M-Estimation in INARCH Models with a special focus on small means - - PowerPoint PPT Presentation

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Hanan El-Saied, Roland Fried Department of Statistics TU Dortmund Germany

M-Estimation in INARCH Models

with a special focus on small means

19th International Conference on Computational Statistics, Paris, August 22nd-27th 2010

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• Motivation: Outliers in IN(G)ARCH models
• M-estimation for i.i.d. Poisson data
• M-estimation for INARCH-model
• Bias correction
• Outlook

Contents

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Motivation: Number of Campylobacterosis Infections

Ferland, Latour, Oraichi (2006)

INGARCH-model:  

 

1 t 1 13 t 1 t t t s , Y t

Y Poi ~ Y

s

  

        

20 40 60 80 100 120 140 10 20 30 40 50 time (4 week periods) number

Level shift at time 84, outlier pattern at time 100

Fokianos, F. (2010)

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• M-estimation of location  for i.i.d. data

f log                

 n 1 t t

y           

 n 1 t t

y Minimize e.g. gives ML-estimator



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• M-estimation of location  for i.i.d. data

f log                

 n 1 t t

y           

 n 1 t t

y Minimize e.g. gives ML-estimator



Huber -function

        k | x | ), x ( sign k k | x | , x ) x (

k

• 4
• 2

2 4

• 3 -2 -1 0

1 2 3 x psi-function

Tukey -function

 

k | x | I k x 1 x ) x (

2 2 k

                  

• 4
• 2

2 4

• 2
• 1

1 2 x psi-function

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M-estimation for i.i.d. Poisson data

Modified Huber -function with bias correction

                        k | a y | ), a y ( sign k k | a y | , a y ) , y (

a , k

with a=a() such that

 

 

, Y E

1 a , k

  



Simpson et al. (1987)

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M-estimation for i.i.d. Poisson data

Modified Huber -function with bias correction

                        k | a y | ), a y ( sign k k | a y | , a y ) , y (

a , k

Modified Tukey -function with bias correction

                                                   k a y I a y k a y ) , y (

2 2 2 a , k

with a=a() such that

 

 

, Y E

1 a , k

  



Simpson et al. (1987)

Initialization by sample median or by estimating P(Y=0)

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Efficiencies: asymptotic and sample size n=50

huberM (robustbase), k=1.8 glmrob, k=1.8 Tukey, k=5 Tukey, k=6 Tukey, adaptive k

Asymptotic efficiency of Huber M-est. for several k Finite sample efficiency of Huber & Tukey M-est., n=50

5 10 15 20 25 .80 .85 .90 .95 1.00



asymptotic efficiency

k=1 k=2

5 10 15 20 .0 .2 .4 .6 .8 1.0 relative efficiency 25

Cadigan & Chen (2001)

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Efficiencies: asymptotic and sample size n=50

huberM (robustbase), k=1.8 glmrob, k=1.8 Tukey, k=5 Tukey, k=6 Tukey, adaptive k

Asymptotic efficiency of Huber M-est. for several k Finite sample efficiency of Huber & Tukey M-est., n=50

5 10 15 20 25 .80 .85 .90 .95 1.00



asymptotic efficiency

k=1 k=2

5 10 15 20 .0 .2 .4 .6 .8 1.0 relative efficiency 25

Cadigan & Chen (2001)

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Robustness for =0.5 and =5

Efficiency relatively to sample mean in case of increasing number of outliers of increasing size, n=50, log-scale

5 10 15 20 0.1 1 10 100 1000 number and size of outliers relative efficiency 5 10 15 20 number and size of outliers relative efficiency 0.1 1 10 100 1000

huberM (robustbase), k=1.8 glmrob, k=1.8 Tukey, k=5 Tukey, k=6 Tukey, adaptive k

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• Conditional likelihod estimation for INARCH

                                 

   

y y 1 1 y

n 1 p t p t 1 t t t t t

   Conditioning on first p observations y1, …, yp:

INARCH-model:

 

 

p t p 1 t 1 t t t s , Y t

Y Y , Poi ~ Y

s

  

         

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• Conditional likelihod estimation for INARCH

                                 

   

y y 1 1 y

n 1 p t p t 1 t t t t t

   Conditioning on first p observations y1, …, yp:

INARCH-model:

 

 

p t p 1 t 1 t t t s , Y t

Y Y , Poi ~ Y

s

  

                     

t t t

y

M-estimation:

                                           

  p t 1 t

y y 1 

, 2 marginal mean & variance

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Efficiencies: INARCH(1), 0=1,several 1, n=100

Efficiency for 0

0.0 0.2 0.4 0.6 0.8 .0 .2 .4 .6 .8 1.01.2

1

relative efficiency

Efficiency for 1

0.0 0.2 0.4 0.6 0.8 .0 .2 .4 .6 .8 1.01.2 relative efficiency

Huber, k=1.8, Huber, k=2.5 Tukey, k=5, Tukey, k=7 Tukey, adaptive k

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Robustness: INARCH(1) with 0=1, 1=.4

