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. 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. . MLE February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang i.i.d. Point Estimation : Ingredients . . Summary 6 / 24 Method of Moments . . . . Likelihood Function . . . . . . . . . . . . . . . . . . . . . . • Data: x = ( x 1 , · · · , x n ) - realizations of random variables ( X 1 , · · · , X n ) . • X 1 , · · · , X n ∼ f X ( x | θ ) . • Assume a model P = { f X ( x | θ ) : θ ∈ Ω ⊂ R p } where the functional form of f X ( x | θ ) is known, but θ is unknown. • Task is to use data x to make inference on θ

  2. X n is x n is called the estimate of • X • Suppose n • Define w • Define w . . x , and x i.i.d. , where X n . . . . X . X n n n i X i X . X X n X . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . . . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Point Estimation . Definition . w x . Example . . w x . The realization of the estimation, . called a point estimator of w X Then w X 7 / 24 . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ .

  3. x n is called the estimate of • X • Suppose n • Define w • Define w . X . . X n i.i.d. , where . , and x x . X n . n n i X i X . X X n X . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . . . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Point Estimation . Definition . . The realization of the estimation, w x w x . . Example . . 7 / 24 . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ . Then w ( X ) = w ( X 1 , · · · , X n ) is called a point estimator of τ ( θ ) .

  4. • Suppose n • Define w • Define w . i , and x x . X X n n n X i . X . X X n X . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 i.i.d. . . Example . . . . . . . . Likelihood Function Method of Moments MLE . Summary Point Estimation . Definition . . . 7 / 24 . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ . Then w ( X ) = w ( X 1 , · · · , X n ) is called a point estimator of τ ( θ ) . The realization of the estimation, w ( x ) = w ( x 1 , · · · , x n ) is called the estimate of τ ( θ ) . • X 1 , · · · , X n ∼ N ( θ, 1) , where θ ∈ Ω ∈ R .

  5. • Define w • Define w . X i . i.i.d. X X n n n i X . . X X n X . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . Example . Point Estimation . . . . . . . . Likelihood Function Method of Moments MLE . Summary . Definition . . 7 / 24 . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ . Then w ( X ) = w ( X 1 , · · · , X n ) is called a point estimator of τ ( θ ) . The realization of the estimation, w ( x ) = w ( x 1 , · · · , x n ) is called the estimate of τ ( θ ) . • X 1 , · · · , X n ∼ N ( θ, 1) , where θ ∈ Ω ∈ R . • Suppose n = 6 , and ( x 1 , · · · , x 6 ) = (2 . 0 , 2 . 1 , 2 . 9 , 2 . 6 , 1 . 2 , 1 . 8) .

  6. • Define w . . Example . . i.i.d. n X . X n X . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 7 / 24 . Likelihood Function Method of Moments . . . . . . . MLE . . Summary Point Estimation . Definition . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ . Then w ( X ) = w ( X 1 , · · · , X n ) is called a point estimator of τ ( θ ) . The realization of the estimation, w ( x ) = w ( x 1 , · · · , x n ) is called the estimate of τ ( θ ) . • X 1 , · · · , X n ∼ N ( θ, 1) , where θ ∈ Ω ∈ R . • Suppose n = 6 , and ( x 1 , · · · , x 6 ) = (2 . 0 , 2 . 1 , 2 . 9 , 2 . 6 , 1 . 2 , 1 . 8) . • Define w 1 ( X 1 , · · · , X n ) = 1 ∑ n i =1 X i = X = 2 . 1 .

  7. . Definition February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang n i.i.d. . . Example . . . . 7 / 24 . . Summary . MLE . . Method of Moments Likelihood Function Point Estimation . . . . . . . . . . . . . . . . . . . . . . . If we use a function of sample w ( X 1 , · · · , X n ) as a ”guess” of τ ( θ ) , where τ ( θ ) is a function of true parameter θ . Then w ( X ) = w ( X 1 , · · · , X n ) is called a point estimator of τ ( θ ) . The realization of the estimation, w ( x ) = w ( x 1 , · · · , x n ) is called the estimate of τ ( θ ) . • X 1 , · · · , X n ∼ N ( θ, 1) , where θ ∈ Ω ∈ R . • Suppose n = 6 , and ( x 1 , · · · , x 6 ) = (2 . 0 , 2 . 1 , 2 . 9 , 2 . 6 , 1 . 2 , 1 . 8) . • Define w 1 ( X 1 , · · · , X n ) = 1 ∑ n i =1 X i = X = 2 . 1 . • Define w 2 ( X 1 , · · · , X n ) = X (1) = 1 . 2 .

  8. . is obtained by solving equations like this. n n i X i E X . . . . . . Point estimator of T m E X m . . . . . . m k k Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 m X i . i . . . . . . . . Likelihood Function Method of Moments MLE . Summary Method of Moments A method to equate sample moments to population moments and solve equations. Sample moments Population moments m n n i X i E X m n n 8 / 24 . . . . . . . . . . . . . . . . . .

