ON A GENERALIZED EULERIAN DISTRIBUTION

Transcription

1 A. Ist. Statist. Math. Vol. 43, No. 1, (1991) ON A GENERALIZED EULERIAN DISTRIBUTION CH. A. CHARALAMBIDES Statistical Uit, Uiversity of Athes, Paepisteraiopolis, Athes , Greece (Received April 24, 1989; revised February 13, 1990) Abstract. The distributio with probability fuctio pk(, ~, t3) = A,~,k(a, fl)/(a + fl)[] k = 0, 1, 2,...,, where the parameters a ad /3 are positive real umbers, A,k (~, fl) is the geeralized Euleria umber ad (a + fl)[] = (a + fl)(a +/3 + 1)--. (c~ + fl + - 1), itroduced ad discussed by Jaarda (1988, A. Ist. Statist. Math., 40, ), is further studied. The probability geeratig fuctio of the geeralized Euleria distributio is expressed by a geeralized Euleria polyomial which, whe expaded suitably, provides the factorial momets i closed form i terms of o-cetral Stirlig umbers. Further, it is show that the geeralized Euleria distributio is uimodal ad asymptotically ormal. Key words ad phrases: Euleria umbers, Euleria polyomials, Stirlig umbers, radom permutatios, uimodality, asymptotic ormality. 1. Itroductio Carlitz ad Scoville (1974) itroduced a geeralized (symmetric) Euleria umber A(r, s [ a,/3) i coectio with the problem of eumeratig (a,/3)- sequeces (geeralized permutatios). Recurrece relatios ad other algebraic properties of these umbers were developed. It ca be show that (1.1) pk(, a,/3) = A,k(a, /3)/(a +/3)[~], k = 0, 1, 2,...,, where the parameters a ad /3 are positive real umbers ad A,k(a, /3) = A(k, - k I a,/3) is a legitimate probability fuctio. The distributio with probability fuctio (1.1) may be called a geeralized Euleria distributio. Morisita (1971), after a series of experimetal studies with at lios, suggested a model i which each at lio was allowed to settle i fie sad (or i coarse sad) with a probability proportioal to the evirometal desity. He the provided a recurrece relatio for the probability that k out of at lios settled i fie sad. Jaarda (1988), i a iterestig mathematical ad statistical treatmet of Morisita's model, proved that a explicit solutio of this recurrece ca be give i terms of the geeralized Euleria umber E~,k(a, b). This umber is related to the geeralized (symmetric) Euleria umber A(r, s I a,/3) of Carlitz ad Scoville (1974) by E,k(a, b) = A(- k, k I a, b) = A(k, - k I b, a). 197

2 198 CH. A. CHARALAMBIDES I the preset paper, which is motivated by Jaarda's work, it is show that the problem of derivig the probability fuctio of the umber L of at lios choosig fie sad to settle (i Morisita's model) whe the evirometal desities are positive itegers is equivalet to the problem of derivig the probability fuctio of the umber X~+I of rises i a radom (c~,/3)-sequece of Z+l = {1, 2,..., + 1}. The otio of a (a, ~)-sequece, itroduced by Carlitz ad Scoville (1974), costitutes a geeralizatio of the otio of a permutatio. From the above equivalece the probability fuctio of L~ is deduced (Sectio 2). The probability geeratig fuctio of this distributio, whe the evirometal desities are positive real umbers, is expressed i terms of a geeralized Euleria polyomial. Further, this polyomial, if suitably expaded, provides a explicit expressio of the factorial momets i terms of o-cetral Stirlig umbets (Sectio 3). It is also show that the -th geeralized Euleria polyomial has distict o-positive real roots. Usig this result, oe ca prove that the geeralized Euleria distributio is uimodal ad asymptotically ormal (Sectio 4). 2. Morisita's model, radom permutatios ad (c~,/~)-sequeces Morisita (1971) cosidered at lios, each of which was allowed to choose to settle either i fie or coarse sad, ad postulated that (2.1) (2.2) (2.3) Pr(the first at lio to choose coarse sad) = 1 - Pr(the first at lio to choose fie sad) = a/(a + b), Pr(the -th at lio to choose coarse sad give that k at lios are i fid sad) = ( a + k )/ ( a + b + - 1), Pr(the -th at lio to choose fie sad give that - k - 1 at lios are i coarse sad) = (b + - k - 1)/(a + b + - 1) where the parameters a ad b are positive real umbers. Cosider a arbitrary permutatio cr = (al, a2,..., a+l) of the set Z+l = {1, 2,..., + 1}. A pair of cosecutive elemets (a~, a~+l) i cr is called a rise if a~ < ai+l ad a fall if al > ai+l. If k(a) is the umber of rises of the permutatio (r, the clearly 0 < k(a) < ad the umber of falls of the same permutatio is - k(a). The umber of permutatios of Z+l with k rises (ad - k falls) is equal to the Euleria umber A+l,k+l. Suppose that a permutatio is radomly chose from the set of the ( + 1)! permutatios of Z+l ad let X+l be the umber of its rises. The the probability fuctio of the radom variable X,~+I is give by (2.4) pk() -- Pr(X+l --/ ) -- A+l,k+l/( + 1)!, k ---- O, 1, 2,...,. I order to relate Morisita's model with a radom permutatio model, cosider the followig costructio of a radom permutatio of Z+l. Startig with the umber 1, the remaiig umbers, 2, 3,..., + 1, are placed oe after the other