Bias for 0

5 10 15 20

• .3
• .2
• .1

.0 .1

number and size of outliers

bias

Increasing number k of outliers of size k at end of time series

Conditional ML Huber, k=1.8, Huber, k=2.5 Tukey, k=5, Tukey, k=7 Tukey, adaptive k

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Robustness: INARCH(1) with 0=1, 1=.4

Bias for 0

5 10 15 20

• .3
• .2
• .1

.0 .1

number and size of outliers

bias

Bias for 1

5 10 15 20

number and size of outliers

bias .0 .1 .2 .3 .4 .5

Increasing number k of outliers of size k at end of time series

Conditional ML Huber, k=1.8, Huber, k=2.5 Tukey, k=5, Tukey, k=7 Tukey, adaptive k

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• Bias correction for INARCH(p) model

M-estimator with bias correction: with ao,…, ap depending on 0, …, p such that expectation of left hand side equals 0.                                                                                                          

   

a a y y 1 1 y

n 1 p t p p t 1 t t t t t

    

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Bias for INARCH(1) in dependence on 1

n=100 0 1

.1 .3 .5 .7 .9

• .1

.0 .1 .2 .3 Bias .1 .3 .5 .7 .9

• .04 -.02 .00 .02

.04

1

Bias

1

Conditional ML Tukey, k=7 Tukey, k=5 Tukey, k=5, corrected

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Bias for INARCH(1) in dependence on 1

n=100 0 1

.1 .3 .5 .7 .9

• .1

.0 .1 .2 .3 Bias .1 .3 .5 .7 .9

• .04 -.02 .00 .02

.04

1

Bias

n=200

.1 .3 .5 .7 .9

• .1

.0 .1 .2 .3 Bias .1 .3 .5 .7 .9

• .04 -.02 .00 .02 .04

Bias

1

Conditional ML Tukey, k=7 Tukey, k=5 Tukey, k=5, corrected

Bias correction effective only in large samples

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Conclusions

Tukey M-estimators more robust against many large outliers Needs good robust initialization - from median or P(Y=0) Adaptive choice of the tuning constant k gives M-estimators with good efficiencies irrespective of the true Poisson parameter M-estimators provide robustness also in INARCH case Bias correction works for long time series Ongoing work: extend to INGARCH, prove asymptotic normality

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References

Cadigan, N.G., Chen, J. (2001). Properties of Robust M-estimators for Poisson and Negative Binomial Data.

• J. Statist. Comput. Simul. 70, 273-288.

Davis, R.A., Dunsmuir, W.T.M., Street, S.B. (2003). Observation driven models for Poisson counts. Biometrika 90, 777-790. Ferland, R.A., Latour, A., Oraichi, D. (2006). Integer-valued GARCH processes. Journal of Time Series Analysis 27, 923-942. Fokianos, K., Fried, R. (2010). Outliers in INGARCH Processes. Journal of Time Series Analysis 31, 210-225. Fokianos, K., Rahbek, A., Tjøstheim, D. (2009). Poisson Autoregression. Journal of the American Statistical Association 104, 1430-1439. Simpson, D.G., Carroll, R.J., Ruppert, D. (1987). M-Estimation for Discrete Data: Asymptotic Distribution Theory & Implications. Annals of Statistics 15, 657-669.

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Efficiencies: asymptotic and n=50

huberM (robustbase), k=1.8 glmrob, k=1.8 Tukey, k=5 Tukey, k=6 Tukey, adaptive k

Asymptotic efficiency of Huber M-est. for several k Finite sample efficiency of Huber & Tukey M-est., n=50

5 10 15 20 25 .80 .85 .90 .95 1.00



asymptotic efficiency 5 10 15 20 mu relative efficiency .0 .2 .4 .6 .8 1.0

k=1 k=2

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Outliers in the INGARCH(p,q)-Model

 

 

         

     q 1 j j t j p 1 i i t i t t t s , Y t

Y Poi ~ Y

s

 

 

) t ( I Z Poi ~ Z

t q 1 j j t j p 1 i i t i t t t s , Z t

s

             

      

                  

       q 1 j t j t j i t p 1 i i t i t t t t t t t t

C ) ( ) ( Poi ~ C , C Y Z

Ergodicity for INGARCH(1,1): Fokianos, Rahbek & Tjøstheim (2009)

Process contaminated by outlier of size  at time : Clean process: Equivalently for >0 and t :



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Different Types of Outliers

=0 spiky outlier (SO)

50 100 150 200 5 10 15 time y y

=0.8 transient shift (TS)

50 100 150 200 5 10 15 time y

=1 level shift (LS)

50 100 150 200 5 10 15 20 25 time

Goal: Detect and classify different outliers

underlying mean process (t)