Symbolic computation of moments of sampling distributions

Sym b olic computation of momen ts of sampling distribu tions E. Di Nardo ∗ , G. Guarino † , D. Senato ‡ No v ember 4, 2018 Abstract By means of th e n otion o f um brae indexed by m ultisets, a general metho d to express estimators and their pro ducts in terms of p ow er sums is d erived. A connection b etw een the notion of multiset and integer partition leads immediately to a w ay to sp eed up the pro cedures. Comparisons of com- putational times with known p ro cedu res show h ow this approach turn s out to b e more efficient in eliminating muc h unnecessary computation. keyw ords: Umbral calculus; sy mmetric functions; moments of moments; sampli n g distributions; U -statistics AMS 2000 Sub ject Classification: primary 05A40, 65C 60, secondary 62H12, 68W30 1 In tro d uction It is acknowledged that an appropria te c hoice of la nguage a nd notation can simplify and clar ify many statistical ca lculations. In recent years, most of the work has b een done by sy mbo lic computation, main references are [14], [2]. These b o o ks o ffer a v arie ty of applicatio ns of symbolic metho ds, from as ymptotic expansions to the Edgeworth series, from likelihoo d functions to the saddlep oint approximations. In [25], Zeilb erger describ es a methodolog y for using computer algebra systems to automatically derive momen ts, up to order 4 , of interesting combinatorial r andom v aria bles. Such a metho dolo gy is applied to pattern statistics o f p er m uta tions. F or differ ent a pplications of computer algebr a in statistics, see also [1 8]. The a im of this pa p er is to s how the co mputational efficiency of umbral calculus in manipulating expressions inv olving ra ndom v aria bles. The umbral ca lc ulus to which we r efer is the version feature d by Rota a nd T aylor in [20]. The bas ic device is the representation of a unital sequence of n umbers by a sym b ol α, calle d an umbr a , tha t is, the sequence 1 , a 1 , a 2 , . . . is represented by the sequence 1 , α, α 2 , . . . of powers of α via an oper a tor E resembling the exp ectation o p e r ator of random v ariables. This approa ch has led to a finely ada pted language for random v aria bles by Di Nardo and Senato, see [7]. In [8 ], a tten tio n is fo cused on cumulan ts since a r andom v ariable is often better describ ed by its cumulan ts than b y its moments, as it happens for the family of Poisson rando m v ar iables. Moreov er, due to the pr op erties o f additivity and inv ar iance under translatio n, cumulan ts are not necessa r ily connected with moments of any pro bability distribution. As a matter of fact, an umbra seems to ha ve the str ucture of a ra ndom v ariable but with no r eference to a pro bability spa c e , bringing us clo ser to statistica l metho ds. In [9], it is shown that classica l umbral ∗ Dipartiment o di Matematica, Universit` a degli Studi dell a Basil icata C.da Macchia Romana, I-85100 Pote nza, E-mail: elvira.d inardo@unibas.it † Medical Sc ho ol, Uni v er sit` a Cattolica del Sacro Cuore (Rome branc h), Largo Agostino Gemelli 8, I-00168, Roma, Italy ., E-mail: giuseppe.guari no@rete.basilicata.it ‡ Dipartiment o di Matematica, Universit` a degli Studi dell a Basil icata C.da Macchia Romana, I-85100 Pote nza, E-mail: domenico .senato@unibas.it 1 2 calculus pr ovides a unifying framework for unbiased estimator s of cumulan ts , called k -statistics, and their m ultiv ar ia te gener alizations. Moreov er , within the umbral framework, a statistical result do es not r equire a chec k of background details by ha nd, but b ecomes a corolla ry o f a more gener al theorem. Here, we fo cus attention on the mo re genera l problem o f ca lculations o f algebr aic expres s ions such a s the v aria nce