Comparison of methods to extract an asymmetry parameter from data

Comparison of metho ds to extract an asymmetry parameter from data J¨ org Pretz ∗ Physikalisches Institut, Universit¨ at Bonn, 5 3115 Bonn, German y Abstract Sev eral metho ds to extract an asymmetry parameter in an even t d istribution func- tion are d iscussed and compared in terms of statistical precision and applicabilit y . These metho ds are: simple counting rate asymmetries, ev ent w eighti ng pro cedures and the unbinned extended maxim um lik eliho o d metho d. I t is kno wn th at w eight- ing metho ds reac h the same figure of merit (FOM) as the likeli ho od metho d in the limit of v anishing asymmetries. T h is article pr e sen ts an improv ed we ighting pro ce- dure reac h i ng th e F O M of the lik eliho od metho d for arbitrary asymmetries. Cases where the maxim u m lik eliho o d metho d is not app lic able are also d iscussed. Key wor ds: ev ent w eighti ng, min ima l v ariance b ound, Cram ´ er-Rao inequalit y, asymmetry extraction, optimal observ ables, parameter d et ermination, maximum lik eliho od P ACS: 02.70.Rr, 13.88.+e 1 In tr o duction W e consider t w o differen tial ev ent distributions n ± ( x ) f o llo wing the functional form n ± ( x ) = α ( x )(1 ± β ( x ) A ) . (1) In a typical exp erime n tal situation encoun tered in pa rticle ph ysics α ( x ) in- cludes a flux and acceptance factor and β ( x ) is an analyzing p o w er. Both dep en d o n a set of kinematic v ariables here denoted by x . Concrete examples ∗ corresp onding author Email addr ess: j org.pretz@cern. ch (J¨ org Pretz). Preprint submitted to Elsevier 29 Octob er 2018 are spin cross section asymmetries and muon deca y . In the latter case the asymmetry corresp onds to the muon p olarization. The tw o data sets (+ and − ) are for example o bta ine d by c hanging the sign of a p o larization. The goal is to extract the parameter A by measuring t he eve nt distributions n ± ( x ). This w ould be an easy task if b oth α ( x ) and β ( x ) we re know n. Ho we v er in man y applications the factor α ( x ) is not kno wn, or not accurately enough kno wn and o nly β ( x ) is giv en. Section 2 presen ts v arious metho ds to extract the parameter A : The simplest metho d, based on coun ting rate asymmetries, the more efficien t extended un- binned maxim um lik eliho o d (EML) metho d and finally metho ds based on ev en t w eighting are discussed. While in Section 2 we assume that the fa c t o r α ( x ) is the same f or b oth data sets, Section 3 extends the discussion to the case where one has t wo different factors. Up to Section 3 w e a s sume that the n umber of observ ed ev en ts is large enough so that av erages can b e replaced b y the corresp onding exp ec t a tion v alues. Effects o ccurring at low statistics ev en t samples are discussed in Section 4. A summary and conclusions are give n in Section 5. 2 Differen t Metho ds to extract A 2.1 Determining A fr om c ounting r ate a s ymm etry The exp ectation v alue of the nu m b er of eve nts for the tw o data sets reads D N ± E = Z n ± ( x )d x = (1 ± h β i A ) Z α ( x )d x (2) with h β i = R αβ d x/ R α d x . The in tegrals run o ver the kinematic range of x . The a s ymmetry A can b e extracted without the kno wledge of R α ( x )d x : A = 1 h β i h N + i − h N − i h N + i + h N − i . (3) Eq. (3) leads to fo llowing estimator for A : ˆ A cnt = N + + N − P + β i + P − β i N + − N − N + + N − = N + − N − P + β i + P − β i where β i ≡ β ( x i ), N + and N − are the num b ers of o bs erved ev en ts. The sums P + and P − run ov er all ev en ts in the corresp onding data set (+ o r − ). 