Statistics of weighted Poisson events and its applications

Statistic s of weigh ted P oisson ev ents and i ts applications G. Bohm Deutsches Elektr onen-Synchr o tr on D ESY, D-15738 Zeuthen, Germany and G. Zec h 1 Universit¨ at Sie gen, D-57068 Sie gen, Germany Abstract The statistics of the s um of r andom weig h ts wher e the num b er of w eights is P ois- son distributed has i mp ortant a p plications in n uclear physics, particle p hysic s and astroph ys ics. Even ts are frequentl y weigh ted according to their acceptance or rele- v ance to a certain type of r eaction. The sum is d escrib ed by the co mp oun d P oisson distribution (CPD) whic h is shortly r eview ed. It is sh o wn that the CPD can b e appro x im ated b y a scaled Poisson distrib ution (SPD). Th e SPD is applied to p a- rameter estimation in situations where the data are distorted b y resolution effects. It p erforms considerably b etter than the normal appro x im ation that is usually used. A sp ecial Poisson b o otstrap tec hn ique is pr esen ted whic h p ermits to deriv e confidence limits for observ ations follo wing the CPD. Key wor ds: w eighte d eve nts; comp ou n d P oisson d istribution; P oisson b ootstrap; least square fit; parameter estimation; confi d ence limits. 1 In tr o duction In the analys is o f the data collected in particle exp erimen ts, frequen tly w eighted ev en ts hav e to b e dealt with. F or instance, lo sses due to a limited acceptance of the detector are corrected by w eigh t ing eac h ev en ts b y the inv erse of its detection probabilit y . The sum of the w eights is used to estimate the n umber 1 Corresp on d ing author, email: zec h@physik.uni-siegen.de Preprint submitted to Elsevier 6 Septem b er 2013 of inciden t particles. F requen tly an eve n t cannot b e uniquely a ssigned t o a sig- nal or to a bac kground and is attributed a w eigh t. It is necessary to a sso ciate confidence limits, upp er or low er limits to the sum o f these we igh ts and w e w ant to apply go o dness-of-fit tests to histograms o f w eighted ev en ts. When data a r e compared to differen t theoretical predictions, we igh ting simplifies the computation and sometimes it is unav oidable. Of sp ecial imp ortance is param- eter estimation f r o m histograms where the measured v ariable is distorted b y the limited resolution and acceptance of the detector. T o ev aluate these ef- fects, Monte Carlo sim ulations ha v e to b e p erformed. In the sim ulatio n a p.d.f. f 0 ( x ) is assumed to describ e the data where x represen ts the set of eve n t v ari- ables that are measured. The sim ulatio n often requires considerable computer p ow er. T o c hange the p.d.f. of the simu lation to f ( x ), the sim ulat ed ev en ts are w eighted b y f ( x ) /f 0 ( x ). This is una voidable if f ( x | θ ) dep ends on one or sev eral parameters θ that ha v e to b e estimated. During the fitting pro cedure where t he exp erimental distributions ar e compared to the simulated ones, the parameters and corresp ondingly the w eights f ( x | θ ) /f ( x | θ 0 ) a re v aried. The distribution of the sum x of a Poiss o n distributed num b er of w eights is described b y a c omp ound Poisson distribution (CPD), prov ided the w eights can b e considered as indep enden t and iden tical distributed random v ariables. This condition is realized in the ma jo rit y of exp erimen ta l situatio ns. The CPD applies t o a large n umber of problems. In astrophy sics, for instance, the total energy of random airshow ers follows a CPD. Also outside phys ics the CPD is widely used to mo del pro cesses like the sum of claims in car acciden ts and other