  9. . m n i . . . . . . Point estimator of T is obtained by solving equations like this. m i . . . . . . m k k Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 8 / 24 n MLE n equations. A method to equate sample moments to population moments and solve Method of Moments Summary . Method of Moments Population moments Likelihood Function . . . . . . . . Sample moments . . . . . . . . . . . . . . . . . . m 1 = 1 ∑ n µ ′ 1 = E [ X | θ ] = µ ′ 1 ( θ ) i =1 X i m 2 = 1 i =1 X 2 ∑ n µ ′ 2 = E [ X | θ ] = µ ′ 2 ( θ ) m 3 = 1 i =1 X 3 ∑ n µ ′ 3 = E [ X | θ ] = µ ′ 3 ( θ )

  10. . m . n i . . . . . . m . n . . . . . m k k Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 i 8 / 24 Population moments Summary . A method to equate sample moments to population moments and solve equations. Sample moments n MLE Method of Moments Method of Moments Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . m 1 = 1 ∑ n µ ′ 1 = E [ X | θ ] = µ ′ 1 ( θ ) i =1 X i m 2 = 1 i =1 X 2 ∑ n µ ′ 2 = E [ X | θ ] = µ ′ 2 ( θ ) m 3 = 1 i =1 X 3 ∑ n µ ′ 3 = E [ X | θ ] = µ ′ 3 ( θ ) Point estimator of T ( θ ) is obtained by solving equations like this.

  11. . . . i . . . . . . . n . . . . m k Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 i n 8 / 24 Population moments . . . . . . . . Likelihood Function Method of Moments MLE . Method of Moments A method to equate sample moments to population moments and solve equations. Sample moments Summary n . . . . . . . . . . . . . . . . . . m 1 = 1 ∑ n µ ′ 1 = E [ X | θ ] = µ ′ 1 ( θ ) i =1 X i m 2 = 1 i =1 X 2 ∑ n µ ′ 2 = E [ X | θ ] = µ ′ 2 ( θ ) m 3 = 1 i =1 X 3 ∑ n µ ′ 3 = E [ X | θ ] = µ ′ 3 ( θ ) Point estimator of T ( θ ) is obtained by solving equations like this. µ ′ = 1 ( θ ) m 1 µ ′ = 2 ( θ ) m 2 µ ′ = k ( θ )

  12. . n . E X X E X E X Var X n n i X i X n . i X i Solving the two equations above, X , n i X i X n . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . . . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Examples of method of moments estimator . Problem . . i.i.d. . Solution . . . 9 / 24 . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n

  13. . i E X Var X n n i X i X n n X i . Solving the two equations above, X , n i X i X n . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 E X 9 / 24 . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Examples of method of moments estimator . Problem . . i.i.d. . Solution . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n µ ′ = E X = µ = X 1

  14. . X i . n n i X n n i Solving the two equations above, . X , n i X i X n . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 9 / 24 Solution . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Examples of method of moments estimator . Problem . i.i.d. . . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n µ ′ = E X = µ = X 1 E X 2 = [ E X ] 2 + Var ( X ) = µ 2 + σ 2 = 1 ∑ X 2 µ ′ = 2 i =1

  15. . X i . . n n i n n i Solving the two equations above, Solution X , n i X i X n . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 9 / 24 . Likelihood Function Examples of method of moments estimator . MLE Problem Method of Moments . . . . . . . . . . . i.i.d. Summary . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n µ ′ = E X = µ = X 1 E X 2 = [ E X ] 2 + Var ( X ) = µ 2 + σ 2 = 1 ∑ X 2 µ ′ = 2 i =1 { ˆ µ = X

  16. . Solving the two equations above, . . . n n i n i X , . n i X i X n . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 Solution 9 / 24 . Problem Method of Moments Summary Likelihood Function . . . . i.i.d. . Examples of method of moments estimator . MLE . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n µ ′ = E X = µ = X 1 E X 2 = [ E X ] 2 + Var ( X ) = µ 2 + σ 2 = 1 ∑ X 2 µ ′ = 2 i =1 { ˆ µ = X µ 2 + ˆ σ 2 = 1 i =1 X 2 ∑ n ˆ

  17. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang i n i n n . . . Solution . i.i.d. 9 / 24 . . Method of Moments . Likelihood Function Examples of method of moments estimator Summary . MLE Problem . . . . . . . . . . . . . . . . . . . . . . . . . ∼ N ( µ, σ 2 ) . Find estimator for µ, σ 2 . X 1 , · · · , X n µ ′ = E X = µ = X 1 E X 2 = [ E X ] 2 + Var ( X ) = µ 2 + σ 2 = 1 ∑ X 2 µ ′ = 2 i =1 { ˆ µ = X µ 2 + ˆ σ 2 = 1 i =1 X 2 ∑ n ˆ µ = X , ˆ σ 2 = ∑ n i =1 ( X i − X ) 2 / n . Solving the two equations above, ˆ

  18. p k f X x k p x p x E X k x x k Equating first two sample moments, n n i X i x . . n n i X i E X E X Var X k p kp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 kp . . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Method of moments estimator - Binomial . Problem . . i.i.d. . Solution . . . . . 10 / 24 . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Binomial ( k , p ) . Find an estimator for k , p .