2 As shown in Section 2.3 and App. A, the figure of merit (F OM), i.e. the inv erse of the v a r ianc e on ˆ A cnt , is F OM ˆ A cnt = N h β i 2 1 − A 2 h β 2 i (4) where N = N + + N − denotes t he total n umber o f ev ents. This figure of merit ma y b e increased if a cut is set to remov e some data with lo w v alues of β . How ev er, it will not reac h t he FOM o f the unb inned extended lik eliho o d metho d discussed now, unless β ( x ) is constan t. 2.2 Extende d Maximum Likeliho o d (EML) Metho d W e no w turn to the unbinne d extended maxim um lik eliho o d (EML) metho d[1,2] whic h is known to r each t he Cram ´ er-Ra o limit of the lo we st p ossible statistical error. The log- likelihoo d function derived from Eq. (1 ) reads l = X + ln ( α i (1 + β i A )) − h N + i ( A ) + X − ln ( α i (1 − β i A )) − h N − i ( A ) . Using the expression in Eq. (2) for the exp e ctation v alues h N ± i results in l = X + ln (1 + β i A ) + X − ln (1 − β i A ) − 2 Z α ( x )d x − X + , − ln α i . (5) The last t wo terms do not dep end on A and can b e ignored in the lik eliho o d maximization. The asymmetry A can thus b e determined without know ledge of α ( x ). F or small v alues of β A one can ev en deriv e an analytic expression for ˆ A LH whic h reads ˆ A LH = P + β i − P − β i P + β 2 i + P − β 2 i . (6) F or arbitrary asymmetries the maximization has to b e done num erically . Note, that this requires CPU intensiv e sums ov er all ev en ts in the maximization pro cedure. The fig ure o f merit (F O M ) is giv en by F OM ˆ A LH = − ∂ 2 l ∂ A 2 = X + β 2 i (1 + β i A ) 2 + X − β 2 i (1 − β i A ) 2 . 3 Replacing the sum ov er ev ents by in tegrals o ne finds F OM ˆ A LH = Z α (1 + β A ) β 2 (1 + β A ) 2 d x + Z α (1 − β A ) β 2 (1 − β A ) 2 d x = Z 2 αβ 2 1 − β 2 A 2 d x . (7) Noting, that for an ar bitrary function f ( x ) the av erage is defined b y h f i = R f ( x ) ( n + ( x ) + n − ( x ))d x R n + ( x ) + n − ( x )d x = R α ( x ) f ( x )d x R α ( x )d x , the figure of merit can b e written as F OM ˆ A LH = N * β 2 1 − β 2 A 2 + . (8) 2.3 Weighting Metho d Next, consider the follow ing estimator ˆ A w = P + w i − P − w i P + w i β i + P − w i β i (9) where w i ≡ w ( x i ) is a, for the moment arbitrary , w eight factor assigned to ev ery ev ent. The exp ectation v alue of ˆ A w equals A indep ende ntly of the w eigh t function w ( x ) used. App. A sho ws that F OM ˆ A w = N h w β i 2 h w 2 (1 − A 2 β 2 ) i . (10) Tw o cases a re o f intere st: 1.) Setting w = 1 corresp onds to the coun ting rate asymmetry discussed in Section 2.1 and pro ves Eq. (4) f or the FOM. 2.) In the case w = β the FOM is F OM ˆ A w = β = N h β 2 i 1 − A 2 h β 4 i h β 2 i . A comparison with Eq. (8) indicates tha t FOM ˆ A w = β coincides with the FOM of the likelihoo d metho d fo r v a nishing A . Actually , in this case, the t w o es- 4 timators ar e iden tical as can b e see n b y comparing Eq. (9) with w ≡ β and Eq. (6). Note tha t the estimator ˆ A w can b e applied for arbitrary asymmetries as w ell, accepting a decrease of t he F OM compared the EML estimator as discusse d in Section 2.5. Suc h a weigh t ing pro cedure has b een used for example in Refs. [3,4] to extract spin asymmetries in the case where h β i A ≪ 1. In Ref. [5] a w eigh ting metho d is discussed to sim ultaneously extract signal and background asymmetries . The fact that a weigh ting pro cedure reac hes the same FOM as the EML metho d w as first discussed in Ref. [7] in the con text o f signal and bac kground extractions. The next section sho ws that one can find a w eigh t factor reac hing the FOM of the EML metho d ev en in the case of non- v anishing asymmetries. 