insurance cases whic h are assumed to o ccur randomly with differen t sev erit y . Usually w e do not disp ose of the distribution of the w eigh t s but hav e to base the analysis on the observ ed w eigh ts of a sample of ev ents. The mean v alue and the v ariance of the we ig h ted sum can b e inferred directly f r o m the cor- resp onding empirical v alues, but in many cases a more detailed kno wledge of the distribution is necessary . T o deal with these situations, the a pproximation of the CPD by a scaled P oisson distribution a nd a sp ecial b o otstrap tec hnique are prop o sed. In the first part of this article some prop erties of the CPD and an appro xima- tion of it that is useful in the analysis of exp erimen tal data are discussed. The second part con tains applications where observ ed samples of w eigh ts ha v e to b e analyzed and where the underlying distribution of the w eights is unknow n. P arameter estimation with w eighted Mon te Carlo ev en ts using a scaled P ois- son distribution and the estimation of confidence limits with P o isson b o otstrap are studied. 2 2 The comp ound Poisson distribution 2.1 Distribution of a sum of weighte d Poisson numb ers The distribution of a weigh ted Poiss on n um b er x = w m with m ∼ P λ ( m ) = e − λ λ m /m !, m = 0 , 1 , 2 , . . . and the w eigh t w , a real v alued p ositiv e parameter, w > 0 is W ( x ) = e − λ λ x/w ( x/w )! . T o ev aluate the momen ts of the Poisson distribution, it is conv enien t to use the cum ulan ts. The momen t s µ k of a distribution are p olynomials o f the cum ulants κ 1 , ..., κ k of the distribution. F o r the first t wo moments µ, σ 2 , the ske wness γ 1 and the excess γ 2 the relations are µ = κ 1 , σ 2 = κ 2 , γ 1 = κ 3 /κ 3 / 2 2 , γ 2 = κ 4 /κ 2 2 . The cum ulants κ k of the P oisson distribution P λ are esp ecially simple, they are a ll iden tical and equal to the mean v alue λ and thus γ 1 = 1 /λ 1 / 2 and γ 2 = 1 /λ . F rom the homogeneit y of the cumu lan ts follow s for the cum ulant κ k ( x ) of order k o f the distribution o f x the relation κ k ( x ) = κ k ( w m ) = w k λ . W e consider now tw o Poisson pro cess es with random v ariables n 1 and n 2 and mean v alues λ 1 and λ 2 . W e are intereste d in the distribution of the we igh ted sum x = x 1 + x 2 = w 1 n 1 + w 2 n 2 with p ositiv e w eights w 1 , w 2 . The mean v alue and the v ariance of x are E( x ) = w 1 λ 1 + w 2 λ 2 , (1) V ar( x ) = w 2 1 λ 1 + w 2 2 λ 2 . (2) These results follow from the prop erties of exp ected v alues and a r e in tuitively clear. The cum ulant o f the distribution of the sum of tw o indep enden t ra ndom v ariables x 1 and x 2 is the sum of the tw o cum ulants: κ k = w k 1 λ 1 + w k 2 λ 2 . (3) Relation (3) can b e generalized to N Poiss o n pro cesses with mean v alues λ i : κ k = N X i =1 w k i λ i . (4) W e will see b elow that the case is o f sp ecial intere st where all mean v alues λ i are equal. With λ i = λ/ N , and x = Σ w i the mo dified relat io n for the cum ulan ts is κ k = λ Σ w k i / N = λ D w k E , (5) 3 and µ , σ , γ 1 , γ 2 are: µ = λ Σ i w i / N = λ h w i , (6) σ 2 = λ Σ i w 2 i / N = λ D w 2 E , (7) γ 1 = λ Σ i w 3 i / N σ 3 = h w 3 i λ 1 / 2 h w 2 i 3 / 2 , (8) γ 2 = λ Σ i w 4 i / N σ 4 = h w 4 i λ h w 2 i 2 . (9) Here h v i denotes the mean v alue Σ N i =1 v i / N . So far w e ha ve treated the w eights as parameters, while according to the definition of a comp o und P oisson pro cess, the weigh ts are ra ndom v ariables. 