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Maximum Likelihood Estimator (MLE) for the Process Change Point Using the Step Change Likelihood Function for a Binomial Random Variable

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Maximum Likelihood Estimator (MLE) for the Process Change Point Using the Step Change Likelihood Function for a Binomial Random Variable A robust change point estimator for binomial CUSUM control charts Advisor: Takeshi Emura ( 江 村 剛 志 ) 博士 Presenter : Yi Ting Ho 2014/6/24 1 Outline Introduction Background 1. Binomial CUSUM control chart 2. Maximum likelihood estimator 3. Page’s last zero estimator Method Combine CUSUM estimator and MLE 2014/6/24 2 Outline Simulations Design closely follows those of Perry and Pignatiello (2005) Data analysis Jewelry manufacturing data by Burr (1979) Conclusion 2014/6/24 3 Outline Introduction Background 1. Binomial CUSUM control chart 2. Maximum likelihood estimator 3. Page’s last zero estimator Method Combine CUSUM estimator and MLE 2014/6/24 4 Introduction • What is the change point? • We consider observations come from binomial distribution with the same fraction nonconforming p . CUSUM Chart 15 Out-of-control 10 h=12.043 CumulativeSum 5 Change 0 point 0 10 20 30 40 50 60 2014/6/24 Subgroup number 5 Introduction In SPC, the np-chart is most famous charts used to monitor the number of nonconforming items for industrial manufactures. np-chart When sample size is large and defective 10 upper control limit rate is not too small, the np-chart works 8 6 well. defectives 4 center line 2 lower control limit 0 2014/6/24 0 5 10 15 20 25 30 Subgroup number 6 Introduction Page (1954, 1955) first suggested the CUSUM chart to estimate the change point. The binomial CUSUM control chart is a good alternative when small changes are important. 2014/6/24 7 Introduction Samuel and Pignatiello (2001) proposed maximum likelihood estimator (MLE) for the process change point using the step change Likelihood function for a binomial random variable. Perry and Pignatiello (2005) shows that the performance of the MLE is often better than Page’s last zero estimator. 2014/6/24 8 Introduction The MLE method outperforms CUSUM method when magnitudes of change is large. In order to construct more robust in different parameter setting, this thesis combines CUSUM estimator and MLE. Furthermore, compares the new method with two estimators. 2014/6/24 9 Outline Introduction Background 1. Binomial CUSUM control chart 2. Maximum likelihood estimator 3. Page’s last zero estimator Method Combine CUSUM estimator and MLE 2014/6/24 10 Binomial CUSUM control chart i.i.d i.i.d True X1, X 2 ,, X ~ Bin( n, p0 ), X 1, X 2 ,, XT ~ Bin( n, pa ) True True { ( , pa ) | {1, 2 ,, T }, p0 pa 1} for a fixed value T {1, 2,}. n x nx p0 (1 p0 ) , if i 1, 2,, , x P( X i x ) n ( pTrue )x (1 pTrue )nx , if i 1, 2,, T. a a x p 0 : in-control fraction nonconforming. True pa : out-of-control fraction nonconforming. : the true change point . 