2.4 Impr ove d Weighting Metho d V ariational calculus sho ws (s. App. B) that t he maxim um F OM is reache d using a we ig h t factor w = β 1 − β 2 A 2 0 . (11) Here, A 0 is a first estimate of t he a s ymmetry A obtained fo r ex ample from the w eigh ting metho d presen ted in Section 2 .3 . The w eighting factor defined in Eq. (11) leads to the fo llo wing estimator (the index iw stands for improv ed w eigh t ) ˆ A iw = P + β i 1 − β 2 i A 2 0 − P − β i 1 − β 2 i A 2 0 P + β 2 i 1 − β 2 i A 2 0 + P − β 2 i 1 − β 2 i A 2 0 . (12) Eq. (10) giv es F OM ˆ A iw = N D β 2 1 − β 2 A 2 0 E 2 D β 2 1 − β 2 A 2 (1 − β 2 A 2 0 ) 2 E . (13) Th us giv en a go o d estimate A 0 ≈ A , we get FOM ˆ A iw = FOM ˆ A LH , i.e. the impro ve d w eigh ting metho d reac hes the same F OM a s the EML method for arbitrary a symmetries as we ll with the adv an ta ge that no CPU consuming maximization pro cedure is needed. 5 Count in g rate w eight in g Lik eliho o d Impro ved w eighti ng asymmetry metho d metho d metho d Figure of merit N h β i 2 1 − A 2 h β 2 i h β 2 i 1 − A 2 h β 4 i h β 2 i D β 2 1 − β 2 A 2 E T able 1 The r atio F OM/ N for v arious metho ds d iscu ssed. Before w e mo ve to a comparison of the differen t metho ds, w e note that the estimator ˆ A = P + β + i − P − β − i P + ( β + i ) 2 + P − ( β − i ) 2 + A 0 with β ± = β 1 ± β A 0 (14) reac hes as w ell the FOM of the EML for A ≈ A 0 , In con trast to ˆ A iw its exp ectatio n v alue o nly equals A if A 0 ≈ A . F or τ deca ys the optimal w eight factor in Eq. (14) is discussed in Ref. [6]. 2.5 Comp arison of differ ent metho ds T ab. 1 summarizes the FOM of the v a r io us estimators prop osed. Fig . 1 sho ws the figure of merit o f the differen t estimators vs. A for the c ho ic e α ( x ) = const. = 2 500 and β ( x ) = x, 0 < x < 1 . (15) The curv es are a naly tic calculations. The p oin ts are results of sim ula t io ns . F or eac h v alue of the asymmetry 10000 configurations with α = 2500 , whic h corresp onds on av erage to 50 00 ev ents , w ere sim ulated. One configur a tion consists of a plus and min us data set used to ev aluate an asymmetry . F or each of the 1 0000 configuratio ns sim ulated, the asymmetries we re calculated using the estimators dis cussed abov e. The FOM w as dete rmined from the RMS of the asymmetry distributions. The results are in p erfect agreemen t with the analytic calculations. The sta- tistical errors of the sim ulat ions are of the order of the size of the p oin t s. Note, t ha t for all metho ds no bias w as found for the asymmetry . The question of bias and the range of v alidit y of the expressions for the F OM will be dis- cussed in more detail in Section 4. The weigh ting metho ds are sup erior to the 6 A 0 0.2 0.4 0.6 0.8 1 FOM 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 counting rate weighting improved weighing Likelihood Fig. 1. Figure of mer it for different metho ds as a function of the asymmetry A . metho d using simply the coun ting rates. As exp ec t e d the improv ed weigh ting or the EML metho d reac h a higher FOM than the simple we ig h ting method the la rger the asymmetry A . The results dep end of course o n the shap e of α ( x ) and β ( x ). The gain in F OM using w eigh t ed ev en ts compared to coun ting rates dep ends o n the spread of β ( x ). F or A = 0 for example it is h β 2 i / h β i 2 as can b e deriv ed from Eq. (10). Fig. 2 sho ws the influence of the choic e of A 0 on the FOM in the impro ved w eigh t ing metho d for the factor s α and β as given in Eq. ( 15 ) and an a sym- metry A = 0 . 