2.2 Distribution of the sum of r andom weights In most applications of particle phy sics t he distribution o f the sum o f indi- vidually weigh ted ev en ts is o f interes t. The n um b er n of ev ents is describ ed b y a Pois son distribution and t o eac h ev ent a random we igh t is asso ciated. Instead of the N indep enden t P o isson pr o cesses with mean v alues λ i and ran- dom v ariables n i w e can consider the ra ndom v ariable n = Σ n i as the result of a single Poisson pro cess with λ = Σ λ i . The num b ers n i are then chosen from a multinomial distribution where n is distributed to the N different w eight classes with probabilities ε i = λ i /λ , i.e. a w eigh t w i is chose n with probabilit y ε i : W ( n 1 , ..., n N ) = N Y i =1 P ( n i | λ i ) = P λ ( n ) M n ε 1 ,...,ε N ( n 1 , ..., n N ) , (10) M n ε 1 ,...,ε N ( n 1 , ..., n N ) = n ! N Y i =1 ε n i i , N Y i =1 n i ! . (11) The v alidit y of (1 0) is seen from the following identit y f o r the binomial case, P λ M n λ 1 /λ,λ 2 /λ = e − λ λ n n ! n ! n 1 ! n 2 ! λ n 1 1 λ n 2 2 λ n 1 λ n 2 = e − ( λ 1 + λ 2 ) λ n 1 1 λ n 2 2 n 1 ! n 2 ! = P λ 1 P λ 2 whic h is easily generalized to the multinomial case. It do es not matter whether w e describ e the distribution of x b y indep enden t P oisson distributions or b y the pro duct o f a single P oisson distribution with a m ultinomial distribution. If a ll probabilities are equal, ε i = 1 / N , the m ultino- mial distribution describes a random selection o f the w eigh ts w i out of the N w eigh ts with equal probabilities 1 / N . The form ula s (6) to (9) remain v alid. 4 T o describ e a contin uous w eight distribution f ( w ) with finite v ariance, the limit N → ∞ has to b e considered. Again our form ula s remain v alid with εN = 1. W e get x = Σ n i =1 w i . The mean v alues D w k E in (5), (6), (7), (8), (9) are to b e replaced b y the corresp onding exp ected v alues E( w k ), e.g. t he momen ts of the w eight distribution. 3 Appro ximation b y a scaled Poisson distribution T o analyze the sum of we ig h ts of observ ed samples where the underlying w eigh t distribution is not kno wn, it is necessary to appro ximate t he CPD. According to the central limit theorem, the sum of w eigh ted P oisson rando m num b ers with mean n um b er λ and exp ected w eight E( w ) can asymptotically , for λ → ∞ , b e describ ed b y a normal distribution with mean µ = λ E( w ) and v ariance σ 2 = λ E( w 2 ), prov ided the exp ected v alues exist. The sp eed of con v ergence with λ dep ends o n the distribution of the weigh ts. As is demonstrated b elow , the moments of the CPD are closer to those of a sc ale d Poisson distribution (SPD ) than to the momen ts of the normal distri- bution. Especially , if the w eigh t distribution is narrow, the SPD is a v ery go o d appro ximation of the CPD and in the limit where a ll weigh ts are identic al, it coincides with the CPD. The SPD is fixed by the requiremen t tha t the first tw o momen ts of the CPD ha ve to b e repro duced. W e define an equiv alen t mean v alue ˜ λ , ˜ λ = λ E( w ) 2 E( w 2 ) (12) = µ E( w ) E( w 2 ) = µ s , (13) an equiv alen t r a ndom v ariable ˜ n ∼ P ˜ λ , a scale factor s , s = E( w 2 ) E( w ) , (14) and a scaled r andom v ariable ˜ x = s ˜ n suc h that the exp ected v alue E( ˜ x ) = E( x ) = µ and the v ariance V ar( ˜ x ) = V ar( x ) = σ 2 . The cum ulan ts of the scaled distribution are ˜ κ k = s k ˜ λ . T o ev aluate the quality of the approximation o f the CPD by the SPD, we compare t he cum ulan ts of the tw o distributions a nd form the r a tios κ k / ˜ κ k . P er definition the ratios for k = 1 , 2 agree b ecause the t wo lo we st momen ts agree. 