Oracle Corporation et al v. SAP AG et al

Filing 836

Declaration of Scott W. Cowan in Support of 835 Memorandum in Opposition, filed bySAP AG, SAP America Inc, Tomorrownow Inc. (Attachments: # 1 Exhibit A, # 2 Exhibit B, # 3 Exhibit C, # 4 Exhibit D, # 5 Exhibit E, # 6 Exhibit F, # 7 Exhibit G, # 8 Exhibit H, # 9 Exhibit I, # 10 Exhibit J, # 11 Exhibit K, # 12 Exhibit L, # 13 Exhibit M, # 14 Exhibit N, # 15 Exhibit O, # 16 Exhibit P, # 17 Exhibit Q)(Related document(s) 835 ) (Froyd, Jane) (Filed on 9/9/2010)

Oracle Corporation et al v. SAP AG et al

Doc. 836 Att. 7

EXHIBIT G

Dockets.Justia.com

Springer Series in Statistics
Ande~on:

Carl- E r i k S a r n d a l Jan Wretman

Bengt Swensson

Continuous-Time Markov Chains: An Applications-Oriented Approach. A n d r e w s / H e n b e r g : Data: A Collection of Problems from Many Fields for t h e Student and Research Worker. A n s c o m b e : Computing in Statistical Science through APL. Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition. Bremaud: Point Processes and Queues: Martingale Dynamics. BrockwelljDavis: Time Series: Theory and Methods, 2nd edition. DaleyjVere-Jones: A n Introduction to the Theory o f Point Processes. Dzhaparid.ze: Parameter Estimation and Hypothesis Testing in Spectral Analysis o f Stationary Time Series. Fmrell: Multivariate Calculation. Fienberg/HoaglinjKJuskaJ/Fanur (Eds.): A Statistical Model: Frederick Mosteller's Contnbutions to Statistics, Science, and Public Policy. GoodmanjKJuskaJ: Measures o f Association for Cross Classifications. Grandell: Aspects of Risk Theory. Hall: T h e Bootstrap and Edgeworth Expansion. J/iirdle: Smoothing Techniques: With Implementation in S. J/tlJ1igan: Bayes Theory. Heyer: Theory of Statistical Experiments. Jolliffe: Principal Component Analysis. Kres: Statistical Tables for Multivariate Analysis. L e a d b e t t e r / L i n d g r e n j R o o r z e n : Extremes and R e l a t e d P r o p e r t i e s o f R a n d o m Sequences and Processes. L e Cam: Asymptotic Methods in Statistical Decision Theory. L e C a m j Y a n g : Asymptotics in Statistics: Some Basic Concepts. Manoukian: M o d e m Concepts and Theorems o f Mathematical Statistics. Miller, Jr.: Simultaneous Statistical Inference, 2nd edition. M o s t e l l e r / W a l l a a : Applied Bayesian and Classical Inference: T h e Case o f The Federalist Pape~. Pollard: Convergence of Stochastic Processes. Pratt/GibbOflS: Concepts o f Nonparametric Theory. Read/Cressie: Goodness-of-Fit Statistics for Discrete Multivariate Data. Reiss: Approximate Distributions o f O r d e r Statistics: With Applications to Nonparametric Statistics. Ross: N o n l i n e a r E s t i m a t i o n . Sachs: Applied Statistics: A Handbook of Techniques, 2nd edition. S i i m d a l / S w e n s s o n / W r e t m a n : Model Assisted SUlVey Sampling. Seneta: Non-Negative Matrices and Markov Chains. Siegmund: Sequential Analysis: Tests and Confidence IntelVals. Tong: T h e Multivariate Normal Distribution. Vapnik: Estimation o f Dependences Based on Empirical Data. West/Harrison: Bayesian Forecasting and Dynamic Models. Wolter: Introduction to Variance Estimation.