8. Cho osing A 0 in a range 0.7– 0.86, one reache s a t least 99% of F OM ˆ A LH . The normal w eighting metho d cor r esp onds to A 0 = 0. 3 Differen t acceptance/flux factor in t he t wo data sets W e now turn to the case where the acceptance and flux factor α is not the same in the t wo data se ts. W e ass ume that they differ by a known factor c whic h is indep end en t of x . In this case the differen tial eve nt distributions are giv en by n + ( x ) = 2 c 1 + c α ( x )(1 + β ( x ) A ) and (16) n − ( x ) = 2 1 + c α ( x )(1 − β ( x ) A ) . (17) 7 0 A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 iw A /FOM LH A FOM 0.9 0.92 0.94 0.96 0.98 1 Fig. 2. FOM ˆ A iw / F OM ˆ A LH for A = 0 . 8 as a fu nctio n of A 0 . The factor 2 / (1+ c ) has been in tro duced in order to normalize the distributions to the same num b er of ev en ts for all v alues of c at A = 0. The log lik eliho o d function reads l = X + ln  2 c 1 + c α i (1 + β i A )  − D N + E ( A ) + X − ln  2 1 + c α i (1 − β i A )  − h N − i ( A ) = X + ln (1 + β i A ) + X − ln (1 − β i A ) + X + ln 2 c 1 + c α i + X − ln 2 1 + c α i − 2 Z α d x − 2 A c − 1 1 + c Z αβ d x . (18) Here the last term cannot b e ignored b ecause it con ta ins the parameter A and th us the lik eliho o d metho d cannot b e applied without kno wledge of the factor R αβ d x . The w eighting metho d on the other hand can b e applied with a small mo dification: ˆ A w ,c = P + w i − c P − w i P + w i β i + c P − w i β i . (19) The exp ec tation v alue of ˆ A w ,c equals again A . The figure o f merit reads (deriv a- 8 A 0 0.2 0.4 0.6 0.8 1 FOM 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 c=1 c=2 c=3 Fig. 3. Figure of merit for three different v alues of c as a f unction of the asymmetry A , assu ming the same num b er of ev ents in the case A = 0. tion see s. App. C) F OM ˆ A w,c = N 4 c (1 + c ) 2 h w β i 2 D w 2 (1 − β 2 A 2 )  1 − β A 1 − c 1+ c E . (20) The F OM is show n in Fig. 3 for differen t v alues of c for the improv ed we ig h ting metho d. As in Fig. 1 the lines corresp ond to an analytic calculation using Eq. (20), the points are results of sim ulations. Note, t ha t for arbitrary c the w eigh t reac hing the highest FOM is w = β (1 − β 2 A 2 )  1 − β A 1 − c 1+ c  . The EML method could b e used for c 6 = 1, if one uses an estimate for R αβ d x ≈ P + β ( x i ) + c P − β ( x i ) from data. Sim ulations sho wed that the F O M of this mo dified EML metho d equals the one of the improv ed w eighting metho d. 4 V alidit y at low n um b er of eve n ts In this section w e discuss the v alidity o f t he equations derive d fo r the v arious F OMs and p ossible biases of the estimators. The form ula s w ere deriv ed us- 9 time + - + - + - + - + - + - + - + - + - + - + - + - + - small config. asymmetry larger configuration asymmetry Fig. 4. C ombining data in d iffe r e n t configur ations. T he b o xes denote the plus and min u s data sets. ing usu al error propagation and can th us only