biostatistics 602 statistical inference
play

Biostatistics 602 - Statistical Inference Apil 23rd, 2013 - PowerPoint PPT Presentation

Aug 10, 2023 •242 likes •1.82k views

. . . . .. . . .. . .. . . . .. . . .. . .. . . . . . . . . .. Biostatistics 602 - Statistical Inference Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Apil 23rd, 2013 Hyun Min Kang Final Exam Review &

  1. f X x . .. . .. . . .. . . . Review . .. . . .. .. . . . . . . . . . P2 P1 x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang x . : E Bayes Estimator is a posterior mean of m x Posterior distribution . d f x Marginal distribution m x Bayesian Framework Wrap-up . P4 P3 .. . . . .. . . .. . . .. . .. . . . .. . . .. . . .. 3 / 31 .. . . . .. . . . . .. . . . .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . Prior distribution π ( θ ) Sampling distribution x | θ ∼ f X ( x | θ ) Joint distribution π ( θ ) f ( x | θ )

  2. f X x . . . .. . . .. . . . .. . . .. .. . .. .. Review . . . . . . . . x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang x . : E Bayes Estimator is a posterior mean of m x Posterior distribution . Bayesian Framework Wrap-up . P4 P3 P2 P1 . . .. .. . .. . . .. . . . .. . .. . . .. . . . . . . . .. . . .. . . .. . . 3 / 31 .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . Prior distribution π ( θ ) Sampling distribution x | θ ∼ f X ( x | θ ) Joint distribution π ( θ ) f ( x | θ ) ∫ π ( θ ) f ( x | θ ) d θ Marginal distribution m ( x ) =

  3. . .. .. . . .. . . . . . .. . .. .. . . . .. . Bayesian Framework Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang x . : E Bayes Estimator is a posterior mean of Wrap-up . . P4 P3 P2 P1 Review . . . . . . . . .. . . .. .. . .. .. . . . . . .. . . .. . . . . .. . . . .. . . .. . . .. . 3 / 31 .. . .. . . . . . . . . . . . . . . . . . . . . . . . . Prior distribution π ( θ ) Sampling distribution x | θ ∼ f X ( x | θ ) Joint distribution π ( θ ) f ( x | θ ) ∫ π ( θ ) f ( x | θ ) d θ Marginal distribution m ( x ) = Posterior distribution π ( θ | x ) = f X ( x | θ ) π ( θ ) m ( x )

  4. . .. .. . . .. . . . . . .. . . .. . . .. .. P4 Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Bayesian Framework Wrap-up . P3 . P2 P1 Review . . . . . . . . . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . . .. . . .. . .. .. . . . . . .. 3 / 31 . .. . . . . . . . . . . . . . . . . . . . . . Prior distribution π ( θ ) Sampling distribution x | θ ∼ f X ( x | θ ) Joint distribution π ( θ ) f ( x | θ ) ∫ π ( θ ) f ( x | θ ) d θ Marginal distribution m ( x ) = Posterior distribution π ( θ | x ) = f X ( x | θ ) π ( θ ) m ( x ) Bayes Estimator is a posterior mean of θ : E [ θ | x ] .

  5. . . Review . . . . . . . . . .. . . .. . P2 .. . . .. . . .. .. P1 P3 .. : E R Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang expected loss. minimizes posterior Bayes Rule Estimator minimizes Bayes risk . Bayes Risk is the average risk across all P4 , the risk function is MSE For squared error loss L . E L Risk Function is the average loss : R Bayesian Decision Theory Wrap-up . . . . . . .. . . .. . . .. . . .. . . .. . . .. . . . 4 / 31 . . .. . . .. . .. .. .. . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . Loss Function L ( θ, ˆ θ ) (e.g. ( θ − ˆ θ ) 2 )

  6. . . .. . . .. . . .. . . . . . . . . . .. . . .. .. . .. . Review . : E R Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang expected loss. minimizes posterior Bayes Rule Estimator minimizes Bayes risk . Bayes Risk is the average risk across all P1 , the risk function is MSE For squared error loss L Bayesian Decision Theory Wrap-up . P4 P3 P2 . . .. . . . .. . . .. . .. .. . . .. . . .. . . . 4 / 31 .. .. . . .. . .. . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . Loss Function L ( θ, ˆ θ ) (e.g. ( θ − ˆ θ ) 2 ) Risk Function is the average loss : R ( θ, ˆ θ ) = E [ L ( θ, ˆ θ ) | θ ] .

  7. . . . . .. . . .. . .. . . . .. .. . .. . .. . . . . . . . . .. : E R Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang expected loss. minimizes posterior Bayes Rule Estimator minimizes Bayes risk . Bayes Risk is the average risk across all Review Bayesian Decision Theory Wrap-up . P4 P3 P2 P1 . . . .. . . . . .. . . . .. . .. . . .. . . . .. . . .. . . .. . . .. . . 4 / 31 .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . Loss Function L ( θ, ˆ θ ) (e.g. ( θ − ˆ θ ) 2 ) Risk Function is the average loss : R ( θ, ˆ θ ) = E [ L ( θ, ˆ θ ) | θ ] . For squared error loss L = ( θ − ˆ θ ) 2 , the risk function is MSE

  8. . .. .. . . .. . . . . . .. .. . .. . . . .. . Bayesian Decision Theory Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang expected loss. minimizes posterior Bayes Rule Estimator minimizes Bayes risk Wrap-up . . P4 P3 P2 P1 Review . . . . . . . . .. . . .. .. . . .. .. . . . . . .. . . .. . . . 4 / 31 .. . . . .. . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Loss Function L ( θ, ˆ θ ) (e.g. ( θ − ˆ θ ) 2 ) Risk Function is the average loss : R ( θ, ˆ θ ) = E [ L ( θ, ˆ θ ) | θ ] . For squared error loss L = ( θ − ˆ θ ) 2 , the risk function is MSE Bayes Risk is the average risk across all θ : E [ R ( θ, ˆ θ ) | π ( θ )] .

  9. . . . .. . . .. . .. .. .. . .. . . .. . . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang expected loss. Bayesian Decision Theory Wrap-up P4 . P3 P2 P1 Review . . . . . . . . . .. . . .. .. .. . . .. . . . . . .. . . .. . . . . .. .. . .. . . . .. . . 4 / 31 . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . Loss Function L ( θ, ˆ θ ) (e.g. ( θ − ˆ θ ) 2 ) Risk Function is the average loss : R ( θ, ˆ θ ) = E [ L ( θ, ˆ θ ) | θ ] . For squared error loss L = ( θ − ˆ θ ) 2 , the risk function is MSE Bayes Risk is the average risk across all θ : E [ R ( θ, ˆ θ ) | π ( θ )] . Bayes Rule Estimator minimizes Bayes risk ⇐ ⇒ minimizes posterior

  10. Asymptotic Relative Efficiency ARE V n W n . . .. . . .. . . .. Review . . .. . . .. . .. . . . . . . . . P2 P1 V . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang asymptotically efficient under regularity condition. Asymptotic Efficiency of MLE Theorem 10.1.12 MLE is always is 1. Asymptotically Efficient ARE with CR-bound of unbiased estimator of W . Delta Method Asymptotic Normality Using central limit theorem, Slutsky Theorem, and Consistency Using law of large numbers, show variance and bias Asymptotics Wrap-up . P4 P3 .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . .. . . .. . . .. 5 / 31 . . . . . . . . . . . . . . . . . . . . . converges to zero, for any continuous mapping function τ

  11. Asymptotic Relative Efficiency ARE V n W n . . .. . . .. . . .. Review . . .. . . .. . .. . . . . . . . . P2 P1 V . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang asymptotically efficient under regularity condition. Asymptotic Efficiency of MLE Theorem 10.1.12 MLE is always is 1. Asymptotically Efficient ARE with CR-bound of unbiased estimator of W . Delta Method Asymptotic Normality Using central limit theorem, Slutsky Theorem, and Consistency Using law of large numbers, show variance and bias Asymptotics Wrap-up . P4 P3 .. . . . . . .. . . .. . .. .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . .. . . .. . . .. 5 / 31 . . . . . . . . . . . . . . . . . . . . . converges to zero, for any continuous mapping function τ

  12. . . .. . . .. . . .. . . . . . . . . . .. . . .. . .. .. . Review . V . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang asymptotically efficient under regularity condition. Asymptotic Efficiency of MLE Theorem 10.1.12 MLE is always is 1. Asymptotically Efficient ARE with CR-bound of unbiased estimator of Delta Method P1 Asymptotic Normality Using central limit theorem, Slutsky Theorem, and Consistency Using law of large numbers, show variance and bias Asymptotics Wrap-up . P4 P3 P2 . . .. . . . .. . . .. . .. .. . . .. . . .. . . . 5 / 31 . . .. .. . . . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . converges to zero, for any continuous mapping function τ Asymptotic Relative Efficiency ARE ( V n , W n ) = σ 2 W / σ 2

  13. . . .. . . .. . . .. . . . . . . . . . .. . . .. . .. .. . Review . Delta Method Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang asymptotically efficient under regularity condition. Asymptotic Efficiency of MLE Theorem 10.1.12 MLE is always Asymptotically Efficient ARE with CR-bound of unbiased estimator of V . Asymptotic Normality Using central limit theorem, Slutsky Theorem, and P1 Consistency Using law of large numbers, show variance and bias Asymptotics Wrap-up . P4 P3 P2 . . .. . . . .. . . .. . .. .. . . .. . . .. . . . 5 / 31 . .. . . . .. . . .. . . .. .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . converges to zero, for any continuous mapping function τ Asymptotic Relative Efficiency ARE ( V n , W n ) = σ 2 W / σ 2 τ ( θ ) is 1.