2014/6/24 11 Binomial CUSUM control chart S 0 and Si max{ 0, Xi nk Si1 }, i 1, 2, 0 1 pas ln a 1 p k 0 , where as pa (1 p0 ) ln as p0 (1 pa ) k : The reference value. as p a : out-of-control fraction nonconforming for which to design the CUSUM chart. 2014/6/24 12 Binomial CUSUM control chart When S i exceeds decision interval h 0 the chart signals that an increase in the process fraction nonconforming has occurred. 2014/6/24 13 Maximum likelihood estimator (MLE) i.i.d i.i.d True X1, X 2 ,, X ~ Bin( n, p0 ), X 1, X 2 ,, XT ~ Bin( n, pa ) True True { ( , pa ) | {1, 2 ,, T }, p0 p a 1} for fixed value T {1, 2,}. The likelihood function is given by T n x n True i nxi True xi True nxi L(, pa | p0 , ) p0 (1 p0 ) ( pa ) (1 pa ) i1 xi i 1 xi where ( x1,x2,, xT ). 2014/6/24 14 Maximum likelihood estimator (MLE) The log-likelihood is True log L( , pa | p0 , ) C log( p0 ) xi log( 1 p0 )( n xi ) i1 i1 T T True True log( pa ) xi log( 1 pa ) ( n xi ), i 1 i 1 True where C is a constant. We can rewrite log L(, pa | p 0 , ) as pTrue T 1 pTrue T log L(, pTrue | p , ) C* log a x log a ( n x ), a 0 i i p0 i 1 1 p0 i 1 T T * where C C log( p0 ) xi log( 1 p0 )( n xi ) is a constant. i1 i1 2014/6/24 15 Maximum likelihood estimator (MLE) If is given, the log-likelihood equation becomes T T x ( n x ) i i log L( , pTrue | p , ) i 1 i 1 , pTrue a 0 pTrue 1 pTrue a a a T T T x xi ( n xi ) i i 1 i 1 pˆ True( ) i 1 . pTrue 1 pTrue a T a a n i 1 2014/6/24 16 Maximum likelihood estimator (MLE) T T 2 xi ( n xi ) i 1 i 1 2 log L( , pa | p0 , ) 2 True 2 0. pTrue pTrue (1 p ) a a a pˆ True( ) True Putting a into log L (, p | p , ), we have a profile a 0 log-likelihood for as pˆ True( ) T 1 pˆ True( ) T log L(, pˆ True( ) | p , ) C* log( a ) x log( a ) ( n x ). a 0 i i p0 i 1 1 p0 i 1 2014/6/24 17 Maximum likelihood estimator (MLE) Therefore, the change point estimator ˆ MLE is True True pˆ ( ) T 1 pˆ ( ) T ˆ a a MLE arg max log xi log ( n xi ) . {1, 2,, T } p 1 p 0 i 1 0 i 1 2014/6/24 18 Page’s last zero estimator S 0 and S max{ 0, X nk S }, i 1, 2, Under 0CUSUM controli chart we havei cumulativei1 sum An estimate of the change point is given by ˆCUSUM max{ i : Si 0}. 2014/6/24 19 Outline Introduction Background 1. Binomial CUSUM control chart 2. Maximum likelihood estimator 3. Page’s last zero estimator Method Combine CUSUM estimator and MLE 2014/6/24 20 Proposed method We propose a new change point estimator that combines ˆMLE and ˆCUSUM. ˆNEW ( w ) wˆCUSUM(1 w )ˆMLE , 0 w 1, True pa pTrue p p0 a 0 True as , if pa p0 pa p0 , pas p True as a 0 w w( p | pa ) a True pa as p p p0 a 0 , if pTrue p pas p . pTrue p a 0 a 0 a 0 2014/6/24 21 Proposed method True First, we estimate unknown pa by MLE. T xi iˆMLE 1 pˆ a (ˆMLE ) T , n iˆMLE 1 The estimator of weight function as pˆ a (ˆMLE ) pˆ (ˆ ) p p0 a MLE 0 , if pˆ (ˆ ) p pas p , pas p a MLE 0 a 0 as a 0 wˆ w( pˆ a (ˆMLE ) | pa ) pˆ (ˆ ) a MLE as p p p0 a 0 , if pˆ (ˆ ) p pas p . pˆ (ˆ ) p a MLE 0 a 0 2014/6/24 a MLE 0 22 Proposed method Hence, we obtain the proposed estimator of the change point as ˆNEW ( wˆ ) wˆˆCUSUM(1 wˆ )ˆMLE , 0 wˆ 1. 