of a sample mean or, more generally , moments of s a mpling distr ibutions, whic h hav e a v ariety of applicatio ns within statistica l inference [23]. The field has , in the pa st, b een mar red by the difficulty of manual computations. Symbolic computations ha ve remov ed many of such difficulties, leaving so me issues unreso lved. O ne o f the most intriguing questions is to explain why symbolic pr o cedures, which are straightforward in the multiv ariate ca se, turn out to b e obsc ur e in the s impler univ ariate o ne, see [2]. W e show that the no tion of mult is et is the key for de a ling with symbo lic computation in mult iv ar iate statistics. Actually , a t the r o ot of the ques tio n, there are some asp ects of the c ombinatorics of symmetric functions that would b enefit if the attention is shifted from sets to the mo re g eneral notio n of multiset. On the o ther hand, due to its g enerality , umbral calculus reduces the co mbin a torics of symmetric functions, commonly used by statisticia ns, to few relations which cover a g reat v ariety of c alculations. In par ticula r um br al equiv alences (2 1) smo o th the wa y to handle any kind of pro duct of sums. As example, by using equiv alences (21), we ev aluate the mea n of pro duct of aug ment e d p olynomials in s eparately indep endent and identically distributed random v ariables. The re s ulting um br al str ategy is completely different fro m those re c ent ly pr op osed in the literature, see fo r instance [2 4], and co mputationally more efficient, as we show in the last section. Moreov e r , by means o f the notio n of umbrae indexed by m ultiset, we remove the neces sity to sp ecify if the random v ariables of a vector are identically distributed or not. The basic pro cedure consis ts in finding m ultise t sub divisions, which s uita bly extends the notion of set partitions. The strategy her e pr op osed is a so rt of iterated inclusio n-exclusion rule [1 , 3], whose efficiency is improved taking into ac c ount the structure o f multiset and its relation with integer partitions. The r esult is the a lgorithm mak eTab , given in the app endix. The pap er is structured as follows. Section 2 is pr ovided for readers unaw are of classica l umbral calculus. W e resume terminology , nota tion and so me basic definitions. In Section 3, w e give a general pro cedure for writing do wn U -s tatistics. Re c a ll that many statistics of interest may be exa c tly represented or approximated b y U -s tatistics [1 1]. Suc h a pro cedur e is bas e d on the umbral rela tio n b etw een moments and augmented symmetric functions. The connection with pow er sums is analyzed in Section 4. The effectiveness o f umbral metho ds is shown in s everal ex a mples, pro p osed with the int ention of he lping the reader una ware of um bral calculus to understand the basic algebraic rules necessary to work with this syntax. Section 5 is dev oted to umbral formulae giving power sums in terms of a ug mented symmetric functions. These formulae provide the most na tural wa y to form the pro duct of augmen ted symmetric functions by using a suita ble umbral substitution. The co nsequent r eduction o f the co mputational time is made clear through some exa mples, w hich p o int out the role play ed by the singleton umbra in s e le cting the suitable v ar iables. Section 6 is devoted to computational compar isons with the pro cedures k nown in the literature in dealing with momen ts of s ampling distributions. The sp eed up of umbral metho ds is evident. Some concluding remarks e nd the pap er . Although, by a numerical p oint of view, MAPL E seems to work less efficiently res pe c t to MAT HEMATI CA , we have implement ed o ur alg orithms in M APLE b ecause the syntax is more comfortable for symbolic com- putation. All task s have b een p erfor med on a PC Pentium(R )4 In tel(R), CP U 3.00 Ghz, 480MB Ram with M APLE version 10 .0 and MA THEMA TICA version 4.2 . 