5 The sk ewness a nd excess for the tw o distributions in terms of the exp ected v alues of p ow ers k of w , E( w k ) a re according to (8), (9) and (12): γ 1 = E( w 3 ) λ 1 / 2 E( w 2 ) 3 / 2 , (15) γ 2 = E( w 4 ) λ E( w 2 ) 2 , (16) ˜ γ 1 = 1 ˜ λ 1 / 2 = " E( w 2 ) λ E( w ) 2 # 1 / 2 , (17) ˜ γ 2 = 1 ˜ λ = E( w 2 ) λ E( w ) 2 . (18) F or the ratios w e obtain γ 1 ˜ γ 1 = E( w 3 )E( w ) E( w 2 ) 2 ≥ 1 , (19) γ 2 ˜ γ 2 = E( w 4 )E( w ) 2 E( w 2 ) 3 ≥ 1 . (20) The pro of o f these inequalities is g iv en in the App endix. As γ 1 and γ 2 of the normal distribution a re zero, the v alues γ 1 and γ 2 of the CPD are closer to those of the SPD than to those of the normal distribution. This prop erty suggests t ha t the SPD is a b etter approxim ation to t he CPD than the normal distribution. According to the cen tral limit theorem, CPD and SPD approach the norma l distribution with increasing ˜ λ . The equalities in (19) and (20) hold if all we igh ts are equal. Remark that the ratios do not dep end on the expected n umber λ o f weigh ts, only the momen ts of the w eight distribution en t er. The ratios are close to unity in most pra ctical cases. They can b ecome large if the w eigh t distribution comprises w eights tha t differ considerably and esp ecially if man y small w eigh ts a re com bined with few large w eights. This is the case, for instance, for an expo nential w eight distribution. In Figure 1 the results of a sim ulation of CPDs with differen t we igh t distri- butions is presen ted. The sim ulated ev en ts are collected into histog ram bins but the histog rams are displa y ed as line g r a phs whic h are easier to read than column graphs. Corresp onding SPD distributions are generated with the pa- rameters chosen according t o the relations (12) and (14). They ar e indicated b y dotted lines. The approx imations by normal distributions are sho wn a s dashed lines. D ue to the discrete P oisson distribution the histograms for the comp osite P o isson distribution and the SPD hav e pronounced structures that mak es it difficult to compare the results. T o a v oid at least partially this dis- turbing effect, the binning w as adapted to the steps of the SPD. The w eigh t distribution of the t o p left graph is uniform in the interv al [2 , 3] and t he weigh t 6 0 1 0 2 0 3 0 4 0 5 0 1 0 0 1 5 0 0 1 0 2 0 3 0 4 0 5 0 1 0 0 1 5 0 µ = 2 0 f ( w ) = e xp ( - w ) x µ = 5 0 P ( w = 1 ) = 0 . 9 P ( w = 1 0 ) = 0 . 1 x µ = 2 5 t r u n ca t e d n o r m a l x µ = 5 0 f ( w ) = 1 , [ 2 , 3 ] x Fig. 1. Comparison of a CPD (solid) with a SPD (dotted) and a n orm al d istribution (dashed) distribution of the top righ t graph is a truncated, renormalized normal dis- tribution N t ( x | 1 , 1) = c N ( x | 1 , 1), x > 0 with mean and v ariance equal t o 1 where nega tiv e v alues are cut. In b o t h cases the appro ximatio n by the SPD is hardly distinguishable from the CPD. In the b ottom left graph the w eights are exp o nen tially distributed. This case inhibits larg e w eights with low f requency where the approximation b y the SPD is less go o d. Still it mo dels the CPD reasonably w ell. In the b ott o m right graph t he we ig h t distribution is discrete with the w eight w 1 = 1 chosen with probabilit y 0 . 9 and the w eight w 2 = 1 0 c hosen with probability 0 . 