be appro ximations whic h are the b etter the higher N . In the simulations presen ted in Section 2.5 for ev ery v alue of the asymmetry 100 0 0 configurations we r e sim ulated with on a verage 5000 even ts (corresp onding to α = 2500 in Eq. (15)). In eac h of these config- urations t he asymmetry was determined using the v ario us estimators. In this section w e discuss effects o ccurring if the asymmetries are extracted in smaller configurations, as indicated in Fig. 4. F o r lo w er n umber of eve n ts in o ne configuration one reac hes a p oin t where due to statistical fluctuations the estimated asymmetry in a giv en configuration can b e larger than 1 or smaller than − 1. In this case the EML metho d is no more applicable since the term (1 ± β A ) can get negative. F or α = 250 this happ ens in ab out 1% of the configurations for an asymmetry of A = 0 . 8. Dividing the sample f ur t her in many smaller configurations one reac hes a p oin t where one ha s o nly 0 or 1 ev en t in one configura t ion. This limit can also b e obt a ine d b y dividing the sample in man y narrow bins of β ha ving a t most one ev en t in a bin. In this case the remaining estimators are iden tical: ˆ A cnt ≡ ˆ A w = β ≡ ˆ A iw = ± 1 /β i . The sign dep end s whether the ev en t o ccurred in the plus or the m in us data set. One finds D ˆ A iw E = A a nd D ˆ A 2 iw E = 1 /β 2 i , th us the FOM for this ev en t reads F OM i = 1 D ˆ A iw E 2 − D ˆ A 2 iw E = β 2 i 1 − β 2 i A 2 . Note that A is not the estimated asymmetry from a s ing le ev ent but rather tak en fr o m a larger ev en t sample. Combining all the asymmetries determined on single ev ents leads to P i ± 1 β i · F OM i P i F OM i = P + 1 β i β 2 i 1 − β 2 i A 2 − P − 1 β i β 2 i 1 − β 2 i A 2 P + β 2 i 1 − β 2 i A 2 + P − β 2 i 1 − β 2 i A 2 . (21) 10 Assuming A = A 0 , Eq. (21) is nothing but the estimator of the impro ve d w eigh t ing metho d, ˆ A iw , defined in Eq. (12). In other words, for the impro ve d w eigh t ing metho d it mak es no differences whether the da t a ar e analyz ed in one large configuratio n or in many small o ne s. The impro ved we ig h ting is also equiv alen t to using a n infinite n umber of bins in β and ev alua t ing the asymmetries in eve ry bin and then com bining the results. The adv an tag e of the impro ved w eighting metho d is that this binning has not t o b e p erformed. The obse rv ations discusse d ab o v e are c onfirmed b y sim ula t ions . In tota l 1 0 9 configurations with α = 0 . 025 w ere sim ulated. The sim ulated data w ere a na- lyzed as fo llo ws. First the asymmetries were calculated in the appro ximately 5% of the configurations actually con taining at least one ev ent. Then the w eigh t e d a verage of these asymmetries is calculated. The same data w ere an- alyzed in a differen t w a y by com bining 10 configurations and calculating the asymmetries in these la rger configurations corresp onding to α = 0 . 2 5 . This pro cedure w as rep eated un t il r e a c hing 10 4 configurations with α = 25 0 0. Fig . 4 illustrates the pr o cedure. Thes e simulations w ere p erformed for an asymmetry of 0.8 generating ev en ts in the range 0 . 01 < β ( x ) < 0 . 