  19. . n Equating first two sample moments, n n i X i x E X kp n i . X i E X E X Var X k p kp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 x 10 / 24 . Problem . . . . . . . . Likelihood Function Method of Moments MLE . Summary Method of moments estimator - Binomial . . . i.i.d. . Solution . . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Binomial ( k , p ) . Find an estimator for k , p . ( k ) f X ( x | k , p ) = p x (1 − p ) k − x x ∈ { 0 , 1 , · · · , k }

  20. . i . x Equating first two sample moments, n n X i n n X i . E X E X Var X k p kp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 10 / 24 Solution . Method of moments estimator - Binomial MLE . . Method of Moments Problem . Likelihood Function . . . . . . . i.i.d. . . Summary . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Binomial ( k , p ) . Find an estimator for k , p . ( k ) f X ( x | k , p ) = p x (1 − p ) k − x x ∈ { 0 , 1 , · · · , k } 1 ∑ x = µ ′ = 1 = E X = kp i =1

  21. . n Solution . . . x Equating first two sample moments, n i.i.d. X i n n i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 10 / 24 Summary Likelihood Function Method of Moments . Problem . Method of moments estimator - Binomial . . . . . . . . . . MLE . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Binomial ( k , p ) . Find an estimator for k , p . ( k ) f X ( x | k , p ) = p x (1 − p ) k − x x ∈ { 0 , 1 , · · · , k } 1 ∑ x = µ ′ = 1 = E X = kp i =1 1 2 = E [ X 2 ] = ( E X ) 2 + Var ( X ) = k 2 p 2 + kp (1 − p ) ∑ X 2 µ ′ = i =1

  22. . The method of moments estimators are February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang of k and p . These are not the best estimators. It is possible to get negative estimates k X p n X . k Method of moments estimator - Binomial (cont’d) Summary . . . . . . . . Likelihood Function Method of Moments 11 / 24 MLE . . . . . . . . . . . . . . . . . . . 2 ˆ = X − 1 ∑ n i =1 ( X i − X ) 2

  23. . The method of moments estimators are February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang of k and p . These are not the best estimators. It is possible to get negative estimates k X p n X . k 11 / 24 Method of moments estimator - Binomial (cont’d) Likelihood Function . . . . . . . . Method of Moments MLE . Summary . . . . . . . . . . . . . . . . . . 2 ˆ = X − 1 ∑ n i =1 ( X i − X ) 2 ˆ = ˆ

  24. . The method of moments estimators are February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang of k and p . These are not the best estimators. It is possible to get negative estimates k X p n X . k 11 / 24 Method of moments estimator - Binomial (cont’d) Likelihood Function . . . . . . . . Method of Moments MLE . Summary . . . . . . . . . . . . . . . . . . 2 ˆ = X − 1 ∑ n i =1 ( X i − X ) 2 ˆ = ˆ

  25. . E X p p r p p r X i m i n n m p p p m E X m p February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang p Xp p r m X X i i n n X r X i . i . . . . . . . . Likelihood Function Method of Moments MLE . Summary Examples of MoM estimator - Negative Binomial . Problem . . 12 / 24 n n . Solution . . . . i.i.d. . . . . m . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Negative Binomial ( r , p ) . Find estimator for ( r , p ) .

  26. . m n n i X i E X r p p r p p p m m p X n n i X i X r m p p Xp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 m 12 / 24 . Problem . . . . . . . . Likelihood Function Method of Moments MLE . Summary Examples of MoM estimator - Negative Binomial . . . Solution n n . . i.i.d. . . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Negative Binomial ( r , p ) . Find estimator for ( r , p ) . 1 X i = E X = r (1 − p ) ∑ = m 1 i =1

  27. . n . n n p p m m m X n i n X i X r m p p Xp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 n p 12 / 24 Summary . Problem . . . MLE i.i.d. Method of Moments Likelihood Function . . . . . Solution . . . Examples of MoM estimator - Negative Binomial . . . . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Negative Binomial ( r , p ) . Find estimator for ( r , p ) . 1 X i = E X = r (1 − p ) ∑ = m 1 i =1 ) 2 1 ( r (1 − p ) + r (1 − p ) i = E X 2 = ∑ X 2 = m 2 p 2 i =1

  28. . p . n n p . n n p X Solution n r m p p Xp p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 12 / 24 . Problem Method of Moments . Likelihood Function Summary . . . . . . . Examples of MoM estimator - Negative Binomial MLE . . . . i.i.d. . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Negative Binomial ( r , p ) . Find estimator for ( r , p ) . 1 X i = E X = r (1 − p ) ∑ = m 1 i =1 ) 2 1 ( r (1 − p ) + r (1 − p ) i = E X 2 = ∑ X 2 = m 2 p 2 i =1 m 1 ˆ = = m 2 − m 2 2 1 i =1 X 2 ∑ n i − X 1

  29. . p . . n n p n . p X . n r p p p Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 Solution n 12 / 24 Examples of MoM estimator - Negative Binomial . . . . . . . . Likelihood Function Method of Moments i.i.d. . Summary MLE Problem . . . . . . . . . . . . . . . . . . . . . X 1 , · · · , X n ∼ Negative Binomial ( r , p ) . Find estimator for ( r , p ) . 1 X i = E X = r (1 − p ) ∑ = m 1 i =1 ) 2 1 ( r (1 − p ) + r (1 − p ) i = E X 2 = ∑ X 2 = m 2 p 2 i =1 m 1 ˆ = = m 2 − m 2 2 1 i =1 X 2 ∑ n i − X 1 m 1 ˆ X ˆ ˆ = p = 1 − ˆ 1 − ˆ