  14. . . .. . . .. . . .. . . . . . . . . . .. . . .. . .. .. . Review . Delta Method Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang asymptotically efficient under regularity condition. Asymptotic Efficiency of MLE Theorem 10.1.12 MLE is always Asymptotically Efficient ARE with CR-bound of unbiased estimator of V . Asymptotic Normality Using central limit theorem, Slutsky Theorem, and P1 Consistency Using law of large numbers, show variance and bias Asymptotics Wrap-up . P4 P3 P2 . . .. . . . .. . . .. . .. .. . . .. . . .. . . . 5 / 31 . .. . . . .. . . .. . . .. .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . converges to zero, for any continuous mapping function τ Asymptotic Relative Efficiency ARE ( V n , W n ) = σ 2 W / σ 2 τ ( θ ) is 1.

  15. . P4 Power function c when R Pr X Type II error Hypothesis Testing Wrap-up . P3 R P2 P1 Review . . . . . . . . . .. . . .. . Pr X represents Type I error under H , and power (=1-Type .. x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x log c rejects H when II error) under H . x L x L x LRT test sup Level test sup Size .. . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 6 / 31 . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0

  16. . Review Pr X Power function Hypothesis Testing Wrap-up . P4 P3 P2 P1 . . . . . . . . represents Type I error under H , and power (=1-Type . .. . . .. . . .. . R II error) under H . .. c Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x log x Size rejects H when x L x L x LRT test sup Level test sup . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . .. . . .. . . 6 / 31 . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0

  17. . . . . . . . . . Hypothesis Testing Wrap-up . P4 P3 P2 P1 Review . II error) under H . .. . . .. . . .. .. . represents Type I error under H , and power (=1-Type Size . c Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x log x test sup rejects H when x L x L x LRT test sup Level .. . . . . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . .. 6 / 31 . .. . .. . . .. . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ )

  18. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. Size . . .. . . .. .. . .. Hypothesis Testing test sup . log Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x c Level x rejects H when x L x L x LRT test sup . . .. .. . .. . . .. . . .. . . . . . .. . . .. . . .. . . . 6 / 31 .. . . .. . . .. . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 .

  19. . . P4 P3 P2 P1 Review . . . . . . . . . .. . Wrap-up .. . . .. . .. .. . . Hypothesis Testing .. log Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x c Level x rejects H when x L x L x LRT test sup . . . .. . . . .. . . .. . . . . . .. . . .. . . .. . . .. 6 / 31 . . .. . . .. . . .. . . .. . . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 . Size α test sup θ ∈ Ω β ( θ ) = α

  20. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . .. .. . . P4 Wrap-up . log Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c x c Hypothesis Testing x rejects H when x L x L x LRT .. . . . . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . 6 / 31 . . . .. . . .. . . .. . . .. . .. . . . .. . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 . Size α test sup θ ∈ Ω β ( θ ) = α Level α test sup θ ∈ Ω β ( θ ) ≤ α

  21. . . . . .. . . .. . .. . .. . .. . . .. . .. . . . . . . . . .. x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient c log c log Review Hypothesis Testing Wrap-up . P4 P3 P2 P1 . . . .. .. . . . .. . . . . . .. . . .. . . . 6 / 31 .. .. .. . . . .. . . .. . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 . Size α test sup θ ∈ Ω β ( θ ) = α Level α test sup θ ∈ Ω β ( θ ) ≤ α LRT λ ( x ) = L (ˆ θ 0 | x ) rejects H 0 when λ ( x ) ≤ c L (ˆ θ | x )

  22. . . . . .. . . .. .. . .. . . .. . . .. .. . . Wrap-up Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient Hypothesis Testing . .. P4 P3 P2 P1 Review . . . . . . . . . . . .. .. . . .. . . . . . . .. . . .. . . .. 6 / 31 . . .. . .. . . .. . . .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 . Size α test sup θ ∈ Ω β ( θ ) = α Level α test sup θ ∈ Ω β ( θ ) ≤ α LRT λ ( x ) = L (ˆ θ 0 | x ) rejects H 0 when λ ( x ) ≤ c L (ˆ θ | x ) ⇒ − 2 log λ ( x ) ≥ − 2 log c = c ∗ ⇐

  23. . . . . .. . . .. .. . .. . . .. . . .. .. . . Wrap-up Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang statistics are identical. LRT based on sufficient statistics LRT based on full data and sufficient Hypothesis Testing . .. P4 P3 P2 P1 Review . . . . . . . . . . . .. .. . . .. . . . . . . .. . . .. . . .. 6 / 31 . . .. . .. . . .. . . .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . Type I error Pr ( X ∈ R | θ ) when θ ∈ Ω 0 Type II error 1 − Pr ( X ∈ R | θ ) when θ ∈ Ω c 0 Power function β ( θ ) = Pr ( X ∈ R | θ ) β ( θ ) represents Type I error under H 0 , and power (=1-Type II error) under H 1 . Size α test sup θ ∈ Ω β ( θ ) = α Level α test sup θ ∈ Ω β ( θ ) ≤ α LRT λ ( x ) = L (ˆ θ 0 | x ) rejects H 0 when λ ( x ) ≤ c L (ˆ θ | x ) ⇒ − 2 log λ ( x ) ≥ − 2 log c = c ∗ ⇐

  24. c and . . Test UMP test in the class of all the level UMP level . with a class of test of every other test for every UMP Test UMP Wrap-up P4 Type II error given the upper bound of Type I error) P3 P2 P1 Review . . . . . . . . . .. . . .. test. (smallest Neyman-Pearson For H . T Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. is an UMP level t T T or R t T vs. H R Karlin-Rabin If T is sufficient and has MLR, then test rejecting . is an increasing function of t for every g t MLR g t test for its size. k is a UMP level f x region f x , a test with rejection . .. .. . . .. . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . . . . . . .. . . .. . . .. . . 7 / 31 .. . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 .

  25. . P2 Type II error given the upper bound of Type I error) test. (smallest Test UMP test in the class of all the level UMP level UMP Wrap-up . P4 P3 P1 vs. H Review . . . . . . . . . .. . . .. .. . .. Neyman-Pearson For H , a test with rejection . t Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. is an UMP level t T T or R T region f x T R Karlin-Rabin If T is sufficient and has MLR, then test rejecting . is an increasing function of t for every g t MLR g t test for its size. k is a UMP level f x . . .. . . .. . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . . . . . . .. . . .. . . .. . . .. . . .. . . .. 7 / 31 . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 . UMP Test β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and β ′ ( θ ) of every other test with a class of test C .

  26. . Review Neyman-Pearson For H Type II error given the upper bound of Type I error) UMP Wrap-up . P4 P3 P2 P1 . . . . . . . . , a test with rejection . .. . . .. . .. .. . vs. H region f x .. t Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. is an UMP level t T T or R T f x T R Karlin-Rabin If T is sufficient and has MLR, then test rejecting . is an increasing function of t for every g t MLR g t test for its size. k is a UMP level . . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . . 7 / 31 .. . .. . . .. . . .. . . .. . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 . UMP Test β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and β ′ ( θ ) of every other test with a class of test C . UMP level α Test UMP test in the class of all the level α test. (smallest

  27. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . .. .. . . P4 Wrap-up . or R Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. is an UMP level t T T t UMP T T R Karlin-Rabin If T is sufficient and has MLR, then test rejecting . is an increasing function of t for every g t MLR g t Type II error given the upper bound of Type I error) .. . . .. . .. . .. . . .. . . . . . .. . . .. . . .. . . .. 7 / 31 . . . . .. . . .. . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 . UMP Test β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and β ′ ( θ ) of every other test with a class of test C . UMP level α Test UMP test in the class of all the level α test. (smallest Neyman-Pearson For H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , a test with rejection region f ( x | θ 1 )/ f ( x | θ 1 ) > k is a UMP level α test for its size.

  28. . . Review . . . . . . . . . .. . . .. . P2 .. . . .. . . .. . P1 P3 .. or R Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. is an UMP level t T T t P4 T T R Karlin-Rabin If T is sufficient and has MLR, then test rejecting Type II error given the upper bound of Type I error) UMP Wrap-up . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . . . .. . .. . . .. . . .. . . .. . . .. 7 / 31 . .. . . . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 . UMP Test β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and β ′ ( θ ) of every other test with a class of test C . UMP level α Test UMP test in the class of all the level α test. (smallest Neyman-Pearson For H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , a test with rejection region f ( x | θ 1 )/ f ( x | θ 1 ) > k is a UMP level α test for its size. MLR g ( t | θ 2 )/ g ( t | θ 1 ) is an increasing function of t for every θ 2 > θ 1 .