3 2 The classical um b ral calculus Classical umbral calc ulus is a syntax consisting o f the following data: i) a set A = { α, β , . . . } , called the alphab et , whose ele ment s are named umbr ae ; ii) a commutativ e integral domain R whose quotient field is of characteristic zero ; iii) a linear functional E , called an evaluation , defined on the p olyno mial ring R [ A ] and taking v alues in R such that a) E [1] = 1; b) E [ α i β j · · · γ k ] = E [ α i ] E [ β j ] · · · E [ γ k ] for any set of distinct umbrae in A and for i, j, . . . , k non-negative integers ( unc orr elation pr op erty ); iv) an element ε ∈ A, called an augment ation , s uch that E [ ε n ] = 0 for all n ≥ 1 ; v) an element u ∈ A, ca lled a unity um br a, such that E [ u n ] = 1 for all n ≥ 1 . Note that, for statistical applications, R is the field of rea l num b ers. An u mbr al p olynomial is a p olynomia l p ∈ R [ A ] . T he s uppo rt of p is the s et of all umbrae o cc urring in p. If p and q are tw o umbral p olynomials, then i ) p and q are unc orr elate d if and only if their s uppo rts a re disjoint; ii ) p and q are umbr al ly e quivalent iff E [ p ] = E [ q ] , in symbols p ≃ q . The basic idea of the classical umbral calculus is to asso ciate a sequence of num b ers 1 , a 2 , a 3 , . . . to an indeterminate α, which is said to repre s ent the s equence. This devic e is familiar in statistics, when a i represents the i - th moment of a ra ndo m v ariable X . In this ca se, the sequence 1 , a 1 , a 2 , . . . results from applying the exp ecta tion op erator E to the s equence 1 , X , X 2 , . . . c o nsisting of p ow ers of X . This is why the elements a n ∈ R such tha t E [ α n ] = a n , n ≥ 0 are na med moments of the umbra α and we say that the um br a α r epr esents the sequence of moments 1 , a 1 , a 2 , . . . . The um br a ǫ plays the same role of a r andom v a riable which takes the v a lue 0 with probability 1 and the umbra u plays the same r ole of a random v aria ble which takes the v alue 1 with pr obability 1. The uncorrelation pr op erty among um bra e par allels the ana logue one for random v a riables. In this setting no attention must b e paid to the well-known “mo ment problem” . In parallel with r andom v ariable theory , the factorial moments of an um bra α are the element s a ( n ) ∈ R corres p o nding to the um bral po lynomials ( α ) n = α ( α − 1) · · · ( α − n + 1) , n ≥ 1 , via the ev aluation E , that is E [( α ) n ] = a ( n ) . There are umbrae playing a sp ecia l role in the um br al calculus. Their prop erties hav e b een in vestigated with full particula rs in [7, 8]. Singleton um bra. The singleton umbr a χ is the umbra whose moments a re all zero , except the first E [ χ ] = 1 . Its factorial moments ar e x ( n ) = ( − 1) n − 1 ( n − 1)! As we will see later on, this umbra is the keystone for manag ing symmetric umbral p olynomia ls. 4 Bell um bra. The Bel l umbr a β is the umbra whose facto r ial moments are a ll equal to 1 , that is E [( β ) n ] = 1 for all n ≥ 1 . Its moments are the Bell num b ers , that is the n umber of partitions of a finite nonempty set with n elements, o r the n - th co efficient in the T aylor ser ies ex pansion of the function exp( e t − 1) . So β is the umbral counterpart of a Poisson ra ndom v aria ble with parameter 1 . It is po ssible that tw o distinct umbrae represent the sa me s equence of mo ment s , in such case these are called similar umbr ae . More for mally tw o umbrae α and γ ar e said to b e similar when E [ α n ] = E [ γ n ] ∀ n ≥ 0 , in s y mbols α ≡ γ . F urthermore, given a sequence 1 , a 1 , a 