1. This is ag ain an extreme situation. The SPD a nd the CPD agree reasonably w ell globa lly , but hav e differen t discrete structures whic h result in jumps caused by the binning. The examples sho w, that the appro ximation by the SPD is mostly close to the CPD and that it is a lwa ys sup erior to the approximation b y the normal distribution. In T able 1 w e compare sk ewness γ 1 and excess γ 2 of the SPD to the v alues of the CPD. The mean v alues from 100 0000 sim ulated exp eriments are tak en. The mean nu m b er of w eights is alw ays 50, e.g. n ∼ P 50 ( n ). The w eights used to obtain the first 3 ro ws a re uniformly distributed in the indicated in terv al. The we igh ts of the following row are distributed according to exp( − w ), the w eigh ts of the next row follow the truncated no rmal distribution. The last t wo ro ws corr esp o nd to tw o discrete w eigh ts, w 1 = 1 and w 2 = 10 c ho sen 7 T able 1 Sk ewn ess and excess of the SPD appr o ximation t yp e of w eight ˜ λ γ 1 γ 2 ˜ γ 1 ˜ γ 2 u [0 , 1] 37 . 50 0 . 184 0 . 036 0 . 163 0 . 027 u [1 , 2] 48 . 21 0 . 149 0 . 023 0 . 144 0 . 021 u [2 , 3] 49 . 34 0 . 144 0 . 021 0 . 142 0 . 020 exp ( − w ) 25 . 0 0 0 . 300 0 . 120 0 . 200 0 . 040 N t (1 , 1) 36 . 48 0 . 199 0 . 045 0 . 166 0 . 027 1 ( p = 0 . 5), 10 29 . 94 0 . 197 0 . 039 0 . 182 0 . 033 1 ( p = 0 . 8), 10 19 . 01 0 . 299 0 . 092 0 . 229 0 . 052 with equal pro ba bilit ies and with w 1 = 0 . 8 and w 2 = 0 . 2 , resp ectiv ely . The second column indicates the num b er of equiv alen t ev ents ˜ λ defined in (12) , e.g. the nu m b er of un we ig h ted even ts with the same relative uncertain ty as the weigh ted sum x . F or example, the relative fluctuation δ x/x of the sum x of n ∼ P 50 ( n ) random w eigh ts with w ∼ exp ( − w ) is 1 / √ 25 = 0 . 2. The follo wing columns con tain the v alues of γ 1 and γ 2 of the CPD and those from the scaled Poiss o n distribution. The v a lues of the normal appro ximation a re γ 1 = γ 2 = 0. The first tw o momen ts ar e p er definition equal for the CPD and the SPD. The SPD v alues are close to the no minal v alues if the w eigh t distribution is rather narrow corresp onding to ˜ λ/λ close to one. Remark that in the cases where the ratio ˜ λ/λ is small, sk ewness and exce ss are relativ ely large and corresp ondingly , the normal appro ximatio n with γ 1 = γ 2 = 0 is not very go o d. As in the limit n → ∞ b oth, the CPD a nd the SPD, a pproac h the normal distribution, small eve n t num b ers, or, more precisely , small v alues of ˜ λ a re esp ecially critical. 4 The P oisson b o otst rap In standard b o otstrap ([3]) samples are dra wn from the observ ed observ ations x i , i = 1 , 2 , ..., n , with replacemen t. P oisson b o o t strap is a sp ecial re-sampling tec hnique where to all n observ ation x i P oisson distributed num b ers n i ∼ P 1 ( n i ) = 1 / ( en i !) are asso ciated. More precisely , fo r a b o otstrap sample the v alue x i is tak en n i times where n i is ra ndomly c ho sen fro m the P oisson distribution with mean equal to one. Samples where the sum of outcomes is differen t from the observ ed sample size n , e.g. Σ n i =1 n i 6 = n are rejected. P oisson b o o tstrap is completely equiv alent to the standard b o otstrap. It has attractive theoretical prop erties [4]. 8 In our case the situation is differen t. W e do not disp ose o f a sample of CPD outcomes but o nly of a single o bserv ed v alue of x which is accompanied by a sample of w eights. As the distribution of the num b er of w eigh ts is known up to the Poiss on mean, t he b o o tstrap tec hnique is used to infer parameters dep ending on the w eight distribution, T o generate o bserv a tions x k , w e hav e to generate the num b ers n i ∼ P 1 ( n i ) and form the sum x = Σ n i w i . All results ar e k ept. The resulting Poiss o n b o ot stra p distribution (PBD) p ermits to estimate uncertain ties of parameters and quantiles of the CPD. Mean v alues deriv ed from an infinite n um b er of sim ulated exp erimen ts and the momen ts extracted from the corresp onding PBD s would repro duce exactly