  29. . .. .. . . .. . . . . . .. .. . .. . . . .. . UMP Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang test for one-sided composite hypothesis. Karlin-Rabin If T is sufficient and has MLR, then test rejecting Type II error given the upper bound of Type I error) Wrap-up . . P4 P3 P2 P1 Review . . . . . . . . .. . . .. .. . . .. .. . . . . . .. . . .. . . . 7 / 31 .. . . . .. . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Unbiased Test β ( θ 1 ) ≥ β ( θ 0 ) for every θ 1 ∈ Ω c 0 and θ 0 ∈ Ω 0 . UMP Test β ( θ ) ≥ β ′ ( θ ) for every θ ∈ Ω c 0 and β ′ ( θ ) of every other test with a class of test C . UMP level α Test UMP test in the class of all the level α test. (smallest Neyman-Pearson For H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , a test with rejection region f ( x | θ 1 )/ f ( x | θ 1 ) > k is a UMP level α test for its size. MLR g ( t | θ 2 )/ g ( t | θ 1 ) is an increasing function of t for every θ 2 > θ 1 . R = { T : T > t 0 } or R = { T : T < t 0 } is an UMP level α

  30. Wald Test If W n is a consistent estimator of , and S n is a consistent estimator of Var W n , then Z n S n follows a • Two-sided test : Z n • One-sided test : Z n . Asymptotic Tests and p-Values z standard normal distribution W n d Wrap-up or Z n . P4 P3 P2 P1 Review . . . . . . . . . .. .. . z z . valid p-value Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. is also a valid S x W x S X Pr W X p x p-Value given sufficient statistics For a sufficient statistic S X , is a p-Value A p-value W x Pr W X sup that H is true, p x Constructing p-Value Theorem 8.3.27 : If large W X value gives evidence . and every for is valid if, Pr p X p x .. . . .. .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . . .. . . . .. .. . . . .. . . .. . . .. . . .. . . .. . . .. 8 / 31 . . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition.

  31. • Two-sided test : Z n • One-sided test : Z n . P4 z z standard normal distribution d Asymptotic Tests and p-Values Wrap-up . P3 z P2 P1 Review . . . . . . . . . .. . .. .. . or Z n p-Value A p-value .. valid p-value Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. is also a valid S x W x S X Pr W X p x p-Value given sufficient statistics For a sufficient statistic S X , is a p x W x Pr W X sup that H is true, p x Constructing p-Value Theorem 8.3.27 : If large W X value gives evidence . and every for is valid if, Pr p X . . . .. .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . 8 / 31 . . . .. .. . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition. Wald Test If W n is a consistent estimator of θ , and S 2 n is a consistent estimator of Var ( W n ) , then Z n = ( W n − θ 0 )/ S n follows a

  32. . Review standard normal distribution d Asymptotic Tests and p-Values Wrap-up . P4 P3 P2 P1 . . . . . . . . p x . .. . . .. .. . .. . p-Value A p-value is valid if, Pr p X .. p-Value given sufficient statistics For a sufficient statistic S X , Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. is also a valid S x W x S X Pr W X p x valid p-value for is a W x Pr W X sup that H is true, p x Constructing p-Value Theorem 8.3.27 : If large W X value gives evidence . and every . . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . . . 8 / 31 . . . .. . . .. . . .. .. . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition. Wald Test If W n is a consistent estimator of θ , and S 2 n is a consistent estimator of Var ( W n ) , then Z n = ( W n − θ 0 )/ S n follows a • Two-sided test : | Z n | > z α /2 • One-sided test : Z n > z α /2 or Z n < − z α /2

  33. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . . .. . . P4 Wrap-up . p x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. is also a valid S x W x S X Pr W X p-Value given sufficient statistics For a sufficient statistic S X , Asymptotic Tests and p-Values valid p-value is a W x Pr W X sup that H is true, p x Constructing p-Value Theorem 8.3.27 : If large W X value gives evidence standard normal distribution d .. .. . .. . .. . .. . . .. . . . . . .. . . .. . . .. . . .. 8 / 31 . . . . .. . . .. . .. . . . .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition. Wald Test If W n is a consistent estimator of θ , and S 2 n is a consistent estimator of Var ( W n ) , then Z n = ( W n − θ 0 )/ S n follows a • Two-sided test : | Z n | > z α /2 • One-sided test : Z n > z α /2 or Z n < − z α /2 p-Value A p-value 0 ≤ p ( x ) ≤ 1 is valid if, Pr ( p ( X ) ≤ α | θ ) ≤ α for every θ ∈ Ω 0 and 0 ≤ α ≤ 1 .

  34. . . . . . . . . . . . .. . . .. . .. P1 . .. .. . . .. . . Review P2 . Pr W X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. is also a valid S x W x S X p x P3 p-Value given sufficient statistics For a sufficient statistic S X , valid p-value standard normal distribution d Asymptotic Tests and p-Values Wrap-up . P4 .. . . . . . .. . . .. . .. . . . .. . . .. . . .. .. . . . . .. . . .. . . .. . .. 8 / 31 . .. . . .. . . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition. Wald Test If W n is a consistent estimator of θ , and S 2 n is a consistent estimator of Var ( W n ) , then Z n = ( W n − θ 0 )/ S n follows a • Two-sided test : | Z n | > z α /2 • One-sided test : Z n > z α /2 or Z n < − z α /2 p-Value A p-value 0 ≤ p ( x ) ≤ 1 is valid if, Pr ( p ( X ) ≤ α | θ ) ≤ α for every θ ∈ Ω 0 and 0 ≤ α ≤ 1 . Constructing p-Value Theorem 8.3.27 : If large W ( X ) value gives evidence that H 1 is true, p ( x ) = sup θ ∈ Ω 0 Pr ( W ( X ) ≥ W ( x ) | θ ) is a

  35. . .. .. . . .. . . .. . . .. . . .. . . . .. . Asymptotic Tests and p-Values Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang p-value. valid p-value standard normal distribution d Wrap-up . . P4 P3 P2 P1 Review . . . . . . . . .. . . .. .. . . .. . . . . . .. . . .. . . . .. .. .. . . .. . . .. . . . 8 / 31 . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Distribution of LRT For testing, H 0 : θ = θ 0 vs. H 1 : θ = θ 1 , → χ 2 − 2 log λ ( x ) 1 under regularity condition. Wald Test If W n is a consistent estimator of θ , and S 2 n is a consistent estimator of Var ( W n ) , then Z n = ( W n − θ 0 )/ S n follows a • Two-sided test : | Z n | > z α /2 • One-sided test : Z n > z α /2 or Z n < − z α /2 p-Value A p-value 0 ≤ p ( x ) ≤ 1 is valid if, Pr ( p ( X ) ≤ α | θ ) ≤ α for every θ ∈ Ω 0 and 0 ≤ α ≤ 1 . Constructing p-Value Theorem 8.3.27 : If large W ( X ) value gives evidence that H 1 is true, p ( x ) = sup θ ∈ Ω 0 Pr ( W ( X ) ≥ W ( x ) | θ ) is a p-Value given sufficient statistics For a sufficient statistic S ( X ) , p ( x ) = Pr ( W ( X ) ≥ W ( x ) | S ( X ) = S ( x )) is also a valid

  36. . Review if inf Coverage coefficient is Interval Estimation Wrap-up . P4 P3 P2 P1 . . . . . . . . L X . .. . . .. . . .. . Pr U X .. test, Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang interval). confidence set (or is a A X then C X is the acceptance region of a level Confidence interval L X test If A Inverting a level U X L X Pr inf if is U X .. . . . . . .. . . .. . . .. . .. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . .. . . .. . . .. 9 / 31 . . . . . . . . . . . . . . . . . . . . . Coverage probability Pr ( θ ∈ [ L ( X ) , U ( X )])

  37. . .. . P4 P3 P2 P1 Review . . . . . . . . . . Interval Estimation . .. . . .. . . .. .. Wrap-up Confidence interval L X .. then C X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang interval). confidence set (or is a A X test, U X is the acceptance region of a level test If A Inverting a level U X L X Pr inf if is . . . .. .. . . .. . . .. . . . . . .. . . .. . . .. . . . . .. . .. . . .. . . .. . . .. . . .. . . .. . 9 / 31 . . . . . . . . . . . . . . . . . . . . . Coverage probability Pr ( θ ∈ [ L ( X ) , U ( X )]) Coverage coefficient is 1 − α if inf θ ∈ Ω Pr ( θ ∈ [ L ( X ) , U ( X )]) = 1 − α

  38. . . . . . . . . . . . .. . . .. . .. P1 . . .. . . .. .. . Review P2 . X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang interval). confidence set (or is a A then C X P3 test, is the acceptance region of a level test If A Inverting a level Interval Estimation Wrap-up . P4 .. . . . .. . . .. . . .. . .. . . . .. . . .. . . .. 9 / 31 . . . .. . . .. . . .. .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . Coverage probability Pr ( θ ∈ [ L ( X ) , U ( X )]) Coverage coefficient is 1 − α if inf θ ∈ Ω Pr ( θ ∈ [ L ( X ) , U ( X )]) = 1 − α Confidence interval [ L ( X ) , U ( X )]) is 1 − α if inf θ ∈ Ω Pr ( θ ∈ [ L ( X ) , U ( X )]) = 1 − α

  39. . . . .. . . .. . .. .. . . .. .. . .. . . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang interval). Interval Estimation Wrap-up P4 . P3 P2 P1 Review . . . . . . . . . .. . . .. .. .. . . .. . . . . . .. . . .. . . . . .. . . .. . . .. . . .. . 9 / 31 . . .. . .. . . . . . . . . . . . . . . . . . . . . . . Coverage probability Pr ( θ ∈ [ L ( X ) , U ( X )]) Coverage coefficient is 1 − α if inf θ ∈ Ω Pr ( θ ∈ [ L ( X ) , U ( X )]) = 1 − α Confidence interval [ L ( X ) , U ( X )]) is 1 − α if inf θ ∈ Ω Pr ( θ ∈ [ L ( X ) , U ( X )]) = 1 − α Inverting a level α test If A ( θ 0 ) is the acceptance region of a level α test, then C ( X ) = { θ : X ∈ A ( θ ) } is a 1 − α confidence set (or