Model Assisted Survey Sampling

Springer-Verlag
New York B I· H · Toky e r I n eldelberg L o n d o n P a r i s o H o n g K o n g Barcelona Budapest

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T h i s text o n s u r v e y s a m p l i n g c o n t a i n s h o t h h a s i c a n d a d v a n c e d m a t e r i a l . T h e m a i n t h e m e is e s t i m a t i o n i n s u r v e y s . O t h e r b o o k s o f t h i s k i n d e x i s t . h u t m o s t wcrc w r i t t e n b e f o r e I h e r e c e n t r a p i d a d v a n c e s . T h i s b o o k h a s f o u r i m p n r t a n t ob.iccti\cs: T n d e v e l o p t h e c e n t r a l i d e a s in s u r v e y s a m p l i n g f r o m t h e u n i f i e d p e r s p e c tivc o f u n e q u a l p r o b a h i l i t y s a m p l i n g . I n a m a j , ) r i l y o f s u n e y s , I h e s a m pling units have different probabilities ofselecti'ln. and these prl)hahililies p l a y a ( T u c i a l r o l e in e s t i m a t i o n . , T,> w r i t e a b a s i c s a m p l i n g t e x t t h a t . u n l i k e i t s p r e d c c c s s o r s . is )!lIided b y s t a t i s t i c a l m o d e l i n g in t h e d e r i v a t i o n o f e s t i m a t o r s . T h e m o d e l a s , , , s l e d , I p p r l ) a e h in t h i s b o o k c l a r i f i e s a n d s y s t e m a t i z e s t h e u s c " I ' a u x i l i a r y v a r i a h l e s . w h i c h is a n i m p o r t a n t f e a t u r e o f s u r v e y d e s i g n . J T o C(lVer s i g n i f i c a n t r e c e n t d e v e l o p m e n t s in s p e c i a l a r e a s s u c h anah'Sis o f s u r v ~ y d a t a . d o m a i n e s t i m a t i o n . , a r i a m ' e eslil11ali(\n. m e t h o d s f o r n o n response. and measurement error models. -1. T o p r o v i d e o p p o r t u n i t y f o r s t u d e n t s 10 p r a l ' l i c e s a m p l i n g t e c h n i q u e s o n real d a t a . W e p r o v i d e n u m e r o u s e x e r c i s e s c o n c e r n i n g e s t i m a t i o n for real ( a l b e i t s m a l l ) p o p u l a t i o n s d e s c r i h ~ d in I he a p p e n d i c e s .

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T h i s b o o k g r e w in p a r t o u t o f o u r w o r k a s l e a d e r s o f s u r v e y m e t h o d ( ) I , ) ! ! ) dc, c!opment projects at Statistics Sweden. In supervising younger wlleagues. we repeat~dly f o u n d it m o r e f r u i t f u l t o s t r e s s a few i m p o r t a n t g e n e r a l p r i n c i p l e s I h a n t o c o n s i d e r e v e r y s e l e c t i o n s c h e m e a n d e v e r y e s t i m a l < ' f f o r m u l a ,IS a separate estimation problem. We emphasize a general approach. T h i s b " o k will be useful in t e a c h i n g h a s i c , a s well a s m " r e a d v a n c e d . u n i v e r s i t y c o u r s e s in s u n ' e y s a m p l i n g . O u r s u g g e s t i o n s for s t r u c t u r i n ! ! s u c h c o u r s e s a r c g i v e n heltl\\' T h e m a t e r i a l h a s h e e n t e s t e d in L)Ur ' ' ' v ' n L'l\urses in \11\nlr<"I1,

72

3. Element Sampling Designs

3.4. Systematic Sampling where

73

leads t o (3.3.19) This intuitively s o u n d e s t i m a t o r is simply t h e m e a n o f t h e y-values observed in the domain. T h e variance a n d t h e variance e s t i m a t o r a r e derived l a t e r with the aid o f Result 5.8.1. Remark 3.3.3. We n o t e t h a t several o f t h e p a r a m e t e r s e n c o u n t e r e d in cases 2 a n d 3 c a n be expressed a s

Yos is the ordered sample mean, repeated elements included, Yo.
I ..
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m

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L: Yt,

(3.3.21 )

Defining the o r d e r e d s a m p l e variance as
I SO2 = - - I ~ (Yt, - Yo. )2 B i.J m - jEl