3 ON A GENERALIZED EULERIAN DISTRIBUTION 199 i all possible ways. There are two possible ways of placig 2: either to the left or to the right of 1, iducig a fall: (2,1) or a rise: (1,2). Thus, (2.5) Pr(placemet of 2 iduces a fall) = Pr(placemet of 2 iduces a rise) = 1/2. Further, there are + 1 possible ways of placig + 1. If it is placed betwee the two elemets of a rise or to the left of the elemets already placed, the umber of rises remais uchaged while the umber of falls is icreased by oe. If it is placed betwee the two elemets of a fall or to the right of the elemets already placed, the umber of rises is icreased by oe while the umber of falls remais uchaged. Thus, (2.6) (2.7) Pr(placemet of + 1 iduces a fall give that k rises are already iduced) = (k + 1)/( + 1), Pr(placemet of + 1 iduces a rise give that k falls are already iduced) = ( - k + 1)/( + 1). It is apparet from the precedig aalysis that there is a oe-to-oe correspodece betwee the set of differet choices of the at lios to settle, whe a = b -- 1, ad the set of the differet choices of the umbers 2, 3,..., + 1 to be placed. More specifically, if the j-th at lio chooses fie (or coarse) sad to settle, the the umber j + 1 is iserted i a place iducig a rise (or a fall) ad vice versa. This correspodece implies that i the particular case of Morisita's model with a = b = 1, the probability fuctio Pr(L -- k), k = 0, 1, 2,..., of the umber L of at lios choosig fie sad to settle is give by (2.4). The otio of a (a,/3)-sequece, itroduced by Carlitz ad Scoville (1974) ad costitutig a geeralizatio of the otio of a permutatio, ca be related to Morisita's model whe the parameters a ad b are positive itegers. A (a,/3)- sequece, of Z+I, i additio to the + 1 elemets of Z+I, icludes a symbols 0 ad/3 symbols 01 subject to the coditios that there is at least oe symbol 0 o the extreme left ad at least oe symbol 01 o the extreme right ad that the umber 1 has all a symbols 0 to its left ad all/3 symbols 01 to its right. There is oe (a,/3)- sequece of ZI: (0,..., 0, 1, 0',..., 0') ad ths (a,/3)-sequeces of Z+I ca be obtaied by isertig the umbers 2, 3,..., + 1 oe after the other i all differet ways. Sice there are a +/3 j - 2 differet ways of isertig the umber j for j -- 2, 3,..., + 1, it follows that the umber of (a,/3)-sequeces of Z~+I is equal to (a+/3) I]. Cosider a arbitrary (a, /3)-sequece s = (Sl, 82,..., 8Wa-}-13-l-1) of Z~+I. A rise is defied as a pair of cosecutive elemets (si, si+l) with si < Si+l where si may be 0. Similarly a fall is defied as a pair of cosecutive elemets (si, si+l) with si > si+l where si+l may be 0'. If k(s) 1 is the umber of rises of a (a,/3)-sequece s of Z+l, the 0 _< k(s) <_ ad the umber of falls of the same (a,/3)-sequece is - k(s) + 1. The umber of (a, fl)-sequeces of Z+l with k + 1 rises (ad - k + 1 falls) is equal to A(k, - k I a,/3) = A( - k, k I/3, a), the geeralized symmetric Euleria umber studied by Carlitz ad Scoville (1974).