  40. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . . .. . . P4 Wrap-up . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . H vs. for testing H (c) Show that the test in part (b) is UMP size versus H Practice Problem 1 (continued from last week) H test of (b) Based on one observation X , find the most powerful size (a) Show that this family has an MLR . . Problem . .. .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . 10 / 31 . . . .. . . .. . . .. . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Let f ( x | θ ) be the logistic location pdf e ( x − θ ) f ( x | θ ) = − ∞ < x < ∞ , −∞ < θ < ∞ (1 + e ( x − θ ) ) 2

  41. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . . .. . . P4 Wrap-up . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . H vs. for testing H (c) Show that the test in part (b) is UMP size versus H Practice Problem 1 (continued from last week) H test of (b) Based on one observation X , find the most powerful size (a) Show that this family has an MLR . . Problem . .. .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . 10 / 31 . . . .. . . .. . . .. . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Let f ( x | θ ) be the logistic location pdf e ( x − θ ) f ( x | θ ) = − ∞ < x < ∞ , −∞ < θ < ∞ (1 + e ( x − θ ) ) 2

  42. . . . . . . . . . . . .. . . .. . .. P1 . . .. . .. .. . . Review P2 . (c) Show that the test in part (b) is UMP size Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . H vs. for testing H (a) Show that this family has an MLR P3 . . Problem . Practice Problem 1 (continued from last week) Wrap-up . P4 .. . . . .. . . .. . . .. . .. . . . .. . . .. . . .. 10 / 31 . . .. .. . . .. . . . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Let f ( x | θ ) be the logistic location pdf e ( x − θ ) f ( x | θ ) = − ∞ < x < ∞ , −∞ < θ < ∞ (1 + e ( x − θ ) ) 2 (b) Based on one observation X , find the most powerful size α test of H 0 : θ = 0 versus H 1 : θ = 1 .

  43. . . . .. . . .. . .. .. . . .. . . .. . . . .. . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang (a) Show that this family has an MLR . . Problem Practice Problem 1 (continued from last week) . . . . . . . . Wrap-up . P4 P3 P2 P1 Review . .. . .. . .. . . .. . . . .. . .. . . .. . . . . . .. .. . . .. . . .. . . . . . .. . 10 / 31 .. . . . . . . . . . . . . . . . . . . . . . . Let f ( x | θ ) be the logistic location pdf e ( x − θ ) f ( x | θ ) = − ∞ < x < ∞ , −∞ < θ < ∞ (1 + e ( x − θ ) ) 2 (b) Based on one observation X , find the most powerful size α test of H 0 : θ = 0 versus H 1 : θ = 1 . (c) Show that the test in part (b) is UMP size α for testing H 0 : θ ≤ 0 vs. H 1 : θ > 0 .

  44. e x e x e x e x e x e x e x e x e x e x . Review . . . . . . . . . .. . . .. . P2 . .. .. . .. . P1 e P3 r x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. x x e x P4 e x Let r x .. Solution for (a) Wrap-up . . . . .. .. . . .. . . .. . . . . . .. . . .. . . .. . . . 11 / 31 . . .. . .. . .. .. . . . .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2

  45. e x e x e x e x e x e x e x e x . . . . . . . . . .. . . .. . . .. P1 .. . . .. . . Review Wrap-up P2 r x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. x x e x P3 e x Let r x Solution for (a) . . P4 .. . . .. . . .. .. . . .. . . . . . .. . . .. . . .. . . .. 11 / 31 . . . . . .. . . .. . . .. . . .. . . .. .. . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2 ) 2 ( 1 + e ( x − θ 1 ) e ( θ 1 − θ 2 ) = 1 + e ( x − θ 2 )

  46. e x e x e x e x e x e x e x e x . . .. . . .. . . . . .. . . .. . .. P3 . . . . . . . . r x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. x x Solution for (a) Review Wrap-up . P4 .. P2 P1 . .. . . .. . . . .. . . .. . . .. . . .. . . .. . . . 11 / 31 . . .. . .. . .. . . .. . .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2 ) 2 ( 1 + e ( x − θ 1 ) e ( θ 1 − θ 2 ) = 1 + e ( x − θ 2 ) Let r ( x ) = (1 + e x − θ 1 )/(1 + e x − θ 2 )

  47. e x e x e x .. .. . . .. .. . . . . . .. . . .. . . . . . . . . . . .. Solution for (a) Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. x x Wrap-up . . P4 P3 P2 P1 Review .. . . . .. . .. . . .. . . . .. . .. . . .. . . . . . . .. . . .. . . .. . . 11 / 31 .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2 ) 2 ( 1 + e ( x − θ 1 ) e ( θ 1 − θ 2 ) = 1 + e ( x − θ 2 ) Let r ( x ) = (1 + e x − θ 1 )/(1 + e x − θ 2 ) e ( x − θ 1 ) (1 + e ( x − θ 2 ) ) − (1 + e ( x − θ 1 ) ) e ( x − θ 2 ) r ′ ( x ) = (1 + e ( x − θ 2 ) ) 2

  48. . . . .. .. . .. . .. .. . . .. . . .. . . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. Solution for (a) Wrap-up P4 . P3 P2 P1 Review . . . . . . . . . .. . . .. .. .. . . .. . . . . . .. . . .. . . . . .. .. . .. . . . .. . . 11 / 31 . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2 ) 2 ( 1 + e ( x − θ 1 ) e ( θ 1 − θ 2 ) = 1 + e ( x − θ 2 ) Let r ( x ) = (1 + e x − θ 1 )/(1 + e x − θ 2 ) e ( x − θ 1 ) (1 + e ( x − θ 2 ) ) − (1 + e ( x − θ 1 ) ) e ( x − θ 2 ) r ′ ( x ) = (1 + e ( x − θ 2 ) ) 2 e ( x − θ 1 ) − e ( x − θ 2 ) ( ∵ x − θ 1 > x − θ 2 ) = > 0 (1 + e ( x − θ 2 ) ) 2

  49. . . . .. .. . .. . .. .. . . .. . . .. . . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Therefore, the family of X has an MLR. Solution for (a) Wrap-up P4 . P3 P2 P1 Review . . . . . . . . . .. . . .. .. .. . . .. . . . . . .. . . .. . . . . .. .. . .. . . . .. . . 11 / 31 . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . For θ 1 < θ 2 , e ( x − θ 2) f ( x | θ 2 ) (1+ e ( x − θ 2) ) 2 = f ( x | θ 1 ) e ( x − θ 1) (1+ e ( x − θ 1) ) 2 ) 2 ( 1 + e ( x − θ 1 ) e ( θ 1 − θ 2 ) = 1 + e ( x − θ 2 ) Let r ( x ) = (1 + e x − θ 1 )/(1 + e x − θ 2 ) e ( x − θ 1 ) (1 + e ( x − θ 2 ) ) − (1 + e ( x − θ 1 ) ) e ( x − θ 2 ) r ′ ( x ) = (1 + e ( x − θ 2 ) ) 2 e ( x − θ 1 ) − e ( x − θ 2 ) ( ∵ x − θ 1 > x − θ 2 ) = > 0 (1 + e ( x − θ 2 ) ) 2

  50. e x e x , the rejection region of UMP level . .. P4 P3 P2 P1 Review . . . . . . . . . . . Wrap-up .. . . .. . . .. . . e Solution for (b) e x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang log x e x F x test satisfies Because under H , F x .. x X k e x e e x k e x . .. . .. .. . . .. . . .. . . . . . .. . . .. . . .. . . . 12 / 31 . . .. . . .. . . .. . . .. . . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . . The UMP test rejects H 0 if and only if ) 2 f ( x | 1) ( 1 + e x = > k 1 + e ( x − 1) f ( x | 0)

  51. e x , the rejection region of UMP level .. P2 P1 Review . . . . . . . . . .. . . . P4 . . .. . . .. . . P3 Wrap-up . e x Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang log x e x F x test satisfies Because under H , F x . x X k e x e e x e Solution for (b) .. .. . .. . . .. .. . . .. . . . . . .. . . .. . . .. . . .. 12 / 31 . . .. . . .. . . .. . .. . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . The UMP test rejects H 0 if and only if ) 2 f ( x | 1) ( 1 + e x = > k 1 + e ( x − 1) f ( x | 0) 1 + e x k ∗ > 1 + e ( x − 1)

  52. e x , the rejection region of UMP level . . . . . . . . . . .. . . .. . . P1 .. . . .. . . .. . Review P3 P2 test satisfies Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang log x e x F x e x .. Because under H , F x x X e Solution for (b) Wrap-up . P4 . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . . 12 / 31 . . . .. . . .. . . .. . .. . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . The UMP test rejects H 0 if and only if ) 2 f ( x | 1) ( 1 + e x = > k 1 + e ( x − 1) f ( x | 0) 1 + e x k ∗ > 1 + e ( x − 1) 1 + e x k ∗ > e + e x