the moments of the CPD. 5 Applications In most applications we do not kno w the we igh t distribution and ha ve to in- fer it approx imat ely from a sample of w eigh t s, w i , i = 1 , ..., n . T o this end w e replace the moments of the w eigh t distribution b y the empirical v alues. A general approac h to approximate the distribution of a sample starting from the cumulan ts is to apply the Edgeworths [1,2] series. Since this metho d is in v olv ed and not directly related to t he Poiss on distribution, it has not b een in v estigated. The Gr a m-Charlier series B [1] con tains explicitly a Poiss on term, but it is not clear how well the tr uncated series appro ximates the CPD. F ur- thermore the higher empirical cum ulan ts κ 3 , κ 4 , ... in most applications suffer from rather large statistical fluctuations. Therefore it is o ften more precise to use the v alues tied t o the mean and the v ariance in the approximation b y the SPD. In addition to the SPD, we consider the simple normal appro ximation and P oisson b o otstrap. 5.1 Par ameter estimation fr om dis torte d me asur ements An imp ortant application o f the statistics of w eighted ev ents is par a meter es- timation in exp erimen ts where the data are distorted b y resolution effects [5]. T ypically , an exp erimen tal histog ram with m j en tries in bin j ha s to b e com- pared to a theoretical prediction x j ( θ ) dep ending on o ne or sev eral parameters θ . The prediction is o bta ined fro m a Monte Carlo sim ulat ion whic h repro duces the exp erimen tal conditions and especially the smearing by resolution effects. The v ariation of the prediction with the parameter cannot b e implemen ted by rep eating the complete sim ulation for eac h selected parameter. Therefore the sim ulated dat a whic h are generated with the para meter θ 0 according to the p.d.f. f ( θ 0 ) are re-we igh ted b y the ratio w = f ( θ ) /f ( θ 0 ). The prediction for a histogram bin j is t hen x j = Σ n j i =1 w j i for n j generated ev ents in bin j . T o 9 p erform a least square fit of θ to a histog r am with B bins, w e form a χ 2 ex- pression where w e compare P oisson num b ers m j times a known normalization constan t c to comp ound Poiss o n num b ers x j . χ 2 = B X j =1 ( cm j − x j ( θ )) 2 δ 2 j , = B X j =1 cm j − n j X i =1 w j i ! 2 δ 2 j . Here δ 2 j is the exp ected v alue of the nume rator under the h yp o t hesis that the t wo summands in t he brack et ha ve the same exp ected v alue µ . T o estimate δ 2 j first µ has to b e estimated. In the normal approximation, w e compute the w eighted mean of the t w o sum- mands (W e suppress the index j .): ˆ µ N = cm c 2 m + Σ w i Σ w 2 i ! / 1 c 2 m + 1 Σ w 2 i ! . (21) In the approximation based on the SPD, the v alue of µ can b e estimated from an approxim ated lik eliho o d expression. The log lik eliho o d is [5] ln L ( µ ) = m ln µ c − µ c + ˜ n ln ˜ λ − ˜ λ + const. (22) where it is assumed that m follo ws a P o isson distribution with mean µ/c and ˜ n = x/s a P oisson distribution with mean ˜ λ = µ/s , see (12), (14). The maxim um lik eliho o d estimate is [5] ˆ µ S P D = cs ˜ n + m c + s (23) and the corresp onding estimate of δ 2 is ˆ δ 2 S P D = cs ( ˜ n + m ) . T o ev aluate the quality of the tw o approx imations, 10000 00 exp eriments hav e b een sim ula t ed for different combinations of ev en t num b ers and w eight dis- tributions. The results ar e summarized in T able 2. Here λ n and λ m are the exp ected n umbers of data a nd Mon te Carlo ev en t s, µ is the mean v alue of x that has b een used in the sim ulation and that should b e repro duced by the estimates, ˆ µ S P D is the mean v alue of the SPD estimates for µ , σ S P D is t he standard deviation of the estimates and ˆ µ N , σ N are the corresp onding v alues 10 T able 2 Comparison of the SPD and the normal appro x im ations λ n λ m w eight µ ˆ µ S P D σ S P D ˆ µ N σ N ˆ µ S P D − µ µ ˆ µ N − µ µ 20 20 exp ( − x ) 20 19 . 