4 200 CH. A. CHARALAMBIDES Thus, puttig A,k(c~, /~) ---- A(k, - k Is, fl) it follows that (2.8) k Note that (2.9) A, k(1, 1) = A+l,k+l, A, k(1, O) = A=,k. Suppose that a (a, ~)-sequece is radomly chose from the set of the (c~ + ~)[] (a, ~)-sequeces of Z+l ad let X~+I be the umber of its rises. The, the probability fuctio pk(, (~,/3) = Pr(X~+I = k + 1), k = 0, 1, 2,..., is give by (1.1), where the parameters c~ ad/~ are, i this case, positive itegers. Further, the precedig aalysis of the costructio of a (a, fi~)-sequece, by virtue of Morisita's postulates (2.1), (2.2) ad (2.3), with a = b ad ~ = a positive itegers, implies Pr(L = k) = Pr(X+l = k+l) = pk(, a, ~), k = 0, 1, 2,...,. I the geeral case of Morisita's model where the parameters a ad b are positive real umbers, ot ecessarily itegers, the probability fuctio Pr(L = k), k = 0, 1, 2,...,, of the umber L~ of at lios choosig fie sad to settle ca be obtaied as (1.1) with a = b ad/3 = a, by comparig the recurrece relatio for Pr(L = k) deduced from postulates (2.1), (2.2) ad (2.3) with the recurrece relatio for the ratio A,k(a, /~)/(c~ + ~)[] deduced from the followig recurrece relatio of the geeralized Euleria umbers A,k(c~, ~) = A(k, - k I o~, t3) (Carlitz ad Scoville (1974)) (2.10) A+l,k(a,/3) = (/3+k)A,k(a, ~) + (a k)a,k-l(c~,/~) k = 0, 1, 2,..., + 1, = 0, 1, 2,... with Ao,o((~, ~) = 1, Ao,k(a, /3) = O, k # 0 (see also Jaarda (1988) where A~,k(a,/3) = E~,k(~, a)). 3. Geeratig fuctios, factorial momets ad geeralized Euleria polyomials The probability geeratig fuctio of the geeralized Euleria distributio (1.1), o usig the geeralized Euleria polyomials (3.1) A(t; ~,/~) = ~-~A,k(a, ~)t k, = 0, 1, 2,..., may be obtaied as k=0 (3.2) G(t; = k = A(t; + )El. k=o

5 ON A GENERALIZED EULERIAN DISTRIBUTION 201 The factorial momet geeratig fuctio is the give by (3.3) oo F(t; a,/3) = Ett(r)(, a, fl)t~/r! = A(t+ 1; a,/3)/(a + fl) M r~-0 where Thus, (3.4) ~(r)(~, 4,/3) = Z[L(~)], r = 1, 2,...,,(0)(~, 4,/3) = 1, with L (~) = L(L~ - 1)( - 2)... (L - r + 1). The derivatio of the factorial momets by expadig the right-had side of (3.3) is facilitated by the followig brief study of the geeralized Euleria polyomials. Itroducig i (3.1) the expressio (2.8) of the geeralized Euleria umbers A,k (~,/3) ad sice A,k (ct,/3) = 0 for k >, it follows that A(t; a, fl) = Z(_I)i a+/3+ a+z+k-j-1 k=o j=o J k - j (/3 q- k - j)tk = E(_I) j o~+/3-[- tj a+/3+k-j-1 (/3+k_j)tk_ j ~=0 J k=~ k - j = ~(-1)~ ~+/3+~. ~+/3+r- 1 (/3+~)~t ~. j=0 3 v=0 r A(t; 4,/3) = (1 - t) ~+~+ Z ~ +/3 + r - 1 ) (/3 + r)t L r=0?" The geeratig fuctio of the geeralized Euleria polyomials A(t, u; 4,/3) = E =O o usig the expressio (3.4) may be obtaied as A(t; a,/3)u/!, A(t,u; a, fl)=e(1-t) ~+#+~ a+/3+r-1 (/3+r)ffu/! ~O r~o =(l-t) a+~ a+/3+r-1 tre[(/3+r)u(l_t)]/! r~0 rt~0 = ee~(l_0( 1 _ t)~+ ~ ~ a +/3 + r - 1 [te,~(~_~)]~ r ~"~0 = e#~(l-t)(1 _ t)~+#[1 _ te~(~-t)]-~-~. T