  30. a i Y i follows . , in general, the distribution is hard to obtain. k i a i Y i , where a i s are known constants with n i a i r i It is often reasonable to assume that the distribution of . k i approximately. Find a moment-based estimator of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 However, the distribution of r k , respectively. We know that the distribution 13 / 24 Summary . . . . . . . . Likelihood Function Method of Moments MLE . Satterthwaite Approximation . Problem . . . . . . . . . . . . . . . . . . . . Let Y 1 , · · · , Y k are independently (but not identically) distributed random variables from χ 2 r 1 , · · · , χ 2 ∑ n i =1 Y i is also chi-squared with degrees of freedom equal to ∑ k i =1 r i .

  31. a i Y i follows . Problem February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . approximately. Find a moment-based estimator of i k It is often reasonable to assume that the distribution of r k , respectively. We know that the distribution . . . 13 / 24 . Method of Moments Summary . . MLE . . . . . . . Likelihood Function Satterthwaite Approximation . . . . . . . . . . . . . . . . . . Let Y 1 , · · · , Y k are independently (but not identically) distributed random variables from χ 2 r 1 , · · · , χ 2 ∑ n i =1 Y i is also chi-squared with degrees of freedom equal to ∑ k i =1 r i . However, the distribution of ∑ k i =1 a i Y i , where a i s are known constants with ∑ n i =1 a i r i = 1 , in general, the distribution is hard to obtain.

  32. . Summary February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang r k , respectively. We know that the distribution . . . Problem . Satterthwaite Approximation 13 / 24 . MLE Method of Moments Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . Let Y 1 , · · · , Y k are independently (but not identically) distributed random variables from χ 2 r 1 , · · · , χ 2 ∑ n i =1 Y i is also chi-squared with degrees of freedom equal to ∑ k i =1 r i . However, the distribution of ∑ k i =1 a i Y i , where a i s are known constants with ∑ n i =1 a i r i = 1 , in general, the distribution is hard to obtain. It is often reasonable to assume that the distribution of ∑ k i =1 a i Y i follows 1 ν χ 2 ν approximately. Find a moment-based estimator of ν .

  33. . E X i a i r i E X To match the second moments, E k i a i Y i Therefore, the method of moment estimator of a i EY i is k i a i Y i Note that can be negative, which is not desirable. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 k i . k . . . . . . . . Likelihood Function Method of Moments MLE . Summary A Naive Solution E k i a i Y i 14 / 24 . . . . . . . . . . . . . . . . . . To match the first moment, let X ∼ χ 2 ν / ν . Then E ( X ) = 1 , and Var ( X ) = 2/ ν .

  34. . Therefore, the method of moment estimator of k To match the second moments, E k i a i Y i E X is a i Y i k i a i Y i Note that can be negative, which is not desirable. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . k 14 / 24 A Naive Solution . . . . . . . . Likelihood Function E Method of Moments MLE . Summary . . . . . . . . . . . . . . . . . . To match the first moment, let X ∼ χ 2 ν / ν . Then E ( X ) = 1 , and Var ( X ) = 2/ ν . ( k ) ∑ ∑ ∑ = a i EY i = a i r i = 1 = E ( X ) i =1 i =1 i =1

  35. . is . k To match the second moments, E a i Y i E Therefore, the method of moment estimator of k a i Y i i a i Y i Note that can be negative, which is not desirable. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 k 14 / 24 Summary . Method of Moments A Naive Solution Likelihood Function . . . . . MLE . . E . . . . . . . . . . . . . . . . . . . To match the first moment, let X ∼ χ 2 ν / ν . Then E ( X ) = 1 , and Var ( X ) = 2/ ν . ( k ) ∑ ∑ ∑ = a i EY i = a i r i = 1 = E ( X ) i =1 i =1 i =1 ) 2 ( k = 2 ∑ X 2 ) ( = ν + 1 i =1

  36. . A Naive Solution February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang E a i Y i E To match the second moments, . k a i Y i E k Summary Method of Moments . . . . . . . . Likelihood Function 14 / 24 . MLE . . . . . . . . . . . . . . . . . . To match the first moment, let X ∼ χ 2 ν / ν . Then E ( X ) = 1 , and Var ( X ) = 2/ ν . ( k ) ∑ ∑ ∑ = a i EY i = a i r i = 1 = E ( X ) i =1 i =1 i =1 ) 2 ( k = 2 ∑ X 2 ) ( = ν + 1 i =1 Therefore, the method of moment estimator of ν is 2 ˆ ν = i =1 a i Y i ) 2 − 1 ( ∑ k Note that ν can be negative, which is not desirable.