2 , . . . in R, ther e ar e infinitely man y distinct, and thus s imilar um br ae representing the seque nc e . So, the um bra l coun terpa rt o f a univ a riate random sample is a n - vector ( α 1 , α 2 , . . . , α n ) , where α i , i = 1 , 2 , . . . , n a re uncorr elated umbrae, similar to the same umbra α. Thanks to the notion of s imilar um bra e, it is p ossible to extend the alphab et A with the so-called auxiliary umbr ae resulting from oper ations amo ng s imilar um brae. This leads to construct a satur ate d umbr al c alculus in which auxiliar y umbrae ar e handled as elements of the alphab et [20]. In the following, we fo cus the atten tion on auxiliar y umbrae which play a special role. Let { α 1 , α 2 , . . . , α n } b e a set of n uncorrelated um brae similar to a n um bra α. The s y mbol n.α denotes a n auxiliary umbra similar to the sum α 1 + α 2 + · · · + α n . So n.α is the umbral counterpart of a s um of indep endent and ide ntically distributed random v ariables. The symbol α .n is an aux ilia ry umbra denoting the pr o duct α 1 α 2 · · · α n . Moments of α .n can be easily recov ered from its definition. Indeed, if the um br a α repr e sents the sequence 1 , a 1 , a 2 , . . . , then E [( α .n ) k ] = a n k for nonnegative int eg ers k a nd n. Moments of n.α can b e expr essed throug h integer pa rtitions. Recall that a par tition of an integer i is a sequence λ = ( λ 1 , λ 2 , . . . , λ t ) , where λ j are weakly decreas ing integers and P t j =1 λ j = i. The integers λ j are named p arts o f λ. The length of λ is the num ber of its parts and will b e indicated by ν λ . A different notation is λ = (1 r 1 , 2 r 2 , . . . ) , wher e r j is the num b er of parts o f λ e qual to j and r 1 + r 2 + · · · = ν λ . F or example, (1 3 , 2 1 , 3 2 ) is a partition o f the integer 1 1 . W e use the class ical no ta tion λ ⊢ i to deno te that “ λ is a par tition of i ”. By using an umbral version of the well-known multinomial expa ns ion theorem [5], we hav e ( n.α ) i ≃ X λ ⊢ i ( n ) ν λ d λ α λ , (1) where the sum is ov er a ll par titio ns λ = (1 r 1 , 2 r 2 , . . . ) o f the integer i, ( n ) ν λ = 0 for ν λ > n, d λ = i ! r 1 ! r 2 ! · · · 1 (1!) r 1 (2!) r 2 · · · and α λ ≡ ( α j 1 ) .r 1 ( α 2 j 2 ) .r 2 · · · , (2) with { j i } distinct in teg ers chosen in { 1 , 2 , . . . , n } = [ n ] . The rea der interested in pr o ofs of ide ntities involving auxilia ry umbrae is referred to [7 ]. A feature of the class ical umbral calculus is the constructio n of new auxilia r y umbrae by suitable symbolic substitutions. F or example, in n.α replace the in teg er n b y an umbra γ . F rom (1 ), the new auxiliary umbra γ .α has moments ( γ .α ) i ≃ X λ ⊢ i ( γ ) ν λ d λ α λ (3) and it is called dot-pr o duct o f γ a nd α. The auxiliary umbra γ .α is the umbral counterpart of a ra ndom sum. In the following, we reca ll some useful dot-pro ducts o f umbrae, whose prop erties have b een inv estiga ted with full particula rs in [8]. 5 α -factorial um bra. T he um br a α.χ is called the α -factorial um bra . Its momen ts are the factorial mo ment s of α, that is ( α.χ ) i ≃ ( α ) i . If α ≡ χ, then E [( χ.χ ) i ] = E [( χ ) i ] = x ( i ) = ( − 1) i − 1 ( i − 1)! . α -cum ulant um bra. The umbra χ.α, w ith χ the singleton umbra, is calle d the α -cum ulant um br a. By v ir tue of (3), its moments are ( χ.α ) i ≃ X λ ⊢ i x ( ν λ ) d λ α λ ≃ X λ ⊢ i ( − 1) ν λ − 1 ( ν λ − 1)! d λ α λ . (4) Since the second eq uiv alence in (4 ) reca lls the well-known expressio n of cumulan ts in ter ms of moments of a ra ndom v ariable, it is straig htf o rward to r efer the moments of the α -cumulan t um br a χ.α as cumulan ts of the umbra α. 3 U -statistics In the following, we fo cus our attention on tw o kinds of a ux iliary umbrae: n.