6 202 CH. A. CHARALAMBIDES Note, i passig, that lim A(t, u; a,/3) = (1 - u) -~-z, t--~l implyig (3.5) A~,k(a, /3) = (a + ~)[~], k----o i agreemet with the result i Theorem 3.2 of Jaarda (1988). The geeratig fuctio of the geeralized Euleria polyomials may be rewritte i the form A(t, u; c~,/3) = e"~(t-1){1 -[e ~(t-1) - 1]/(t - 1)} -~-~ which, o usig the o-cetral Stirlig umbers (Koutras (1982)) (3.6) S(, r l a) = 1 E(_l)k (~)(c~ + k), k=o r = O, 1, 2,...,, = 0, 1, 2,... with the geeratig fuctio OG ~f r=o, 1,2,..., ca be expaded as 1 / - r r tlt/ i r~0 = (~+#)C~JS(,~[~)(t-1) -~ ~"I~!, =O yieldig for the geeralized Euleria polyomials the expressio (3.7) A(t; ~,/3) : E(~ H-/3)[r]S(, r I c~)(t - 1) -r. Returig to the geeralized Euleria distributio, its factorial momet geeratig fuctio (3.3), by virtue of (3.7), may be expaded i powers of t as E~(t; ~,/3) : ~{(~ +/3)E~-~1/(~ +/3)I~J}s(~, ~ - ~ I ~)t ~ r-~o =~{S(,-r,oO/(o~+#+-1)}tr/rL r~-o 7"

7 ON A GENERALIZED EULERIAN DISTRIBUTION 203 Thus, (3.8) p(r)(,t~,/3)=s(,-rl(x)/(~+fl+-i~, r=0, 1, 2,..., /\ r ] ad #(r)(, ~,/~) = 0 for r = + 1, + 2,... The computatio of the factorial momets (3.8) is facilitated by the followig expressio of the o-cetral Stirlig umbers as polyomials of the o-cetrality parameter: (3.9) S(,-rlc~)=~(k)s(-k,-r)c~ k k----0 where S(, r) = S(, r 10) is the usual Stirlig umbers. From (3.8) with r = 1 ad r = 2, o usig (3.9) ad S(- l, -1) = S(- 2, - 2) =1, S(,-2)=3(4)+(3 ) S(,-1)= () 2 ' it follows that (3.10) [() ]/ #(, ~, /3) -- #(u(, c~, /3) = 2 + us (~ + ~ + - 1), ad (3.11) 2[3(4)+(3)( 3c~+1)+(2) ~2 ] #(2)(, c~, t3) = (c~ + t )(c~ +/ ) 2 (4) + (3)(2o~ + 2~ + 1)+ (2)(a + ~)2 +a/3(c~ + ~-1) (~+/3+- 1)2(a+fl+- 2) i agreemet with the expressios obtaied by Jaarda (1988). Before cocludig this sectio it is worth otig that the expressio (3.4) has the followig direct probabilistic applicatio. The -th (power) momet about a arbitrary poit fl of a radom variable X obeyig a egative biomial, or biomial distributio, ca be expressed i terms of the geeralized Euleria polyomials. 4. The asymptotic behavior of the distributio The probability geeratig fuctios G(t; a, t3) of the geeralized Euleria distributio (1.1) which is give by (3.2) i terms of the geeralized Euleria polyomials A(t; a, ~) ca be writte as (4.1) G(t; c~, 13) = H(qj +pit), qj -- 1-pj, j = 1, 2,...,, j=l