99 . Fig. 5 shows the mean v alue of the asymmetries and the stat istical error f or the v a rious estimators for the different v alues of α . As expected the estimator ˆ A iw giv es the same result indep e nden t of α . No bias is observ ed within the statistical error whic h is of the order of 10 − 4 . The asymmetry for the EML metho d is o nly sho wn for α = 2 5 00 since at low er v alues, as explained ab o v e, the EML metho d is no more a pplicable. Fig. 6 sho ws the distribution of the asymmetry ˆ A iw for differen t v alues of α . The en tries in the histograms in Fig. 6 are w eighted b y their correspo nd ing F OM. In the case α = 2500 this w o uld not b e nece ssary b ecause all entries ha ve essen tially the same F OM for a giv en metho d since t he relativ e v aria t ion of the num b er of ev en ts N a nd t he av erages lik e P ± β 2 i / N entering the FOM v ary only v ery little from configuratio n to configuration. At lo w er v alues of α , ho we ver, this is no more the case. T aking ag ain t he extreme case where the asymmetries is calculated from single ev ents , the FOM dep ends on the v alue of β for this even t. This explains wh y the n umber of en tries is smaller tha n 1 in some bins of the histogra m s. At α = 0 . 025 the num b er of configurations is 10 9 . The corresp onding histogram has o nly appro ximately 4 . 9 · 10 7 en tries reflecting the fact that in most of the configuratio n there is no ev en t. Finally , Fig . 7 sho ws the FOM/ N calculated fr o m the RMS of the asymmetry distributions presen ted in Fig. 6 for differen t v a lue s of α . The three lines corresp ond to F OM ˆ A iw / N , FOM ˆ A w = β / N and FOM ˆ A cnt / N calculated using the expressions g iv en in T ab. 1. F or α ≥ 25 there is go od agreemen t with the F OM deriv ed in Section 2 since the p oin ts coincide with the corresp onding lines. A t low er v alues of α the F OM of the w eigh ting and the coun ting rate 11 α -2 10 -1 10 1 10 2 10 3 10 asym metry A 0.7995 0.7996 0.7997 0.7998 0.7999 0.8 0.8001 0.8002 0.8003 0.8004 0.8005 counting rate weighting improved weighing Likelihood Fig. 5. Results for the asymmetry of the simulatio n s as a function of the a ve r a ge n u m b er of eve nts in one configuration. The p oin ts are at α = 0 . 025 , 0 . 25 , 2 . 5 , 25 , 250 , 2500, resp ectiv ely . F or a giv en v alue of α they are sligh tly disp l aced on the horizonta l axis for b etter r e ad ab ility . Note, that v alues at different v alues of α are correlated since the same data were used. metho d start to increase and finally reach as exp ected FOM ˆ A iw at α = 0 . 0 2 5, where the three estimators are practically identical. 5 Summary & Conclusions W e presen ted sev eral estimators to extract an a symmetry parameter A in a num b er densit y function. These estimators w ere based on coun ting ra tes , ev en t w eighting and the un binned extended maximum lik eliho od method. A w eigh t ing pro cedure w as deriv ed tha t reaches the same figure of merit as the un binned maxim um lik eliho o d metho d, kno wn to reach t he minimal v ar ianc e b ound. This w eighting estimator is giv en as an analytic expression, whereas in the EML metho d the maximization of the lik eliho o d function has to b e do ne n umerically . Moreo ve r t his estimator can b e used (with a small mo dification) 12 β w= A -100 -50 0 50 100 nb. of entries 10 4 10 7 10 =0.025 α Entries = 4.88e+07 Mean = 0.7999 RMS = 1.31 11 β w= A -100 -50 0 50 100 nb. of entries -1 10 10 3 10 5 10 7 10 =0.25 α Entries = 3.93e+07 Mean = 0.7999 RMS = 1.1776 β w= A -100 -50 0 50 100 nb. of entries -2 10 10 4 10 7 10 =2.5 α Entries = 9.93e+06 Mean = 0.7999 RMS = 0.5915 β w= A -1 -0.5 0 0.5 1 1.5 2 2.5 3 nb. of entries 2 10 4 