98 3 . 68 19 . 10 3 . 84 0 . 001 0 . 045 10 10 exp ( − x ) 10 9 . 73 2 . 64 9 . 11 2 . 81 0 . 027 0 . 089 10 50 exp ( − x ) 10 9 . 88 1 . 38 9 . 58 1 . 59 0 . 012 0 . 042 20 50 exp ( − x ) 20 19 . 84 2 . 61 19 . 46 2 . 74 0 . 008 0 . 027 50 50 exp ( − x ) 50 49 . 78 5 . 79 49 . 12 5 . 91 0 . 004 0 . 013 10 10 N t (1 , 1) 12 . 88 12 . 78 3 . 13 12 . 05 3 . 30 0 . 008 0 . 068 20 20 N t (1 , 1) 25 . 7 5 25 . 67 4 . 40 24 . 93 4 . 53 0 . 003 0 . 032 20 50 N t (1 , 1) 25 . 75 25 . 69 3 . 22 25 . 27 3 . 31 0 . 002 0 . 019 10 10 u [2 , 3] 25 . 00 25 . 00 5 . 61 23 . 74 5 . 87 0 . 000 0 . 050 20 20 u [2 , 3] 50 . 00 50 . 00 7 . 94 48 . 74 8 . 13 0 . 000 0 . 025 50 50 u [2 , 3] 125 . 00 125 . 01 12 . 54 123 . 75 12 . 67 0 . 000 0 . 010 for the normal appro ximation. The notation of the w eight distributions is t he same as ab ov e. All estimates of µ are negativ ely biased but as exp ected the SPD v alues are considerably closer to the nominal v alues than those o f the normal a ppro ximation. The bias for the SPD is in all cases b elo w 3% whic h is certainly adequate for the estimation of the uncertain ty δ . The fluctuations are any w ay muc h larg er than the biases b oth for the SPD and the normal distribution. Both approximations are adequate, the a ppro ximation with the SPD is slightly sup erior to that with the normal appro ximation and leads to a simple result of the estimate of the v ariance ˆ δ 2 . Indep enden t of the w eight distribution the biases decrease with increasing n um b er of ev ents . In most cases it will b e p ossible t o generate a sufficien t num b er of ev ents suc h that ˜ λ is of the order of 50 or la r g er. 5.2 Appr oximate c onfid e nc e lim its In searc hes for rare ev ents frequen tly the iden tification is not unique a nd to eac h ev en t is a t tributed a weigh t whic h corresp o nds to the proba bility to b e correctly assigned. The underlying w eigh t distribution is not kno wn. Of in terest is the n um b er of pro duced ev ents x = Σ w i and confidence limits for this n um b er. The limits can b e computed from the P oisson b o otstrap distribution. As an example, a sample of n w eights, with n ∼ P 50 ( n ) has b een generated with a uniform w eight distribution in the in terv al [0 , 1]. The v alue x obs = 11 T able 3 Confidence limits α 0.01 0.05 0.10 0.1585 0.8415 0.90 0.95 0.99 PBD 13.8 16.0 17.2 18.2 25.8 26.9 28.5 31.4 PBD* 14.4 16.5 17.6 18.5 26.2 27.3 28.9 32.1 Σ n i =1 w i = 22 . 01 w as obtained. The frequency plot o f t he corr esp o nding b o ot- strap distribution f ( x ) is displa y ed in Fig . 2 left hand side. This distribution w as used to deriv e the error and confidence limits presen ted in T able 3. The limits corresp onding to the α quan tiles x , defined b y α = F ( x ) = R x 0 f ( x ′ ) dx ′ , indicated in the to p line of the table, are quoted in the second line. The usual standard error in t erv al is 18 . 2 < x < 25 . 