  53. e x , the rejection region of UMP level .. . .. . . .. . .. . Review . . .. . . .. . . . . . . . . . . P2 P1 test satisfies Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang log x e x F x e x . Because under H , F x X e Solution for (b) Wrap-up . P4 P3 .. . . . . . .. . .. . .. . .. . . . .. . . .. . . .. 12 / 31 . . . .. . . . .. . . .. . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . The UMP test rejects H 0 if and only if ) 2 f ( x | 1) ( 1 + e x = > k 1 + e ( x − 1) f ( x | 0) 1 + e x k ∗ > 1 + e ( x − 1) 1 + e x k ∗ > e + e x > x 0

  54. . . .. .. . . .. . .. .. . . .. . . .. . . . .. e Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang log test satisfies e x X Solution for (b) . . . . . . . . Wrap-up . P4 P3 P2 P1 Review . . . . .. . . .. . . .. . . . .. . . .. . . . 12 / 31 .. .. . .. . . . .. . . .. . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . The UMP test rejects H 0 if and only if ) 2 f ( x | 1) ( 1 + e x = > k 1 + e ( x − 1) f ( x | 0) 1 + e x k ∗ > 1 + e ( x − 1) 1 + e x k ∗ > e + e x > x 0 Because under H 0 , F ( x | θ = 0) = 1+ e x , the rejection region of UMP level α 1 1 − F ( x | θ = 0) = 1 + e x 0 = α ( 1 − α ) = x 0 α

  55. . .. .. . . .. . . . . . .. . .. .. . . . .. . Solution for (c) Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang , which is identical to the test defined in (b). log Therefore, x X Wrap-up . . P4 P3 P2 P1 Review . . . . . . . . .. . . .. .. .. . .. . . . . . .. . . .. . . . . .. .. . . .. . . .. . . . 13 / 31 . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . Because the family of X has an MLR, UMP size α for testing H 0 : θ ≤ 0 vs. H 1 : θ > 0 should be a form of > x 0 Pr ( X > x 0 | θ = 0) = α

  56. . . . . .. . . .. . . .. .. . .. . . .. .. . . Wrap-up Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang , which is identical to the test defined in (b). X Solution for (c) . .. P4 P3 P2 P1 Review . . . . . . . . . . . .. .. . . .. . . . . . . .. . . .. . . .. 13 / 31 . . . .. . . .. . .. . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . Because the family of X has an MLR, UMP size α for testing H 0 : θ ≤ 0 vs. H 1 : θ > 0 should be a form of > x 0 Pr ( X > x 0 | θ = 0) = α ( 1 − α ) Therefore, x 0 = log α

  57. X i is a consistent estimator for X i is asymptotically normal and derive its asymptotic . P3 . . Problem . Practice Problem 2 Wrap-up . P4 P2 n P1 Review . . . . . . . . . .. . . .. . .. (a) Show that x n (d) Find an asymptotic Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . and Var X You may use the fact that E X above test by inverting the confidence interval for . . vs. H test for H (c) Derive the Wald asymptotic size distribution x n n (b) Show that . .. . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . . . 14 / 31 .. . .. . . .. . . .. .. . . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . Suppose X 1 , · · · , X n are iid random samples with pdf f X ( x | θ ) = θ exp ( − θ x ) , where x ≥ 0 , θ > 0

  58. X i is asymptotically normal and derive its asymptotic . Review . Practice Problem 2 Wrap-up . P4 P3 P2 P1 . . . . . . . . . . .. . . .. . . .. . Problem (a) Show that . (d) Find an asymptotic Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . and Var X You may use the fact that E X above test by inverting the confidence interval for . .. vs. H test for H (c) Derive the Wald asymptotic size distribution x n n (b) Show that n . .. . .. . .. . . .. . . .. . . . .. . .. . . .. . . .. . . . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. 14 / 31 . . . . . . . . . . . . . . . . . . . . . Suppose X 1 , · · · , X n are iid random samples with pdf f X ( x | θ ) = θ exp ( − θ x ) , where x ≥ 0 , θ > 0 x =1 X i is a consistent estimator for θ . ∑ n

  59. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. . . . .. . . .. .. . .. Practice Problem 2 Problem . (d) Find an asymptotic Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . and Var X You may use the fact that E X above test by inverting the confidence interval for . . vs. H test for H (c) Derive the Wald asymptotic size distribution n (b) Show that n (a) Show that . . . .. .. . . . .. . . .. . . . . . .. . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . .. . . . .. . 14 / 31 . . . . . . . . . . . . . . . . . . . . . Suppose X 1 , · · · , X n are iid random samples with pdf f X ( x | θ ) = θ exp ( − θ x ) , where x ≥ 0 , θ > 0 x =1 X i is a consistent estimator for θ . ∑ n ∑ n x =1 X i is asymptotically normal and derive its asymptotic

  60. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . .. .. . . P4 Wrap-up . confidence interval for Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . and Var X You may use the fact that E X above test by inverting the (d) Find an asymptotic Practice Problem 2 distribution n (b) Show that n (a) Show that . . Problem . .. . . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . . .. 14 / 31 . .. . . .. . . . . . . . . . . . . . . . . . . . . . Suppose X 1 , · · · , X n are iid random samples with pdf f X ( x | θ ) = θ exp ( − θ x ) , where x ≥ 0 , θ > 0 x =1 X i is a consistent estimator for θ . ∑ n ∑ n x =1 X i is asymptotically normal and derive its asymptotic (c) Derive the Wald asymptotic size α test for H 0 : θ = θ 0 vs. H 1 : θ ̸ = θ 0 .

  61. . . . . . . . . . . . .. . . .. . .. P1 . . .. .. . .. . . Review P2 . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang above test distribution n (b) Show that (a) Show that P3 . . Problem . Practice Problem 2 Wrap-up . P4 .. . . . .. . .. . . .. . .. . . . .. . . .. . . .. . . . . . .. . . .. . . . .. .. .. .. . . . 14 / 31 . . . . . . . . . . . . . . . . . . . . . . Suppose X 1 , · · · , X n are iid random samples with pdf f X ( x | θ ) = θ exp ( − θ x ) , where x ≥ 0 , θ > 0 x =1 X i is a consistent estimator for θ . ∑ n ∑ n x =1 X i is asymptotically normal and derive its asymptotic (c) Derive the Wald asymptotic size α test for H 0 : θ = θ 0 vs. H 1 : θ ̸ = θ 0 . (d) Find an asymptotic (1 − α ) confidence interval for θ by inverting the You may use the fact that E X = 1/ θ and Var ( X ) = 1/ θ 2 .

  62. . Review . . Solution (a) - Consistency Wrap-up . P4 P3 P2 P1 . . . . . . . . x exp . .. . . .. .. . .. . E X x .. 3 By Theorem of continuous map, n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . P X X i i n . exp . . E X P 2 By LLN (Law of Large Number), X . x exp x dx . . . . . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . . 15 / 31 . . . .. . . .. . . .. .. . .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . 1 Obtain E X = 1/ θ (Derive yourself if not given) ∫ ∞ ∫ ∞ xf ( x | θ ) dx = θ x exp ( − θ x ) dx = 0 0

  63. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. . . . .. . . .. . . .. Solution (a) - Consistency . . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . P X X i i 3 By Theorem of continuous map, n E X . . . E X P 2 By LLN (Law of Large Number), X . x exp . .. .. .. .. . . . .. . . .. . . . . . .. . . .. . . .. . . . 15 / 31 .. .. .. . . .. . . . . . .. . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . 1 Obtain E X = 1/ θ (Derive yourself if not given) ∫ ∞ ∫ ∞ xf ( x | θ ) dx = θ x exp ( − θ x ) dx = 0 0 ∫ ∞ [ − x exp ( − θ x )] ∞ exp ( − θ x ) dx = 0 + 0

  64. . .. . P4 P3 P2 P1 Review . . . . . . . . .. . Solution (a) - Consistency . .. . . .. . . .. . Wrap-up . .. n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . P X X i i 3 By Theorem of continuous map, n . . . . E X P 2 By LLN (Law of Large Number), X . . E X . . . .. . . . .. . . .. . . . . . .. . . .. . . .. . . .. 15 / 31 . .. .. . .. . . .. .. . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 1 Obtain E X = 1/ θ (Derive yourself if not given) ∫ ∞ ∫ ∞ xf ( x | θ ) dx = θ x exp ( − θ x ) dx = 0 0 ∫ ∞ [ − x exp ( − θ x )] ∞ exp ( − θ x ) dx = 0 + 0 ] ∞ [ − 1 = 1 θ exp ( − θ x ) = 0 + θ 0

  65. . . P3 P2 P1 Review . . . . . . . . . .. . .. . . . .. . . .. . . P4 Wrap-up . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . P X X i i 3 By Theorem of continuous map, n Solution (a) - Consistency . . P 2 By LLN (Law of Large Number), X . . E X . . .. .. . .. . . .. . .. . .. . . . . . .. . . .. . . .. . . .. 15 / 31 . . . . .. . . .. . . .. . . .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . 1 Obtain E X = 1/ θ (Derive yourself if not given) ∫ ∞ ∫ ∞ xf ( x | θ ) dx = θ x exp ( − θ x ) dx = 0 0 ∫ ∞ [ − x exp ( − θ x )] ∞ exp ( − θ x ) dx = 0 + 0 ] ∞ [ − 1 = 1 θ exp ( − θ x ) = 0 + θ 0 → E X = 1/ θ .