  37. . a i Y i k i a i Y i Var k i a i Y i E k i a i Y i Var k i E . k i a i Y i E k i a i Y i Var k i a i Y i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 E 15 / 24 Method of Moments Var MLE . Summary An alternative Solution To match the second moments, E Likelihood Function . . . . a i Y i . . a i Y i . . k . . . . . . . . . . . . . . . . . . ) 2 ] 2 ( k ( k ) [ ∑ ∑ ∑ = + E ( a i Y i ) i =1 i =1 i =1

  38. . a i Y i . k Var k i a i Y i E k i E a i Y i k i a i Y i Var k i a i Y i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 k 15 / 24 . Likelihood Function Method of Moments . MLE . . . a i Y i . . . . E To match the second moments, Var An alternative Solution Summary . . . . . . . . . . . . . . . . . . ) 2 ] 2 ( k ( k ) [ ∑ ∑ ∑ = + E ( a i Y i ) i =1 i =1 i =1 ] 2   [  Var ( ∑ k i =1 a i Y i ) ∑ = E ( a i Y i ) ] 2 + 1   [  E ( ∑ k i =1 a i Y i ) i =1

  39. . a i Y i a i Y i k . E k i Var a i Y i k i a i Y i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 Var k 15 / 24 To match the second moments, MLE Method of Moments Likelihood Function An alternative Solution . . . . Summary . . . E . . . . . . . . . . . . . . . . . . . . ) 2 ] 2 ( k ( k ) [ ∑ ∑ ∑ = + E ( a i Y i ) i =1 i =1 i =1 ] 2   [  Var ( ∑ k i =1 a i Y i ) ∑ = E ( a i Y i ) ] 2 + 1   [  E ( ∑ k i =1 a i Y i ) i =1    Var ( ∑ k i =1 a i Y i )  = 2 = ] 2 + 1 ν + 1   [ E ( ∑ k i =1 a i Y i )

  40. . An alternative Solution February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . k a i Y i Var a i Y i E To match the second moments, k Summary Likelihood Function . . . . . . . . . 15 / 24 MLE Method of Moments . . . . . . . . . . . . . . . . . . ) 2 ] 2 ( k ( k ) [ ∑ ∑ ∑ = + E ( a i Y i ) i =1 i =1 i =1 ] 2   [  Var ( ∑ k i =1 a i Y i ) ∑ = E ( a i Y i ) ] 2 + 1   [  E ( ∑ k i =1 a i Y i ) i =1    Var ( ∑ k i =1 a i Y i )  = 2 = ] 2 + 1 ν + 1   [ E ( ∑ k i =1 a i Y i ) ] 2 [ E ( ∑ k 2 i =1 a i Y i ) = ν Var ( ∑ k i =1 a i Y i )

  41. a i EY i r i Y i . n a i Var Y i n i r i Substituting this expression for the variance and removing expectations, we obtain Satterthwaite’s estimator i k a i Y i n i a i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 i a i Y i . i . . . . . . . . Likelihood Function Method of Moments 16 / 24 MLE . Summary Alternative Solution (cont’d) independent chi-squared random variables. Var n . . . . . . . . . . . . . . . . . . To match the second moments, Finally, use the fact that Y 1 , · · · , Y k are

  42. a i EY i r i Y i . n n i r i Substituting this expression for the variance and removing expectations, we obtain Satterthwaite’s estimator a i Y i i . n i a i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 k 16 / 24 . . . . . . Likelihood Function Method of Moments MLE Summary Alternative Solution (cont’d) independent chi-squared random variables. . . n . . . . . . . . . . . . . . . . . . . To match the second moments, Finally, use the fact that Y 1 , · · · , Y k are ∑ ∑ Var ( a i Y i ) = a i Var ( Y i ) i =1 i =1

  43. r i Y i . i n r i Substituting this expression for the variance and removing expectations, we obtain Satterthwaite’s estimator n n a i Y i k i a i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 16 / 24 n . . . . . MLE Method of Moments Summary Likelihood Function independent chi-squared random variables. . . . . Alternative Solution (cont’d) . . . . . . . . . . . . . . . . . . To match the second moments, Finally, use the fact that Y 1 , · · · , Y k are ∑ ∑ Var ( a i Y i ) = a i Var ( Y i ) i =1 i =1 a 2 i ( EY i ) 2 ∑ = 2 i =1

  44. r i Y i . i n r i Substituting this expression for the variance and removing expectations, we obtain Satterthwaite’s estimator n n a i Y i k i a i Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 16 / 24 n . . . . . MLE Method of Moments Summary Likelihood Function independent chi-squared random variables. . . . . Alternative Solution (cont’d) . . . . . . . . . . . . . . . . . . To match the second moments, Finally, use the fact that Y 1 , · · · , Y k are ∑ ∑ Var ( a i Y i ) = a i Var ( Y i ) i =1 i =1 a 2 i ( EY i ) 2 ∑ = 2 i =1

  45. . independent chi-squared random variables. February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang i i we obtain Satterthwaite’s estimator Substituting this expression for the variance and removing expectations, r i n . k n 16 / 24 Alternative Solution (cont’d) . Summary Likelihood Function . . . . . . Method of Moments . MLE . . . . . . . . . . . . . . . . . . . To match the second moments, Finally, use the fact that Y 1 , · · · , Y k are ∑ ∑ Var ( a i Y i ) = a i Var ( Y i ) i =1 i =1 a 2 i ( EY i ) 2 ∑ = 2 i =1 ∑ n i =1 a i Y i ˆ = ν a 2 r i Y 2 ∑ n i =1