10 6 10 =25 α Entries = 1000000 Mean = 0.7999 RMS = 0.1877 β w= A 0.2 0.4 0.6 0.8 1 1.2 1.4 nb. of entries 3 10 4 10 5 10 6 10 =250 α Entries = 100000 Mean = 0.7999 RMS = 0.0595 β w= A 0.7 0.75 0.8 0.85 0.9 nb. of entries 4 10 5 10 6 10 =2500 α Entries = 10000 Mean = 0.7999 RMS = 0.0187 Fig. 6. Distributions of the estimated asymmetries ˆ A iw for different v alues of α . α -2 10 -1 10 1 10 2 10 3 10 FOM/N 0.3 0.35 0.4 0.45 0.5 0.55 0.6 counting rate weighting improved weighing Likelihood Fig. 7. Figur e of merit p er eve nt for differen t metho ds a s a function of α f or an asymmetry A = 0 . 8. The lines sh o w the exp ect ation calcula ted from the exp r essio n s giv en in T ab. 1 . 13 in cases where the EML metho d cannot b e applied b ecause of an incomplete kno wledge o f ev ent distribution function. Ac kno wledgmen t s : I am g rateful to Jean- Marc Le Goff fo r n umerous dis cus- sions on the sub ject, v erifying the calculations and for carefully reading the man uscript. 14 A Figure of merit of ˆ A w T o calculate the F OM o f ˆ A w defined in Eq. (9), o ne needs σ 2 ( P w i ), σ 2 ( P w i β i ) and co v ( P w i , P w i β i ). F or t wo arbit r a ry quan tities f and g the co v ariance b et w een P f i and P g i is co v( X i f i , X j g j ) (A.1) = * X i = j f i g i + X i 6 = j f i g j + − * X i f i + * X j g j + (A.2) = h N i h f g i + h N ( N − 1) i h f i h g i − h N i 2 h f i h g i (A.3 ) = h N i h f g i + D N 2 E − h N i − h N i 2  h f i h g i . (A.4) If the n umber of ev ents N is P oisson distributed, i.e. h N 2 i − h N i − h N i 2 = 0, one finds co v( X i f i , X j g j ) = h N i h f g i ≈ X i f i g i . (A.5) Setting f = g = w , f = g = w β and f = w , g = w β results in σ 2 ( X w i ) = h N i D w 2 E , (A.6) σ 2 ( X w i β i ) = h N i D ( w β ) 2 E , (A.7) co v( X w i , X w i β i ) = h N i D w 2 β E . (A.8) Simple erro r propa g ation in Eq. (9 ) finally leads to Eq. (1 0 ) for the figure of merit. B Optimal weigh t Denoting the w eight factor whic h maximizes the FOM by w 0 , w e consider small deviation f r o m this optim um b y w ( x ) = w 0 ( x ) + ǫ η ( x ) (B.1) where η ( x ) is arbitr a ry a nd ǫ ≪ 1. 15 Inserting Eq. (B.1) in Eq. ( 1 0 ) ke eping terms of 1st order in ǫ one finds F OM = ( h w 0 β i + ǫ h η β i ) 2 h ( w 2 0 + 2 ǫw 0 η ) (1 − β 2 A 2 ) i . (B.2) The conditio n ∂ FOM /∂ ǫ = 0 giv es w 0 = β 1 − β 2 A 2 . C F OM for the case c 6 = 1 The error for the estimator defined in Eq. ( 19) is obtained by simple error propagation taking in to accoun t t he correlatio ns b et we en P w β and P w .  F OM ˆ A w,c  − 1 = ~ v T C ~ v with ~ v T = ∂ ˆ A w ,c ∂ ( P + w i ) , ∂ ˆ A w ,c ∂ ( P − w i ) , ∂ ˆ A w ,c ∂ ( P + w i β i ) , ∂ ˆ A w ,c ∂ ( P − w i β i ) , ! = 1 P + w i β i + c P − w i β i (1 , − c, − A, − cA ) (C.1) and C =           P + w 2 i 0 P + w 2 i β i 0 0 P − w 2 i 0 P − w 2 i β i P + w 2 i β i 0 P + ( w i β i ) 2 0 0 P − w 2 i β i 0 P − ( w i β i ) 2           . (C.2) This leads to Eq. (20). References [1] R. J. Barlow, Statistics, Wiley 1989 [2] R. J. Barlow, Nucl. Instru m. Meth. A 297 (1990) 496. 16 [3] D. Adams et al. [S p in Muon Collaboration (SMC)], Ph ys. Rev. D 56 (1997 ) 5330 [4] E. S. Ageev et al. [COMP AS S Collab o ration], Ph ys. Lett. B 612 (2005) 154 [5] J. Pretz and J. M. Le Goff, Nucl. Instrum. Meth. A 602 (2009 ) 594 [6] M. Da vier, L. Duflot, F. Le Dib erder and A. Rouge, Phys. Lett. B 306 (1993) 411. [7] R. J. Barlow, J. Comput. Ph y s . 72 (1987) 202. 17