α and n. ( χα ) . Such umbrae, and their pro ducts, ar e similar to some well-known symmetric p olynomia ls . Indeed, by definition we hav e n.α r ≡ α r 1 + · · · + α r n , where α 1 , α 2 , . . . , α n are uncor related umbrae, similar to the um bra α. Since the umbrae α i for i = 1 , 2 , . . . , n c an b e rea rrange d without e ffecting the ev aluatio n E , the a uxiliary um br a n.α r is similar to the r -th p ower sum symmetric p olynomial in the indeterminates α 1 , α 2 , . . . , α n . Moreov e r , since n. ( χα ) ≡ χ 1 α 1 + · · · + χ n α n , powers of n. ( χα ) ar e um br ally equiv alent to the umbral elementary symmetric p oly no mials [ n. ( χα )] k ≃ k ! e k ( α 1 , α 2 , . . . , α n ) , wher e e k ( α 1 , α 2 , . . . , α n ) = X 1 ≤ j 1 1 . 5hh 29.10 23.19 k 3 , 3 k 2 , 2 4.67 n.c. 1.71 k 3 , 3 k 3 , 3 32.16 n.c. 15.87 k 2 , 1 , 1 k 2 , 1 , 1 1.031 n.c. 0 .5 2 T able 6: Comparison o f computationa l times. The a cronym “ n.c .” stands for not calculable. Remark tha t the output expressio ns of Andrews and Staffor d’s co de are unpractical. These a re very different from the o utput ex pressions of Po lyK of Math Static a and polyk in MAPL E . F or example, the output o f Andrews and Stafford’s co de for k 3 is 2 ¯ X 3 ( 1 − 2 n )( 1 − 1 n ) + − 3 1 − 1 n − 6 ( 1 − 2 n )( 1 − 1 n ) n ! ¯ X X 2 + 1+ 4 ( 1 − 2 n )( 1 − 1 n ) n 2 + 3 ( 1 − 1 n ) n ! X 3 , (36) whereas the output of Poly K of M athSt atica and pol yk in M APLE is n 2 S 3 − 3 nS 1 S 2 + 2 S 3 1 n ( n − 1)( n − 2) . In k 12 , the expres sion of Andrews and Sta ffo r d’s co de consis ts o f 6 0 2 terms compare d with 7 7 terms of the expressions obtained b y PolyK of MathSt atica and polyk in M APLE . In order to reco ver the same output of PolyK of MathSta tica and polyk in MAPLE in (36 ), w e m ust group the terms in pa renthesis ov er a co mmon denominator , deleting equal fa ctors in the results. This op era tion increases the o verall computational time. F or example the co mputational time of k 10 grows from 0 . 35 to 2 . 693 , the one of k 12 1 Of course, if si ngle k -statistics ar e enough to b e computed, the function k-stat of MathStati ca is suitable and faster. 2 In the forthcoming MathStatic a , release 2, the pro cedure to handle multiv ari ate p olyk ays is now av ailbale (C. Rose, priv ate comm unication). 18 grows fro m 1 . 191 to 1 4 . 56 . F or k -statistics of order greater than 14 , an error o ccurs since the recurs ion exceeds a depth of 256 . 6.3 Pro duct of augmen ted symmetric functions The MAPLE ro utine Pam implement s equiv alences s uch as (29). Pam calls the routine m akeTa b . Consider ing that there are mono mials inv o lving single to n umbrae in the multiset M , the efficiency of makeT ab improv es, as we have mentioned at the e nd of Subsec tion 6.1 . In T able 7, we co mpare s ome co mputational times o f Pam with those o f the routine SIP , written in M ATHEM ATICA languag e by V rbik [24 ], and exclusively devoted to pro ducts o f augmented symmetric functions. [1 i 2 j 3 k · · · ] SIP MAPLE [5 3 9 1 0][1 2 3 4 5 ] 0.7 0.1 [5 3 8 9 1 0][1 2 3 4 5 ] 5.6 0.4 [6 7 8 9 10 ][1 2 3 4 5] 2.2 0.1 [6 7 8 9 10 ][1 2 ][3 4 5 ] 3.1 0.4 [6 7 ][8 9 10][1 2][3 4 5] 4.7 1.3 [5 6 7 8 9 10][1 2 3 4 5 ] 16.7 0.3 [5 6 7 8 9 10][1 2 3 4 5 6] 348.7 1.5 [6 7 8 9 10 ][6 7 ][3 4 5 ][1 2] 125.6 16.4 T able 7: Comparison o f computationa l times. Note that the computational time of SIP dep ends heavily on the num b er of v ariables inv olved in the brack ets, b eca use it reso rts set p ermutations, whose computation cost is facto rial in its cardina lit y . 