8. Classical confidence interv als with exact cov erage cannot b e computed as the full CPD is know n only approximately . But as w e kno w the type of the distri- bution for the n umber of ev en ts, w e can improv e t he cov erage in the f ollo wing w ay : W e c hange the Pois son distribution used to generate the b o otstrap sam- ples from P 1 to P µ suc h that the fra ction of outcomes x b elo w x obs is equal to α . The upp er limit is then x up = x obs × µ . In a similar wa y t he lo wer limit x low is obtained. The cen tral interv al x low < x < x up should then con tain the unkno wn true v alue with confidence 1 − 2 α . In the limit where a ll w eights are equal and the num b er of b o o t stra p samples tends to infinity the in terv al w ould co ver exactly . The obtained limits are contained in the third line of the table. The t w o pro cedures lead to v ery similar v alues. The mo dified error in tev al is no w 18 . 5 < x < 26 . 2. As exp ected from the prop erties of the Poiss on distribution, the in terv als with improv ed co v era g e are shifted to hig her v alues. The P oisson b o otstrap can b e used to estimate distributions of all kinds of parameters of the distribution. As an example the distribution of the sk ewness deriv ed from the observ ed w eight sample is presen ted in the righ t hand plot of Fig. 2. 6 Summary The sum of random w eigh ts where the n umber o f w eights is P oisson distributed is describ ed b y a comp ound P oisson distribution. Prop erties of the CPD are review ed. The CPD is relev an t for the analysis of w eighted ev ents that ha s to b e p erfo r med in v arious ph ysics applications. It is sho wn that with increasing num b er of ev ents t he distribution of t he sum can b e approxim ated by a scaled P oisson distribution whic h coincides with the CPD in the limit where all w eigh ts are equal. Con trary to the normal distribution it approx imately repro duces a lso the higher the momen ts of the 12 1 0 2 0 3 0 4 0 0 2 0 0 0 0 4 0 0 0 0 n u m b e r o f e n t r i e s x 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 0 2 0 0 0 0 4 0 0 0 0 6 0 0 0 0 sk e w n e ss Fig. 2. Bo otstrap distribution of the r andom v ariable x (left hand )and of γ 1 (righ t hand). CPD. The SPD can b e applied to the pa rameter estimation in situations where the data are distorted by resolution effects. The formalism with the SPD is simpler than that with the normal approx imat ion a nd the results ar e more precise. This ha s b een demonstrated for examples with v arious w eight distributions. A sp ecial b o otstrap metho d is presen ted whic h can b e used to estimate from exp erimental samples parameters of the underlying CPD. An example sho ws ho w it can b e applied t o the estimation of confidence limits. 7 App endix: Pr o of of the Inequalities (19) and (20) W e apply H¨ olders inequalit y , X i a i b i ≤ X i a p i ! 1 /p X i b p/ ( p − 1) i ! ( p − 1) /p , where a i , b i are non-negat iv e a nd p > 1. F or p = 2 we obtain the Cauc hy – Sc h w artz inequality . Setting a i = w 3 / 2 i , respective ly b i = w 1 / 2 i , w e get immedi- ately the relation (1 9) for the sk ewness: X i w 2 i ! 2 ≤ X i w 3 i X i w i . 13 More generally , with p = n − 1 and a i = w n/ ( n − 1) i , b i = w ( n − 2) / ( n − 1) i , the inequalit y b ecomes X i w 2 i ! n − 1 ≤ X i w n i X i w i ! n − 2 . This formula includes also t he relation (20) for n = 4. References [1] M.G. Kendall and A. Stu art, The A dvanc e d The ory of Statistics , C harles Griffin & Co., London, Ed. 4 (1948). [2] E.W. W eisstein, Edgeworth Series , http://math w orld .wolfram.com ; en.wikip edia.org/wiki/Edgew orth series. [3] B. Efron and R.T. Tibsh irani, An Intr o duction to the Bo otstr ap , Chapman& Hall, London (1993). [4] G.J. Babu et al., Se c ond-or der c orr e ctness of the Poisson b o otstr ap , T he Annals of S tatistics V ol 27, No. 5 (1999) 1666. [5] G. Bohm and G. Zec h, Comp aring statististic al data to Monte Carlo simulation with weig hte d events , Nucl. In s tr. and Meth A691 (2012) 171. 14