(3.3.22)

e = Lv CtYt
where C I ' . . . , CN are c o n s t a n t s . F o r the p o p u l a t i o n m e a n d e a l t with in case 2, Ct has t h e k n o w n value l i N for all k. I n case 3, e s t i m a t i o n o f t h e d o m a i n t o t a l L Vd Yt, we have ct = I for k E Ud a n d Ct = 0 otherwise. N o r m a l l y , the domain.' membership i n d i c a t o r C t is n o t k n o w n b e f o r e h a n d for all k E U. B u t for ele-, ments k in t h e sample s, it m a y be possible to d e t e r m i n e C t . This is sufficient for the n-estimation m e t h o d to function. By c o n t r a s t , for e s t i m a t i o n o f the d o m a i n mean, Yud , in case 3, we have Ct = liNd for k E Ud a n d C t = 0 otherwise. I f Nd is u n k n o w n , t h e Ct remain u n k n o w n , even for elements k a p p e a r i n g in the sample. This is why t h e n - e s t i m a t i o n m e t h o d fails when Nd is unknown.

we have the following result, which follows from Result 2.9.1. Result 3.3.4. Under t h e S I R design, the pwr e s t i m a t o r o f the population t o t a l Yt rakes the f o r m o f equation (3.3.20). T h e variance is given b y t ==

Lv

VSlR(ipwr) = N ( N - I ) S ; v l m An unbiased variance e s t i m a t o r is

(3.3.23)

VSlR(ipwr)

= N 2 S;.lm

(3.3.24)

Let us c o m p a r e s a m p l i n g with a n d w i t h o u t replacement. I f n, t h e sample size in the w i t h o u t - r e p l a c e m e n t case, e q u a l s m, the o r d e r e d sample size in t h e w i t h - r e p l a c e m e n t case, t h e n
VS1R(Nyo.) I - N- 1 · I Vs/(Ny.) = I - f = I -

3.3.2. Simple R a n d o m Sampling with Replacement
Simple r a n d o m s a m p l i n g with replacement ( S I R), c o n s i d e r e d briefly in Examples 2.9.1 a n d 2.9.2, is t h e o r d e r e d design t h a t gives the same selection proba-' bility l i N " to every o r d e r e d sample os = (k l ,
... ,

f

kj ,

... ,

k .. )

where k j is the element o b t a i n e d in the ith draw. An element m a y be d r a w n m o r e t h a n once. T h e o r d e r e d s a m p l e c o n t a i n s a t m o s t m d i s t i n c t e l e m e n t s . T h e S I R design c a n be implemented by the d r a w - s e q u e n t i a l scheme given' in Example 2.9.1 with m i n d e p e n d e n t with-replacement draws, such t h a t each d r a w gives a n y o n e element a c h a n c e o f l i N t o be d r a w n . I n m draws, a n y given element will a p p e a r in the o r d e r e d s a m p l e a c e r t a i n n u m b e r o f times, say r, such t h a t r is a binomially d i s t r i b u t e d r a n d o m variable with m e a n miN a n d variance

This shows t h a t t h e two e s t i m a t i o n strategies a r e roughly o f e q u a l efficiency when the sampling fraction f = n l N is very small. O n the o t h e r h a n d , if f is subStantial, c o n s i d e r a b l e efficiency is lost in with-replacement sampling. F o r example, if f = 50%, the with-replacement variance is d o u b l e the withoutreplacement variance. Remark 3.3.4. F o r the S I R design, there exist o t h e r unbiased e s t i m a t o r s t h a n Ny.s· These are discussed in Section 3.8.

3.4. Systematic Sampling
3.4.1. Definitions a n d M a i n Result

~(I-~)=~
When N is m o d e r a t e t o large, t h e P o i s s o n d i s t r i b u t i o n with m e a n m i N will be a n excellent a p p r o x i m a t i o n t o t h e d i s t r i b u t i o n o f r. U n d e r t h e S I R design, the p w r e s t i m a t o r (see Section 2.9) for t h e p o p u l a t i o n t o t a l t = Yt takes t h e form

Lv

Systematic sampling refers t o a set o f p r o c e d u r e s t h a t offer several p r a c t i c a l advantages, p a r t i c u l a r l y its simplicity o f exec.ution. We c o n c e n t r a t e o n systematic sampling in its basic form. A first element is d r a w n a t r a n d o m , a n d With equal probability, a m o n g the first a elements in t h e p o p u l a t i o n list. T h e Positive integer a is fixed in a d v a n c e a n d is called the sampling interval. N o