  46. • let • L • More formally, L • and February 7th, 2013 x x attains its maximum. x x L x where . Biostatistics 602 - Lecture 09 x be the value such that x is called the maximum likelihood estimate of based on data x , X is the maximum likelihood estimator (MLE) of . Hyun Min Kang • . . Method of Moments . . . . . . . . Likelihood Function 17 / 24 MLE . Summary Maximum Likelihood Estimator . Definition . . . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) ,

  47. • L • More formally, L • and . . x x L x where x x is called the maximum likelihood estimate of • based on data x , X is the maximum likelihood estimator (MLE) of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 x attains its maximum. 17 / 24 . Method of Moments . . . . . . . . Likelihood Function MLE . Summary Maximum Likelihood Estimator . Definition . . . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) , • let ˆ θ ( x ) be the value such that

  48. • More formally, L • and • x x L x where x . . . based on data x , X is the maximum likelihood estimator (MLE) of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 x is called the maximum likelihood estimate of 17 / 24 MLE . . . . . Likelihood Function Method of Moments . . Summary Maximum Likelihood Estimator . Definition . . . . . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) , • let ˆ θ ( x ) be the value such that • L ( θ | x ) attains its maximum.

  49. • and . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . X is the maximum likelihood estimator (MLE) of based on data x , x is called the maximum likelihood estimate of • . . Definition . Maximum Likelihood Estimator Summary . . . . . . . . Likelihood Function Method of Moments 17 / 24 MLE . . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) , • let ˆ θ ( x ) be the value such that • L ( θ | x ) attains its maximum. • More formally, L (ˆ θ ( x ) | x ) ≥ L ( θ | x ) ∀ θ ∈ Ω where ˆ θ ( x ) ∈ Ω .

  50. • and . Maximum Likelihood Estimator February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . X is the maximum likelihood estimator (MLE) of . . Definition . . Summary . . . . . . . . . Likelihood Function 17 / 24 Method of Moments MLE . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) , • let ˆ θ ( x ) be the value such that • L ( θ | x ) attains its maximum. • More formally, L (ˆ θ ( x ) | x ) ≥ L ( θ | x ) ∀ θ ∈ Ω where ˆ θ ( x ) ∈ Ω . • ˆ θ ( x ) is called the maximum likelihood estimate of θ based on data x ,

  51. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . . Definition . Maximum Likelihood Estimator Summary . MLE . . . . 17 / 24 Likelihood Function . . . . Method of Moments . . . . . . . . . . . . . . . . . . • For a given sample point x = ( x 1 , · · · , x n ) , • let ˆ θ ( x ) be the value such that • L ( θ | x ) attains its maximum. • More formally, L (ˆ θ ( x ) | x ) ≥ L ( θ | x ) ∀ θ ∈ Ω where ˆ θ ( x ) ∈ Ω . • ˆ θ ( x ) is called the maximum likelihood estimate of θ based on data x , • and ˆ θ ( X ) is the maximum likelihood estimator (MLE) of θ .

  52. f X x f X x i n exp . n . . . L x n i e i . x i n i x i where . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . . . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary Example of MLE - Exponential Distribution . Problem . . i.i.d. . Solution . 18 / 24 . . . . . . . . . . . . . . . . . . Let X 1 , · · · , X n ∼ Exponential ( β ) . Find MLE of β .

  53. n exp . . . n n i e x i n . i x i where . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . Solution i.i.d. Summary . . . . . . . . Likelihood Function Method of Moments MLE . Example of MLE - Exponential Distribution Problem . . . 18 / 24 . . . . . . . . . . . . . . . . . . Let X 1 , · · · , X n ∼ Exponential ( β ) . Find MLE of β . ∏ L ( β | x ) = f X ( x | θ ) = f X ( x i | θ ) i =1

  54. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x i n n n . . . Solution . i.i.d. . 18 / 24 Problem . Likelihood Function . . . . Method of Moments . . . . MLE . Summary Example of MLE - Exponential Distribution . . . . . . . . . . . . . . . . . . Let X 1 , · · · , X n ∼ Exponential ( β ) . Find MLE of β . ∏ L ( β | x ) = f X ( x | θ ) = f X ( x i | θ ) i =1 [ 1 ( ) ] = 1 ∏ β e − x i / β ∑ = − β n exp β i =1 i =1 where β > 0 .

  55. n exp . n i x i n log l n i x i n i x i x i n n i x i n x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 n i . n . . . . . . . . Likelihood Function Method of Moments MLE . Summary Use the derivative to find potential MLE log-likelihood function l x log L x log 19 / 24 . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the

  56. . n n i x i n log l n i x i n i . x i n n i x i n x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 x i 19 / 24 n Summary . . . . . . . . Likelihood Function Method of Moments MLE . Use the derivative to find potential MLE log-likelihood function . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the [ ( )] 1 ∑ l ( β | x ) log L ( β | x ) = log − = β n exp β i =1

  57. . x i x i l n i x i n n i n n n i x i n x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 19 / 24 . . . . Likelihood Function MLE . Summary Use the derivative to find potential MLE log-likelihood function . . . . Method of Moments . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the [ ( )] 1 ∑ l ( β | x ) log L ( β | x ) = log − = β n exp β i =1 ∑ n i =1 x i = − − n log β β