7 Concluding remarks This pa p e r foc uses attention on a s ymbolic ca lc ulation of pr o ducts of sta tistics related to cumulan ts or moments. Undoubtedly , an enjoy a ble challenge is to find efficient pro ce dur es to deal with the necessar ily hu g e a mo unt of a lgebraic a nd symbo lic computations inv o lved in suc h a kind of calculatio ns. The methods we pr op ose r esult more efficient compare d with those av aila ble in the litera ture. Note that high order statistics have a v ariety o f applications. Recen tly , Rao [1 7] hav e shown applications o f high or der cumulan ts in statistical inferenc e and time ser ies. Indeed, there are differ ent area s, such a s astrono m y (see [16] a nd references therein), astrophysics [10] and bioph ysics [15], where one c omputes high or der k -statistics in order to r ecognize a gaus sian p opula tion or characterizes asymptotic b ehavior o f high o rder k -statistics if the p opula tion is gaussian. Indeed, k -sta tistics are indep endent from the sa mple mean if and only if the po pulation is gaus sian [13] and in such a case k -statistics o f orde r grea ter than 2 should b e nearly to zero . F or such applicatio ns , increa sing sp eed and efficiency is a significa nt inv estment. As w e hav e shown, the co des of Andrews and Stafford are quite inefficient fo r the problems p osed here . This pap er has p ointed out the role played by the notio n of sub divisio n in sp eeding up the calcula tions resulting by mult iply ing sums of rando m v ariables and the ro le play ed by the umbra χ in selec ting the inv olved v ar iables. The sym b olic algorithm we prop ose, in or der to ev alua ting the mea n o f pro duct of augmented p oly nomials in r andom v ariables, relies o n this innov ative strategy . In clo sing, we would like to emphasize that clas sical umbral calculus not only decre a ses the co m- putational time, but o ffers a theory to prove more general r esults. Recen tly , L-moments and trimmed L-moments have b een noticed as app ea ling alterna tives to co nv entional moments, s e e [12] a nd [6]. W e belie ve that the handling of these num b er se quences would b enefit by an umbral a pproach. 19 8 App endix In the following, we present the MAP LE co de of the pro c e dure giving sub divisions of a multiset. In orde r to hav e the results of T able 3 and 4, the calling syntax is ma keTab( 3,2) . nRep := proc(u) mul(x[2] !,x=convert(u,mu l tiset)); end: #- - - - - - - - - - - - - - - - - - - - - - - - - - - - -# URv := proc(u,v) local U,ou,i,ptr _i,vI; ou:=NULL ; U:=[]; vI:=indets(v ); for ptr_i from nops(u) by -1 to 2 do if has(u[ptr _i],v) then break; fi; od; for i from ptr_i to nops(u) do if not (u[i]=ou or has(u[i],vI) ) then ou:=u[i]; U:=[op(U ),[op(u[1..(i-1)]),u[i]*v,op(u[(i+1)..-1])]]; fi; od; op(U),[o p(u),v]; end: #- - - - - - - - - - - - - - - - - - - - - - - - - - - - -# URV := proc() local U,V,i; U:=[args [1,1]]; V:=args[2,1]; for i from 1 to nops(V) do U := [ seq( URv(u,V[i]) , u=U ) ]; od; seq([x,a rgs[1,2]*args[2,2]/nRep(x)],x=U) ; end: #- - - - - - - - - - - - - - - - - - - - - - - - - - - - -# URmV := proc() local U,i,nb in; if nargs =1 then U:=args; else U:=URV( args[1 ], args[2]); for i from 3 to nargs do U:=seq( URV( u, args[i]), u=[U] ); od; fi; U; end: #- - - - - - - - - - - - - - - - - - - - - - - - - - - - -# comb := proc(V,p tr,Y) if ptr=nops( V)+1 then return(Y); fi; seq( comb(V, ptr+1, [ op(Y), L ] ), L=V[ptr]); end: #- - - - - - - - - - - - - - - - - - - - - - - - - - - - -# makeTab := proc() local U; U:=[seq( [seq( [[seq(P| |i^z,z=y)], combinat [’multinomial’](args[i],seq(r,r=y))], y=combin at[’partition’](args[i]))], i=1..nar gs)]; if nops(U)=1 then [seq([x[1 ],x[2]/nRep(x[1])],x=op(U))]; else [seq(UR mV(op(x)),x=[comb(U,1,[])])]; fi; end: 9 Ac kno wledgmen ts The authors tha nk the referees for v alua ble comments and s ug gestions, whic h impro ved the presentation of the pa p er. 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