8 204 CH. A. CHARALAMBIDES 0 < pl < p2 < "'" < p = 1. This represetatio is show by provig that the geeralized Euleria polyomial A(t; a, ~) has distict real o-positive roots for all = 1, 2,... This proof may be carried out by iductio as follows: The geeralized Euleria polyomials A(t; a,/3) satisfy the differecedifferetial equatio d (4.2) A+l(t; a,/3) = t(1 - t)~a(t; a,/3) + t(a +/3 + )A(t; a,/3), =0, 1, 2,... with A0(t; a,/3) = 1, which may be deduced from (3.1) o usig the triagular recurrece relatio (2.10) of the geeralized Euleria umbers A, k(c~,/3). From (4.2) it follows that At(t; a,/3) = (a +/3)t, A2(t; a,/3) = (a +/3)t[(a +/3)t + 1 t that is the statemet holds for = 1, 2. Now suppose that A(t; a,/3) has distict real o-positive roots ad cosider the fuctio B~(t; a,/3) = (1 - t)-(~+~+)a(t; a,/3). The B(t; a,/3) has exactly the same fiite zeroes as those of A(t; a,/3) ad the idetity (4.2) becomes d B B+l(t; a,/3) = t~ (t; a,/3), = 0, 1, 2,... B(t; a, 13) also has a zero at t = -oo, ad by Rolle's theorem, betwee ay two zeroes of B(t; a,/3), the derivative db(t; a, ~3)~dr has a zero. This implies that B+l (t; a,/3) has distict zeroes o the egative axis; obviously t = 0 is aother zero, makig + 1 altogether. Sice A+l(t; a,/3) is of degree + 1 by iductio, we have foud all roots ad the statemet is proved. As a cosequece of this property of the geeralized Euleria polyomials, the geeralized Euleria umber A,,,k (a,/3) is a strictly logarithmic cocave fuctio of k, that is (4.3) [A,k(a, /3)]2 > A,k+l(a, /3)A,k-l(C~, /3). Further, it follows that the geeralized Euleria distributio (1.1) is uimodal either with a peak or with a plateau of two poits (see, for example, Comtet (1974), p. 270). Let us, ow, cosider the sequece of idepedet zero-oe radom variables Zj, j = 1, 2,...,, with P(X,j = O) = qj, P(X,3 = 1) = pj, j = 1, 2,...,. The the probability geeratig fuctio of the sum S = ~ X,j is give by j=l (4.1) ad hece P(S = k) = A,k(a,/3)/(a +/3)["], k = 0, 1, 2,...,.

9 ON A GENERALIZED EULERIAN DISTRIBUTION 205 From (3.11) it follows that Var(S) --+ oc as -+ c~. Therefore, lettig Y,j -- [Var(S)]-l/2[Z,j - E(Z,j)], j -- 1, 2,..., ad F,j (y) = Pr(Y,j < y), it follows that for a give e > 0 there exists o such that [Y, k[ < e for all > o, implyig that the Lideberg coditio lim ~ f y2df~,j(y)=o, of the bouded variace ormal covergece criterio is fulfilled. Hece [s az(s ) ] (4.4) lim P L ~ <x =~(x) with beig the distributio fuctio of the stadard ormal distributio. Note that (4.4) still holds whe E(S~) ad Var(S) are replaced by approximate values (as --* cx)). The r-th factorial momet (3.8) may be approximated as follows: Itroducig the expressio r--k j=o r-k+j where S2(m, j) is the associated Stirlig umber of secod kid (see Comtet (1974), p. 226) ito (3.9), (3.8) may be writte as -k ak a+fl+-1,(r) (, /3) = ()( k r+j-k ) /( r ) " j=0 k=0 Sice $2(2r, r) = (2r)!/(r!2r), $2(2r - 1, r - 1) = (2r - 1)!/[3(r - 2)!2r-1], it follows that p(r)(, (~, /3) = (2~)/2r + [ra + r(r- 1)/3](2~-1)/2 ~-1 (c~ + fl + - 1)(~) A further approximatio of the factorial polyomials of leads to Therefore, + (-~+2)" p(r)(, o~,/3) ---- (/2) r + [rol + r(r - 1)/3](/2) r-1 + o(-r+2). #(, a,/3) = /2 + o(1), #(~)(, a,/3) = (/2) 2 + (a + 1/3) + o(1). Istead of usig these approximate values to fid a approximate value of the variace, it is better to derive such a value by approximatig its exact value (3.11). I this way, we fid a2(, a,/3) = il2 + o(1). Thus i (4.4) we may use (4.5) E(S) = /2, Var(S~) = il2.

10 206 CH. A. CHARALAMBIDES Ackowledgemets The author wishes to thak both referees for their valuable commets whe revisig this paper. REFERENCES Carlitz, L. ad Scoville, R. (1974). Geeralized Euleria umbers: combiatorial applicatios, J. Reie Agew. Math., 265, Comtet, L. (1974). Advaced Combiatorics, Reidel, Dordrecht, Hollad. Jaarda, K. G. (1988). Relatioship betwee Morisita's model for estimatig the evirometal desity ad the geeralized Euleria umbers, A. Ist. Statist. Math., 40, Koutras, M. (1982). No-cetral Stirlig umbers ad some applicatios, Discrete Math., 42, Morisita, M. (1971). Measurig of habitat value by evirometal desity method, Statistical Ecology (eds. G. P. Patil et al.), , The Pesylvaia State Uiversity Press.