  58. . n February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x n x i i n n x i i n . x i 19 / 24 . Summary MLE Method of Moments Likelihood Function . . . . . . . Use the derivative to find potential MLE . log-likelihood function . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the [ ( )] 1 ∑ l ( β | x ) log L ( β | x ) = log − = β n exp β i =1 ∑ n i =1 x i = − − n log β β ∑ n ∂ l i =1 x i − n = β = 0 β 2 ∂β

  59. . log-likelihood function February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x n x i i n x i n . x i n 19 / 24 MLE . . . . . . . . Use the derivative to find potential MLE Likelihood Function Summary Method of Moments . . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the [ ( )] 1 ∑ l ( β | x ) log L ( β | x ) = log − = β n exp β i =1 ∑ n i =1 x i = − − n log β β ∑ n ∂ l i =1 x i − n = β = 0 β 2 ∂β ∑ = n β i =1

  60. . Use the derivative to find potential MLE February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang n x i n . x i n log-likelihood function 19 / 24 Summary Likelihood Function . . . . . Method of Moments . . . . MLE . . . . . . . . . . . . . . . . . . To maximize the likelihood function L ( β | x ) is equivalent to maximize the [ ( )] 1 ∑ l ( β | x ) log L ( β | x ) = log − = β n exp β i =1 ∑ n i =1 x i = − − n log β β ∑ n ∂ l i =1 x i − n = β = 0 β 2 ∂β ∑ = n β i =1 ∑ n ˆ i =1 x i β = = x

  61. . n i x i n x x nx x x . n Therefore, we can conclude that X X is unique local maximum on the interval Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 n 20 / 24 Likelihood Function Method of Moments MLE . Summary Use the double derivative to confirm local maximum . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ 2 l � ∑ n � � i =1 x i � − 2 = + n � � ∂β 2 β 3 β 2 � � β = x β = x

  62. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang the interval X is unique local maximum on X Therefore, we can conclude that n x n x nx x 20 / 24 Summary . . MLE Method of Moments Likelihood Function . . . . Use the double derivative to confirm local maximum . . . . . . . . . . . . . . . . . . . . . ∂ 2 l � ∑ n � � i =1 x i � − 2 = + n � � ∂β 2 β 3 β 2 � � β = x β = x ( − 2 ∑ n )� 1 i =1 x i � = + n � β 2 β � β = x

  63. . Use the double derivative to confirm local maximum February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang the interval X is unique local maximum on X Therefore, we can conclude that n x x . 20 / 24 Summary MLE Method of Moments Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ 2 l � ∑ n � � i =1 x i � − 2 = + n � � ∂β 2 β 3 β 2 � � β = x β = x ( − 2 ∑ n )� 1 i =1 x i � = + n � β 2 β � β = x 1 ( − 2 nx ) = + n x 2

  64. . Summary February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang the interval X is unique local maximum on X Therefore, we can conclude that x . Use the double derivative to confirm local maximum 20 / 24 . Method of Moments Likelihood Function MLE . . . . . . . . . . . . . . . . . . . . . . . . . . ∂ 2 l � ∑ n � � i =1 x i � − 2 = + n � � ∂β 2 β 3 β 2 � � β = x β = x ( − 2 ∑ n )� 1 i =1 x i � = + n � β 2 β � β = x 1 ( − 2 nx ) = + n x 2 1 = x 2 ( − n ) < 0

  65. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang the interval x . Use the double derivative to confirm local maximum Summary 20 / 24 MLE . . . . Likelihood Function . . . . Method of Moments . . . . . . . . . . . . . . . . . . ∂ 2 l � ∑ n � � i =1 x i � − 2 = + n � � ∂β 2 β 3 β 2 � � β = x β = x ( − 2 ∑ n )� 1 i =1 x i � = + n � β 2 β � β = x 1 ( − 2 nx ) = + n x 2 1 = x 2 ( − n ) < 0 Therefore, we can conclude that ˆ β ( X ) = X is unique local maximum on

  66. . x i x l x n i x i n log n i n , use log x n i x i n L x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 lim If . x . . . . . . . . Likelihood Function Method of Moments MLE . Summary Check boundary and confirm global maximum l x n i x i n log L 21 / 24 . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞

  67. . i lim x l x n i x i n log n x i If n n i x i n L x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 , use log x x . L . . . . . . . . Likelihood Function Method of Moments MLE . Summary Check boundary and confirm global maximum 21 / 24 . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β

  68. . i lim x l x n i x i n log n x i If n n i x i n L x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 , use log x 21 / 24 . . . . . . . . . . Likelihood Function Method of Moments MLE Summary Check boundary and confirm global maximum . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0

  69. . n x n i x i n log n i x i n . i x i n L x Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 l 21 / 24 Summary Check boundary and confirm global maximum . . . . Method of Moments MLE . Likelihood Function . . . . . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0 β ( x β − 1) If β → 0 , use log ( x ) = lim β → 0 1

  70. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x L n x i i n n x i i n 21 / 24 . . Summary Check boundary and confirm global maximum Method of Moments Likelihood Function . . . . . MLE . . . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0 β ( x β − 1) If β → 0 , use log ( x ) = lim β → 0 1 ∑ n i =1 x i l ( β | x ) = − − n log β β