  66. . . . . . . . . . . . .. .. . .. . .. P1 . . .. . . .. . . Review P2 . 2 By LLN (Law of Large Number), X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang P . . P . P3 . E X . . Solution (a) - Consistency Wrap-up . P4 .. . . . . . .. . . .. . .. .. . . . .. . . .. . . .. 15 / 31 . . .. . . .. . . .. . . .. . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . 1 Obtain E X = 1/ θ (Derive yourself if not given) ∫ ∞ ∫ ∞ xf ( x | θ ) dx = θ x exp ( − θ x ) dx = 0 0 ∫ ∞ [ − x exp ( − θ x )] ∞ exp ( − θ x ) dx = 0 + 0 ] ∞ [ − 1 = 1 θ exp ( − θ x ) = 0 + θ 0 → E X = 1/ θ . 3 By Theorem of continuous map, n / ∑ n i =1 X i = 1/ X → θ .

  67. . . . . . . . . . Solution (b) - Asymptotic Distribution Wrap-up . P4 P3 P2 P1 Review . . .. . . .. . . .. . . . . . g X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X n n n g g X 2 Apply CLT(Central Limit Theorem), n X i y . y , then g y 3 Apply Delta method. Let g y . . n X .. .. . .. . .. . . .. . . .. . . . .. . .. . . .. . . .. . . . .. . . . . .. . . .. . . .. . .. . . . .. . . .. 16 / 31 . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here).

  68. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. . . . .. . . .. .. . .. Solution (b) - Asymptotic Distribution . . g X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X n n n g g X . n X i y . y , then g y 3 Apply Delta method. Let g y . . 2 Apply CLT(Central Limit Theorem), . . . .. .. . .. . . .. . . .. . . . . . .. . . .. . . .. . . . 16 / 31 .. . .. . . . .. .. . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here). ( 1 θ, 1 ) X ∼ AN θ 2 n

  69. . . P4 P3 P2 P1 Review . . . . . . . . . .. . Wrap-up .. . . .. . .. .. . . Solution (b) - Asymptotic Distribution .. g Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X n n n g g X . X n X i . . 2 Apply CLT(Central Limit Theorem), . . . . . . .. .. . . .. . . .. . . . . . .. . . .. . . .. . . . . .. .. .. . . .. . . .. . . .. . . . . .. . . 16 / 31 . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here). ( 1 θ, 1 ) X ∼ AN θ 2 n 3 Apply Delta method. Let g ( y ) = 1/ y , then g ′ ( y ) = − 1/ y 2 .

  70. . . Review . . . . . . . . . .. . . .. . P2 .. .. . .. . . .. . P1 P3 .. . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X n n n . P4 2 Apply CLT(Central Limit Theorem), . . . . Solution (b) - Asymptotic Distribution Wrap-up . . . . . .. . . . . . .. . .. . . . .. . . .. . . . .. 16 / 31 . .. .. . . .. .. . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here). ( 1 θ, 1 ) X ∼ AN θ 2 n 3 Apply Delta method. Let g ( y ) = 1/ y , then g ′ ( y ) = − 1/ y 2 . g (1/ θ ) , [ g ′ (1/ θ )] 2 ∑ X i ( ) = 1/ X = g ( X ) ∼ AN θ 2 n

  71. . . Review . . . . . . . . . .. . . .. .. P2 .. . . .. . . .. . P1 P3 .. . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X n n n . P4 2 Apply CLT(Central Limit Theorem), . . . . Solution (b) - Asymptotic Distribution Wrap-up . . . . . .. . . .. . . .. . .. . . . .. . . .. . . . . 16 / 31 . .. . . .. . . .. .. . .. . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here). ( 1 θ, 1 ) X ∼ AN θ 2 n 3 Apply Delta method. Let g ( y ) = 1/ y , then g ′ ( y ) = − 1/ y 2 . g (1/ θ ) , [ g ′ (1/ θ )] 2 ∑ X i ( ) = 1/ X = g ( X ) ∼ AN θ 2 n θ, θ 2 ( ) AN =

  72. . .. . .. .. . .. . . . Review . .. . . .. . . . . . . . . . . P1 . 2 Apply CLT(Central Limit Theorem), Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang n n . . . P2 . . . Solution (b) - Asymptotic Distribution Wrap-up . P4 P3 .. . . . . . .. . . .. . .. . . . .. . . .. . . .. .. 16 / 31 . . . .. . . .. . . .. . . .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . 1 Obtain Var ( X ) = 1/ θ 2 (Derive if needed, omitted here). ( 1 θ, 1 ) X ∼ AN θ 2 n 3 Apply Delta method. Let g ( y ) = 1/ y , then g ′ ( y ) = − 1/ y 2 . g (1/ θ ) , [ g ′ (1/ θ )] 2 ∑ X i ( ) = 1/ X = g ( X ) ∼ AN θ 2 n θ, θ 2 ( ) AN = ( 1 ⇒ √ n ) 0 , θ 2 ) ( ⇐ X − θ = N

  73. . Wrap-up . . n n X i i n W X . . . n P4 P3 P2 P1 Review . . . . . . . . . .. . .. 2 Obtain a constant estimator of Var W n . S Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Slutsky’s Theorem P X X i i n n Continuous Map Theorem i P X X i i n n CLT Var X P X X i .. . . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . 17 / 31 . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ :

  74. . P4 2 Obtain a constant estimator of Var W . . n n . . Wrap-up . P3 n P2 P1 Review . . . . . . . . . .. . . .. . n i .. S Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Slutsky’s Theorem P X X i i n n Continuous Map Theorem X i P X X i i n n CLT Var X P X . .. . .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . . .. . . . 17 / 31 . . . . . .. . . . .. .. . . .. . . . .. .. . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ : θ, θ 2 ∑ n ( ) i =1 X i W ( X ) = ∼ AN

  75. . P3 . . n n . . Wrap-up . P4 P2 n P1 Review . . . . . . . . . .. . . .. . . n i . S Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Slutsky’s Theorem P X X i i n n Continuous Map Theorem X i P X X i i n n CLT Var X P X .. .. . .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . . .. . . . 17 / 31 .. . . .. . . . .. . . . .. . . .. .. . . . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ : θ, θ 2 ∑ n ( ) i =1 X i W ( X ) = ∼ AN 2 Obtain a constant estimator of Var ( W )

  76. . . . . . . . . . . . Wrap-up . P4 P3 P2 P1 Review .. n .. . . .. . . .. . . n . . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Slutsky’s Theorem P X X i i n S . Continuous Map Theorem P X X i i n n P n .. . . . . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . .. 17 / 31 .. . . . .. . . . .. . . .. .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ : θ, θ 2 ∑ n ( ) i =1 X i W ( X ) = ∼ AN 2 Obtain a constant estimator of Var ( W ) 1 Var ( X ) = 1 ∑ ( X i − X ) 2 → ( CLT ) θ 2 n − 1 i =1

  77. . . P3 P2 P1 Review . . . . . . . . .. .. . .. . . . .. . . .. . . P4 Wrap-up . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Slutsky’s Theorem P X X i i n . S P P n . . n n . .. . . .. . . .. .. . . .. . . . . . .. . . .. . . .. . . .. 17 / 31 . .. . . .. . . . .. .. . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ : θ, θ 2 ∑ n ( ) i =1 X i W ( X ) = ∼ AN 2 Obtain a constant estimator of Var ( W ) 1 Var ( X ) = 1 ∑ ( X i − X ) 2 → ( CLT ) θ 2 n − 1 i =1 n − 1 θ 2 → ( Continuous Map Theorem ) . i =1 ( X i − X ) 2 ∑ n

  78. . . . . . . . . . . .. .. . . .. . .. P1 . . .. . . .. . . Review P2 . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang P n P P . P3 . n n . . Wrap-up . P4 .. . . . . . .. . . .. .. . .. . . . .. . . .. . . .. 17 / 31 . . .. . .. . . .. . . . .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald asymptotic size α test 1 Obtain a consistent estimator of θ : θ, θ 2 ∑ n ( ) i =1 X i W ( X ) = ∼ AN 2 Obtain a constant estimator of Var ( W ) 1 Var ( X ) = 1 ∑ ( X i − X ) 2 → ( CLT ) θ 2 n − 1 i =1 n − 1 θ 2 → ( Continuous Map Theorem ) . i =1 ( X i − X ) 2 ∑ n S 2 = θ 2 → ( Slutsky’s Theorem ) . i =1 ( X i − X ) 2 ∑ n

  79. . . . . . . . . . . Wrap-up . P4 P3 P2 P1 Review . region is .. . . .. . . .. . .. . Z X . X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang z X X i i n n X W X X i i n n X i i n n S n .. . . .. . .. . . .. . . .. . . . .. . .. . . .. . . .. . . . . .. .. . . .. . . .. . . .. . . . . .. . . .. . 18 / 31 . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald Asymptotic size α test (cont’d) 3 Construct a two-sided asymptotic size α Wald test, whose rejection

  80. . .. . P4 P3 P2 P1 Review . . . . . . . . . . . . .. .. . .. . . .. . Wrap-up . .. n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang z X X i i n X region is X X i i n n X i i n n . . . .. . . . .. . . .. . . . . . .. . . .. . . .. . . .. 18 / 31 . . .. . . .. . . .. . . .. . . .. . .. . . .. . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald Asymptotic size α test (cont’d) 3 Construct a two-sided asymptotic size α Wald test, whose rejection � W ( X ) − θ 0 � � � | Z ( X ) | = � � S / n � �

