer-Rao bound be used to find the best unbiased estimator • When Cramer-Rao bound is attainable, can Cramer-Rao bound be used for find best unbiased estimator for any τ ( θ ) ? If not, what is the restriction on τ ( θ ) ? • What is another way to find the best unbiased estimator?

. Bayes Estimator March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang for any distribution? If not, in which cases? Last Lecture Summary . Conjugate Family . 2 / 26 . Bayesian Statistics Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . • Can Cramer-Rao bound be used to find the best unbiased estimator • When Cramer-Rao bound is attainable, can Cramer-Rao bound be used for find best unbiased estimator for any τ ( θ ) ? If not, what is the restriction on τ ( θ ) ? • What is another way to find the best unbiased estimator? • Describe two strategies to obtain the best unbiased estimators for τ ( θ ) , using complete sufficient statistics.

. . March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang its expected value. . . Theorem 7.3.23 . Recap - The power of complete sufficient statistics Summary . Conjugate Family Bayes Estimator Bayesian Statistics Recap . . . . . . . . 3 / 26 . . . . . . . . . . . . . . . . . . . . Let T be a complete sufficient statistic for parameter θ . Let φ ( T ) be any estimator based on T . Then φ ( T ) is the unique best unbiased estimator of

• It helps to confirm an estimator is the best unbiased estimator of • If an unbiased estimator of • There may be unbiased estimators of . I n has variance greater than the CR-bound, it does NOT mean that it is not the best unbiased estimator. . . 2 When ”regularity conditions” are not satisfied, valid lower bound. is no longer a that have variance smaller than I n . Hyun Min Kang Biostatistics 602 - Lecture 15 March 12th, 2013 if it happens to attain the CR-bound. 1 If ”regularity conditions” are satisfied, then we have a Cramer-Rao . Bayes Estimator . . . . . . . . Recap Bayesian Statistics 4 / 26 Conjugate Family . Summary Finding UMUVE - Method 1 . . . . . . . . . . . . . . . . . . . . . . . . Use Cramer-Rao bound to find the best unbiased estimator for τ ( θ ) . bound for unbiased estimators of τ ( θ ) .

• If an unbiased estimator of • There may be unbiased estimators of . I n has variance greater than the CR-bound, it does NOT mean that it is not the best unbiased estimator. . . 2 When ”regularity conditions” are not satisfied, valid lower bound. is no longer a . that have variance smaller than I n . Hyun Min Kang Biostatistics 602 - Lecture 15 March 12th, 2013 if it happens to attain the CR-bound. 1 If ”regularity conditions” are satisfied, then we have a Cramer-Rao . Bayes Estimator . . . . . . . . Recap Bayesian Statistics 4 / 26 Conjugate Family . Summary Finding UMUVE - Method 1 . . . . . . . . . . . . . . . . . . . . . . . Use Cramer-Rao bound to find the best unbiased estimator for τ ( θ ) . bound for unbiased estimators of τ ( θ ) . • It helps to confirm an estimator is the best unbiased estimator of τ ( θ )

• There may be unbiased estimators of . is no longer a if it happens to attain the CR-bound. CR-bound, it does NOT mean that it is not the best unbiased estimator. . . 2 When ”regularity conditions” are not satisfied, I n valid lower bound. . that have variance smaller than I n . Hyun Min Kang Biostatistics 602 - Lecture 15 March 12th, 2013 . 1 If ”regularity conditions” are satisfied, then we have a Cramer-Rao . Bayes Estimator . . . . . . . . Recap Bayesian Statistics 4 / 26 Conjugate Family . Summary Finding UMUVE - Method 1 . . . . . . . . . . . . . . . . . . . . . . Use Cramer-Rao bound to find the best unbiased estimator for τ ( θ ) . bound for unbiased estimators of τ ( θ ) . • It helps to confirm an estimator is the best unbiased estimator of τ ( θ ) • If an unbiased estimator of τ ( θ ) has variance greater than the

• There may be unbiased estimators of valid lower bound. . if it happens to attain the CR-bound. CR-bound, it does NOT mean that it is not the best unbiased estimator. . . is no longer a . . that have variance smaller than I n . Hyun Min Kang Biostatistics 602 - Lecture 15 March 12th, 2013 . 1 If ”regularity conditions” are satisfied, then we have a Cramer-Rao 4 / 26 . Bayesian Statistics Bayes Estimator . . . . . . Conjugate Family Summary . Finding UMUVE - Method 1 . . . Recap . . . . . . . . . . . . . . . . . . . . Use Cramer-Rao bound to find the best unbiased estimator for τ ( θ ) . bound for unbiased estimators of τ ( θ ) . • It helps to confirm an estimator is the best unbiased estimator of τ ( θ ) • If an unbiased estimator of τ ( θ ) has variance greater than the 2 When ”regularity conditions” are not satisfied, [ τ ′ ( θ )] 2 I n ( θ )

. . March 12th, 2013 Biostatistics 602 - Lecture 15 Hyun Min Kang valid lower bound. is no longer a . . estimator. CR-bound, it does NOT mean that it is not the best unbiased if it happens to attain the CR-bound. 1 If ”regularity conditions” are satisfied, then we have a Cramer-Rao . . . . Finding UMUVE - Method 1 Bayesian Statistics . . . . . . . . Recap Summary 4 / 26 Bayes Estimator Conjugate Family . . . . . . . . . . . . . . . . . . . . . Use Cramer-Rao bound to find the best unbiased estimator for τ ( θ ) . bound for unbiased estimators of τ ( θ ) . • It helps to confirm an estimator is the best unbiased estimator of τ ( θ ) • If an unbiased estimator of τ ( θ ) has variance greater than the 2 When ”regularity conditions” are not satisfied, [ τ ′ ( θ )] 2 I n ( θ ) • There may be unbiased estimators of τ ( θ ) that have variance smaller than [ τ ′ ( θ )] 2 I n ( θ ) .

• Guess a function • Guess an unbiased estimator h X of . . 2 Obtain T , an unbiased estimator of using either of the following two ways T such that E T . . . Construct T E h X T , then E T E h X . Hyun Min Kang Biostatistics 602 - Lecture 15 March 12th, 2013 . 1 Find complete sufficient statistic T for . . . . . . . . . . Recap Bayesian Statistics Bayes Estimator 5 / 26 Conjugate Family . Summary Finding UMVUE - Method 2 . . Use complete sufficient statistic to find the best unbiased estimator for . . . . . . . . . . . . . . . . . . . . . τ ( θ ) .