2014/6/24 23 Outline Simulations Design closely follows those of Perry and Pignatiello (2005) Data analysis Jewelry manufacturing data by Burr (1979) Conclusion 2014/6/24 24 Simulations i.i.d i.i.d True X1, X 2 ,, X ~ Bin( n, p0 ), X 1, X 2 ,, XT ~ Bin( n, pa ) • We assume the true change point 100. • The in-control fraction nonconforming p0 0.1. • The out-of-control fraction nonconforming pTrue { 0.11, 0.12,, 0.25, 0.30 }. a 25 2014/6/24 Simulations • Consider CUSUM charts designed to detect 30% increase in as by setting pa 1.3 p0 0.13. • Choose h 6. 57 or h 11 . 42 such that in-control process average run lengths (ARL) is close to 150 or 370. 2014/6/24 26 as p0 0.1, pa 0.13, Simulations True pa 0.1, 0.11, 0.15, 0.20, 0.25, 0.30 Cusum Chart Cusum Chart Cusum Chart 12 12 12 h=6.57 h=6.57 h=6.57 10 10 10 8 8 8 6 6 6 4 4 4 Cumulative Sum Cumulative Sum Cumulative Sum Cumulative 2 2 2 0 0 0 0 50 100 150 200 250 300 0 20 40 60 80 100 120 0 20 40 60 80 100 i i i Cusum Chart Cusum Chart Cusum Chart 12 12 12 h=6.57 h=6.57 h=6.57 10 10 10 8 8 8 6 6 6 4 4 4 Cumulative Sum Cumulative Sum Cumulative Sum Cumulative 2 2 2 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 i i i 27 2014/6/24 as p0 0.1, pa 0.13, Simulations True pa 0.1, 0.11, 0.15, 0.20, 0.25, 0.30 Maximum likelihood estimator Maximum likelihood estimator Maximum likelihood estimator 20 20 20 15 15 15 h=6.57 h=6.57 h=6.57 10 10 10 Log - Likelihood Log - Likelihood Log - Likelihood Log 5 5 5 0 0 0 0 50 100 150 200 250 300 0 20 40 60 80 100 0 20 40 60 80 100 i i i Maximum likelihood estimator Maximum likelihood estimator Maximum likelihood estimator 20 20 20 15 15 15 h=6.57 h=6.57 h=6.57 10 10 10 Log - Likelihood Log - Likelihood Log - Likelihood Log 5 5 5 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 i i i 28 2014/6/24 as p0 0.1, pa 0.13, Simulations True pa 0.1, 0.11, 0.15, 0.20, 0.25, 0.30 Cusum Chart Cusum Chart Cusum Chart 20 20 20 h=11.42 h=11.42 h=11.42 15 15 15 10 10 10 Cumulative Sum Cumulative Sum Cumulative Sum Cumulative 5 5 5 0 0 0 0 50 100 150 200 250 300 0 50 100 150 0 20 40 60 80 100 i i i Cusum Chart Cusum Chart Cusum Chart 20 20 20 h=11.42 h=11.42 h=11.42 15 15 15 10 10 10 Cumulative Sum Cumulative Sum Cumulative Sum Cumulative 5 5 5 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 i i i 29 2014/6/24 as p0 0.1, pa 0.13, Simulations True pa 0.1, 0.11, 0.15, 0.20, 0.25, 0.30 Maximum likelihood estimator Maximum likelihood estimator Maximum likelihood estimator 20 20 20 h=11.42 h=11.42 h=11.42 15 15 15 10 10 10 Log - Likelihood Log - Likelihood Log - Likelihood Log 5 5 5 0 0 0 0 50 100 150 200 250 300 0 20 40 60 80 100 0 20 40 60 80 100 i i i Maximum likelihood estimator Maximum likelihood estimator Maximum likelihood estimator 20 20 20 h=11.42 h=11.42 h=11.42 15 15 15 10 10 10 Log - Likelihood Log - Likelihood Log - Likelihood Log 5 5 5 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 i i i 30 2014/6/24 Simulations Table 4 Simulation results for estimating the change point under n 50, p0 0.1 as and pa 0 .
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