  81. . . .. . .. . . .. . .. P1 . . .. . . .. . . Review P2 . n Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang z X X i i n P3 X n region is . . Wrap-up . P4 .. . . . . . . . . . . . . .. . . .. .. . .. . . . .. . . .. . . .. 18 / 31 . . . .. . . . .. . . . .. . . . .. . .. .. . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald Asymptotic size α test (cont’d) 3 Construct a two-sided asymptotic size α Wald test, whose rejection � W ( X ) − θ 0 � � � | Z ( X ) | = � � S / n � � � � i =1 X i − θ 0 � � ∑ n � � = � � √ 1 � � n ∑ n i =1 ( X i − X ) 2 � �

  82. . . . .. . . .. . .. .. . . .. . . .. . . . .. . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang n n region is . Wrap-up . . . . . . . . . P4 P3 .. P2 P1 Review . 18 / 31 . . . .. . . .. . . . .. . . . .. .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . Solution (c) - Wald Asymptotic size α test (cont’d) 3 Construct a two-sided asymptotic size α Wald test, whose rejection � W ( X ) − θ 0 � � � | Z ( X ) | = � � S / n � � � � i =1 X i − θ 0 � � ∑ n � � = � � √ 1 � � n ∑ n i =1 ( X i − X ) 2 � � � � � � 1 ∑ ( X i − X ) 2 ≥ z α /2 � � � = X − θ 0 � � � n � � i =1

  83. . P2 C X By inverting the acceptance region, the confidence interval is n A The acceptance region is Wrap-up . P4 P3 .. n P1 Review . . . . . . . . . .. . . .. . . X n . X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X X i i n n z X X i i i n n z X C X which is equivalent to z X X i .. 19 / 31 . . . .. . . .. . . .. . .. .. .. . . . .. . . .. . . . . .. . . . .. . . .. . . .. . . .. . . . . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . Solution (d) - Asymptotic 1 − α confidence interval  �  � � � 1   ( x i − x ) 2 ≥ z α /2 � ∑ � � =  x : x − θ 0 � � � n � �  i =1

  84. . P2 C X By inverting the acceptance region, the confidence interval is n A The acceptance region is Wrap-up . P4 P3 .. n P1 Review . . . . . . . . . .. . . .. . . X n . X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X X i i n n z X X i i i n n z X C X which is equivalent to z X X i .. 19 / 31 . . . .. . . .. . . .. . .. .. .. . . . .. . . .. . . . . .. . . . .. . . .. . . .. . . .. . . . . .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . Solution (d) - Asymptotic 1 − α confidence interval  �  � � � 1   ( x i − x ) 2 ≥ z α /2 � ∑ � � =  x : x − θ 0 � � � n � �  i =1

  85. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. A . . .. . . .. . . .. .. n . X Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang X X i i n n z X By inverting the acceptance region, the confidence interval is X i i n n z X C X which is equivalent to n . The acceptance region is .. .. . .. . . .. . . .. . . . . . .. . . .. . . .. . . . 19 / 31 .. . . . . . .. . . .. .. . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . Solution (d) - Asymptotic 1 − α confidence interval  �  � � � 1   ( x i − x ) 2 ≥ z α /2 � ∑ � � =  x : x − θ 0 � � � n � �  i =1  �  � � � 1   ( X i − X ) 2 ≥ z α /2 ∑ � � � C ( X ) =  θ : X − θ � � � n � �  i =1

  86. . . . . .. . . .. . .. . . . .. . . .. . .. . . . . . . . . .. .. Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang which is equivalent to n By inverting the acceptance region, the confidence interval is n A Review The acceptance region is Wrap-up . P4 P3 P2 P1 . 19 / 31 . .. . .. . . .. . . . .. . . .. . . .. . . . . . . . . .. . . .. . .. . . . . .. .. .. . . . . . . . . . . . . . . . . . . . . . Solution (d) - Asymptotic 1 − α confidence interval  �  � � � 1   ( x i − x ) 2 ≥ z α /2 � ∑ � � =  x : x − θ 0 � � � n � �  i =1  �  � � � 1   ( X i − X ) 2 ≥ z α /2 ∑ � � � C ( X ) =  θ : X − θ � � � n � �  i =1      1 , 1   z α z α C ( X ) =  θ ∈ X − X +  √ √ i =1 ( X i − X ) 2 i =1 ( X i − X ) 2 n ∑ n n ∑ n 

  87. . . . . . . . . . . Practice Problem 3 Wrap-up . P4 P3 P2 P1 Review . . .. . . .. . . .. . . Problem . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . with confidence coefficient , find the upper confidence limit for is a known constant 3 When . . . vs. H , construct a LRT testing H is a known constant 2 When . . and 1 Find the MLEs of . .. .. . .. .. .. . . .. . . .. . . . .. . .. . . .. . . .. . . . . . . . . .. . . .. . . .. . . .. 20 / 31 . . .. . . .. . . . . . . . . . . . . . . . . . . . . . The independent random variables X 1 , · · · , X n have the following pdf β x β − 1 f ( x | θ, β ) = 0 < x < θ, β > 0 θ β

  88. . . Wrap-up . P4 P3 P2 P1 Review . . . . . . . . .. . . . .. . . .. . . .. Practice Problem 3 Problem . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . with confidence coefficient , find the upper confidence limit for is a known constant 3 When . . . vs. H , construct a LRT testing H is a known constant 2 When . . . . . . .. .. .. . . . .. . . .. . . . . . .. . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . 20 / 31 . . . . . . . . . . . . . . . . . . . . . The independent random variables X 1 , · · · , X n have the following pdf β x β − 1 f ( x | θ, β ) = 0 < x < θ, β > 0 θ β 1 Find the MLEs of β and θ

  89. . .. P2 P1 Review . . . . . . . . . .. . . . P4 . .. . .. .. . . .. P3 . . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . with confidence coefficient , find the upper confidence limit for is a known constant 3 When . Wrap-up . . . . . Problem . Practice Problem 3 . . .. . . . . . .. . . .. . .. .. . . .. . . .. . . . .. . . . .. . . .. . . .. . . .. . 20 / 31 .. . .. . . . . . . . . . . . . . . . . . . . . . . . The independent random variables X 1 , · · · , X n have the following pdf β x β − 1 f ( x | θ, β ) = 0 < x < θ, β > 0 θ β 1 Find the MLEs of β and θ 2 When β is a known constant β 0 , construct a LRT testing H 0 : θ ≥ θ 0 vs. H 1 : θ < θ 0 .

  90. . . . . . . . . . . . .. . . .. . .. P1 . . .. . . .. . . Review P2 . . Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang . . . . . P3 . . Problem . Practice Problem 3 Wrap-up . P4 .. .. . . . .. .. . . .. . .. . . . .. . . .. . . .. . . . . . .. . . .. . . .. 20 / 31 . . .. . . .. . .. . . . . . . . . . . . . . . . . . . . . . The independent random variables X 1 , · · · , X n have the following pdf β x β − 1 f ( x | θ, β ) = 0 < x < θ, β > 0 θ β 1 Find the MLEs of β and θ 2 When β is a known constant β 0 , construct a LRT testing H 0 : θ ≥ θ 0 vs. H 1 : θ < θ 0 . 3 When β is a known constant β 0 , find the upper confidence limit for θ with confidence coefficient 1 − α .

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

STATS 8: Introduction to Biostatistics Statistical Inference for the Relationship Between Two

STATS 8: Introduction to Biostatistics Statistical Inference for the Relationship Between Two

STATS 8: Introduction to Biostatistics Statistical Inference for the Relationship Between Two Variables Babak Shahbaba Department of Statistics, UCI Objective We now discuss hypothesis testing regarding possible relationships between two

341 views • 12 slides
Biostatistics 602 - Statistical Inference March 19th, 2013 Biostatistics 602 - Lecture 16 Hyun

Biostatistics 602 - Statistical Inference March 19th, 2013 Biostatistics 602 - Lecture 16 Hyun

. .. .. . . .. . . . . . .. . . .. . . . .. .. Biostatistics 602 - Statistical Inference March 19th, 2013 Biostatistics 602 - Lecture 16 Hyun Min Kang March 19th, 2013 Hyun Min Kang Asymptotic Evaluation of Point Estimators

1.58k views • 118 slides
Biostatistics 602 - Statistical Inference February 28th, 2013 Biostatistics 602 - Lecture 14

Biostatistics 602 - Statistical Inference February 28th, 2013 Biostatistics 602 - Lecture 14

. .. .. . . .. . . . . . .. . . .. . . . .. .. Biostatistics 602 - Statistical Inference February 28th, 2013 Biostatistics 602 - Lecture 14 Hyun Min Kang February 28th, 2013 Hyun Min Kang Obtaining Best Unbiased Estimator

1.46k views • 96 slides
Biostatistics 602 - Statistical Inference Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun

Biostatistics 602 - Statistical Inference Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun

. . . . .. . . .. . .. . . . .. . . .. . .. . . . . . . . . .. Biostatistics 602 - Statistical Inference Apil 23rd, 2013 Biostatistics 602 - Lecture 26 Hyun Min Kang Apil 23rd, 2013 Hyun Min Kang Final Exam Review &amp;

1.78k views • 155 slides
Biostatistics 602 - Statistical Inference April 18th, 2013 Biostatistics 602 - Lecture 25 Hyun

Biostatistics 602 - Statistical Inference April 18th, 2013 Biostatistics 602 - Lecture 25 Hyun

. . . . .. . . .. . .. . . . .. . . .. .. .. . . . . .. Biostatistics 602 - Statistical Inference April 18th, 2013 Biostatistics 602 - Lecture 25 Hyun Min Kang April 18th, 2013 Hyun Min Kang Bayesian Test &amp; Practice