  71. . Check boundary and confirm global maximum February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x L n x i i n . 21 / 24 Summary . . Likelihood Function . . . . . . . MLE Method of Moments . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0 β ( x β − 1) If β → 0 , use log ( x ) = lim β → 0 1 ∑ n i =1 x i l ( β | x ) = − − n log β β ( 1 ∑ n ) β β β − 1 i =1 x i = − − n β

  72. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang x L . Check boundary and confirm global maximum Summary 21 / 24 MLE . . . . . . . Likelihood Function . Method of Moments . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0 β ( x β − 1) If β → 0 , use log ( x ) = lim β → 0 1 ∑ n i =1 x i l ( β | x ) = − − n log β β ( 1 ∑ n ) β β β − 1 i =1 x i = − − n β i =1 x i − n ( β β − 1) ∑ n = − → −∞ β

  73. . MLE February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . Check boundary and confirm global maximum Summary . 21 / 24 Method of Moments . Likelihood Function . . . . . . . . . . . . . . . . . . . . . . . . . β ∈ (0 , ∞ ) . If β → ∞ ∑ n i =1 x i l ( β | x ) = − − n log β → −∞ β L ( β | x ) → 0 β ( x β − 1) If β → 0 , use log ( x ) = lim β → 0 1 ∑ n i =1 x i l ( β | x ) = − − n log β β ( 1 ∑ n ) β β β − 1 i =1 x i = − − n β i =1 x i − n ( β β − 1) ∑ n = − → −∞ β L ( β | x ) → 0

  74. . Therefore l at x . 3 L x (lowest bound) when approaches the boundary x and L 2 x attains the global maximum when x X X is the MLE of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 l . . Method of Moments . . . . . . . . Likelihood Function MLE . Summary Putting Things Together . . 1 22 / 24 . . . . . . . . . . . . . . . . . . ∂β = 0 at ˆ ∂ l β = x

  75. . x and L 2 . 3 L x (lowest bound) when approaches the boundary Therefore l x attains the global maximum when . x X X is the MLE of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . . 22 / 24 MLE . . . . . . . . Likelihood Function Method of Moments . Summary Putting Things Together . . 1 . . . . . . . . . . . . . . . . . . ∂β = 0 at ˆ ∂ l β = x ∂β 2 < 0 at ˆ ∂ 2 l β = x

  76. . x attains the global maximum when . 2 . . Therefore l x and L x . X X is the MLE of . Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 . 22 / 24 . . . . . Likelihood Function Method of Moments MLE . . Summary . . Putting Things Together . . 1 . . . . . . . . . . . . . . . . . . ∂β = 0 at ˆ ∂ l β = x ∂β 2 < 0 at ˆ ∂ 2 l β = x 3 L ( β | x ) → 0 (lowest bound) when β approaches the boundary

  77. . . February 7th, 2013 Biostatistics 602 - Lecture 09 Hyun Min Kang . . 2 . . . 1 . 22 / 24 Putting Things Together . . . . . MLE . . Method of Moments Likelihood Function . . Summary . . . . . . . . . . . . . . . . . . ∂β = 0 at ˆ ∂ l β = x ∂β 2 < 0 at ˆ ∂ 2 l β = x 3 L ( β | x ) → 0 (lowest bound) when β approaches the boundary Therefore l ( β | x ) and L ( β | x ) attains the global maximum when ˆ β = x ˆ β ( X ) = X is the MLE of β .

  78. • For one-dimensional parameter, negative second order derivative • For two-dimensional parameter, suppose L • Check boundary points to see whether boundary gives global maximum. • Use numerical methods • Or perform directly maximization, using inequalities, or properties of L is the likelihood function. Then we need to show (a) L or . . (b) Determinant of second-order derivative is positive If the function is NOT differentiable with respect to . the function. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 implies local maximum. 2 Check second-order derivative to check local maximum. . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary How do we find MLE? . . 1 Find candidates that makes first order derivative to be zero . 23 / 24 . . . . . . . . . . . . . . . . . . If the function is differentiable with respect to θ .

  79. • For one-dimensional parameter, negative second order derivative • For two-dimensional parameter, suppose L • Check boundary points to see whether boundary gives global maximum. • Use numerical methods • Or perform directly maximization, using inequalities, or properties of L is the likelihood function. Then we need to show (a) L or . . (b) Determinant of second-order derivative is positive If the function is NOT differentiable with respect to . the function. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 implies local maximum. 2 Check second-order derivative to check local maximum. . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary How do we find MLE? . . 1 Find candidates that makes first order derivative to be zero . 23 / 24 . . . . . . . . . . . . . . . . . . If the function is differentiable with respect to θ .

  80. • For one-dimensional parameter, negative second order derivative • For two-dimensional parameter, suppose L • Check boundary points to see whether boundary gives global maximum. • Use numerical methods • Or perform directly maximization, using inequalities, or properties of L is the likelihood function. Then we need to show (a) L or . . (b) Determinant of second-order derivative is positive If the function is NOT differentiable with respect to . the function. Hyun Min Kang Biostatistics 602 - Lecture 09 February 7th, 2013 implies local maximum. 2 Check second-order derivative to check local maximum. . . . . . . . . . . Likelihood Function Method of Moments MLE . Summary How do we find MLE? . . 1 Find candidates that makes first order derivative to be zero . 23 / 24 . . . . . . . . . . . . . . . . . . If the function is differentiable with respect to θ .

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