1.47k views • 134 slides
Biostatistics 602 - Statistical Inference March 14th, 2013 Biostatistics 602 - Lecture 16 Hyun

Biostatistics 602 - Statistical Inference March 14th, 2013 Biostatistics 602 - Lecture 16 Hyun

. .. .. . . .. . . . . . .. . . .. . . . .. .. Biostatistics 602 - Statistical Inference March 14th, 2013 Biostatistics 602 - Lecture 16 Hyun Min Kang March 14th, 2013 Hyun Min Kang Evaluation of Bayes Estimator Lecture 16

982 views • 96 slides
Biostatistics 602 - Statistical Inference April 16th, 2013 Biostatistics 602 - Lecture 24 Hyun

Biostatistics 602 - Statistical Inference April 16th, 2013 Biostatistics 602 - Lecture 24 Hyun

. . . . .. . . .. . .. . . . .. . . .. . .. . . . . . .. Biostatistics 602 - Statistical Inference April 16th, 2013 Biostatistics 602 - Lecture 24 Hyun Min Kang April 16th, 2013 Hyun Min Kang E-M Algorithm &amp; Practice

1.91k views • 147 slides
Biostatistics 602 - Statistical Inference February 19th, 2013 Biostatistics 602 - Lecture 12

Biostatistics 602 - Statistical Inference February 19th, 2013 Biostatistics 602 - Lecture 12

. .. .. . . .. . . . . . .. . . .. . . . .. .. Biostatistics 602 - Statistical Inference February 19th, 2013 Biostatistics 602 - Lecture 12 Hyun Min Kang February 19th, 2013 Hyun Min Kang Cramer-Rao Theorem Lecture 12 .

1.18k views • 90 slides
Biostatistics 602 - Statistical Inference February 26th, 2013 Biostatistics 602 - Lecture 13

Biostatistics 602 - Statistical Inference February 26th, 2013 Biostatistics 602 - Lecture 13

. .. .. . . .. . . . . . .. . . .. . . . .. .. Biostatistics 602 - Statistical Inference February 26th, 2013 Biostatistics 602 - Lecture 13 Hyun Min Kang February 26th, 2013 Hyun Min Kang Rao-Blackwell Theorem Lecture 13

1.41k views • 115 slides
Likelihood and Point Estimation Lecture 09 Biostatistics 602 - Statistical Inference . . . .

Likelihood and Point Estimation Lecture 09 Biostatistics 602 - Statistical Inference . . . .

. Summary February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang February 7th, 2013 Hyun Min Kang Likelihood and Point Estimation Lecture 09 Biostatistics 602 - Statistical Inference . . . . MLE Method of Moments Likelihood

1.44k views • 109 slides
Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . .

Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . .

. Summary January 22th, 2013 Biostatistics 602 - Lecture 04 Hyun Min Kang January 22th, 2013 Hyun Min Kang Ancillary Statistics Lecture 04 Biostatistics 602 - Statistical Inference . . . . Location-scale Family Ancillary Statistics

1.58k views • 84 slides
Evaluation of Point Estimators Lecture 11 Biostatistics 602 - Statistical Inference . .

Evaluation of Point Estimators Lecture 11 Biostatistics 602 - Statistical Inference . .

. . February 14th, 2013 Biostatistics 602 - Lecture 11 Hyun Min Kang February 14th, 2013 Hyun Min Kang Evaluation of Point Estimators Lecture 11 Biostatistics 602 - Statistical Inference . . Summary Cramer-Rao . Evaluation MLE Recap

2.02k views • 136 slides
Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical Inference . Summary . .

Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical Inference . Summary . .

. . February 5th, 2013 Biostatistics 602 - Lecture 08 Hyun Min Kang February 5th, 2013 Hyun Min Kang Data Reduction - Summary Lecture 08 Biostatistics 602 - Statistical Inference . Summary . . Review Exponential Family . . . . . . .

1.48k views • 94 slides
Wald Test Asymptotics of LRT Lecture 21 Biostatistics 602 - Statistical Inference . . . .

Wald Test Asymptotics of LRT Lecture 21 Biostatistics 602 - Statistical Inference . . . .

. Summary April 2nd, 2013 Biostatistics 602 - Lecture 21 Hyun Min Kang April 2nd, 2013 Hyun Min Kang Wald Test Asymptotics of LRT Lecture 21 Biostatistics 602 - Statistical Inference . . . . Wald Test Asymptotics of LRT Karlin-Rabin

1.46k views • 95 slides
Bayes Estimator Lecture 15 Biostatistics 602 - Statistical Inference . . Summary Conjugate

Bayes Estimator Lecture 15 Biostatistics 602 - Statistical Inference . . Summary Conjugate

. . 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

1.35k views • 88 slides
Likelihood Ratio Test Lecture 19 Biostatistics 602 - Statistical Inference . . . . Unbiased

Likelihood Ratio Test Lecture 19 Biostatistics 602 - Statistical Inference . . . . Unbiased

. Summary March 26th, 2013 Biostatistics 602 - Lecture 19 Hyun Min Kang March 26th, 2013 Hyun Min Kang Likelihood Ratio Test Lecture 19 Biostatistics 602 - Statistical Inference . . . . Unbiased Test LRT Recap . . . . . . . . 1

1.76k views • 144 slides
Rocketdyne Follow-On Health Study 6-8 April 2005 Overview Who was in the study? 54,384

Rocketdyne Follow-On Health Study 6-8 April 2005 Overview Who was in the study? 54,384

Rocketdyne Follow-On Health Study 6-8 April 2005 Overview Who was in the study? 54,384 Rocketdyne Workers from 1948-1999 6,601 Short Term (&lt; 6 mo.) 46,970 813 Insufficient Eligible Workers Identifying Information or Not Employee

1.03k views • 33 slides
VINTAGE DERBY WFTDA: DERBY 101 https://www.youtube.com/watch?v=OId6gTd2LCM 2012 - Yellowknife

VINTAGE DERBY WFTDA: DERBY 101 https://www.youtube.com/watch?v=OId6gTd2LCM 2012 - Yellowknife

VINTAGE DERBY WFTDA: DERBY 101 https://www.youtube.com/watch?v=OId6gTd2LCM 2012 - Yellowknife Roller Girls founded by Mad Trapper &amp; Bludgeoning Betsy 2013 - Rebranded into Diamond City Roller Derby 2014 - Proudly opened our

421 views • 24 slides
A Comparison Study of Transection Rate and Wound Surface Size Caused by a Manual and Motorized

A Comparison Study of Transection Rate and Wound Surface Size Caused by a Manual and Motorized

A Comparison Study of Transection Rate and Wound Surface Size Caused by a Manual and Motorized Sharp Punch in FUE Georgios Zontos MD, PhD Medical Director of Harklinikken, Denmark- University of Patras, School of Medicine, Patra, Greece

410 views • 14 slides
Acquiring dramatic gains in image quality: GPU-accelerated beamforming Andre Lehovich, Dustin

Acquiring dramatic gains in image quality: GPU-accelerated beamforming Andre Lehovich, Dustin

Acquiring dramatic gains in image quality: GPU-accelerated beamforming Andre Lehovich, Dustin Kruse, James Hayes, Allan Wegner 19 March 2015 Decision Sciences Medical Company, LLC dsmedco.com Talk overview An apology about the title: in this

575 views • 15 slides
2020 303(d) Listing Methodology Workgroup Kick-off meeting September 14, 2018 CDPHE - C1A,

2020 303(d) Listing Methodology Workgroup Kick-off meeting September 14, 2018 CDPHE - C1A,

2020 303(d) Listing Methodology Workgroup Kick-off meeting September 14, 2018 CDPHE - C1A, 9-11 am Agenda Time Agenda Topic Lead 9:00 am Introductions Skip Feeney 9:15 am 2020 Listing Methodology Workgroup Purpose Skip Feeney

784 views • 42 slides
Use of the results of the SPICE project on marine litter Marta Ruiz HELCOM Secretariat HOLAS

Use of the results of the SPICE project on marine litter Marta Ruiz HELCOM Secretariat HOLAS

Use of the results of the SPICE project on marine litter Marta Ruiz HELCOM Secretariat HOLAS II 8-2017 Helsinki, Finland, 28-29 November 2017 Microlitter Report on information collected on microlitter. It includes a map showing where

610 views • 16 slides
ASIA PACIFIC INFORMATION SUPERHIGHWAY (AP-IS) Atsuko Okuda Chief, ICT and Development Section

ASIA PACIFIC INFORMATION SUPERHIGHWAY (AP-IS) Atsuko Okuda Chief, ICT and Development Section

ASIA PACIFIC INFORMATION SUPERHIGHWAY (AP-IS) Atsuko Okuda Chief, ICT and Development Section ICT and Disaster Risk Reduction Division ESCAP 2 Part 1: Digital Divide in Asia and the Pacific 3 52% of broadband subscribers in Asia and the

797 views • 15 slides
Community Survey Highlights Survey Conducted December 6-17, 2015 Methodology Survey of 600

Community Survey Highlights Survey Conducted December 6-17, 2015 Methodology Survey of 600

L ONG B EACH C ITY C OLLEGE Community Survey Highlights Survey Conducted December 6-17, 2015 Methodology Survey of 600 randomly selected Long Beach Community College District registered voters The margin of sampling error for all

806 views • 24 slides

More recommend