Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type

Expansions for Quan tiles and Multiv ariate Momen ts of Extremes for Distribu tions of P areto T yp e b y Christopher S. Withers Applied Mathematics Group Industr ial Re search Limited Lo w er Hutt, NEW ZEALAND Saralees Nadara jah Sc ho ol of Mathematics Univ ersit y of Manc hester Manc hester M13 9PL, UK Abstract: Let X nr b e the r th large st of a rand om sample of s ize n from a distribu tion F ( x ) = 1 − P ∞ i =0 c i x − α − iβ for α > 0 and β > 0. An in v ersion theorem is pro ve d and us ed to deriv e an expan s ion for the quant ile F − 1 ( u ) and p o wers of it. F rom this an expansion in p o wers of ( n − 1 , n − β /α ) is giv en for the m ultiv ariate momen ts of the extremes { X n,n − s i , 1 ≤ i ≤ k } /n 1 /α for fi xed s = ( s 1 , . . . , s k ), where k ≥ 1. Examples in clude the Cauc hy , Student t , F , second extreme distributions and stable la w s of index α < 1. AMS 2000 Sub ject C lassification: Primary 62E15; Secondary 62E1 7. Keyw ords and Phrases: Bell p olynomials; Extremes; In v ersion theorem; Momen ts; P areto; Quant iles. 1 In tro duction and Summary F or 1 ≤ r ≤ n , let X nr b e the r th largest of a rand om sample of size n fr om a con tin u ou s distribution F o n R , th e real n umb ers. Le t f denote the den sit y of F when it exists. T he study of the asymptotics of the momen ts of X nr has b een of considerable interest. McCord (1964 ) ga ve a first appro ximation to the moments of X n 1 for three classes. This sho we d that a momen t of X n 1 can b eha v e lik e any p ositiv e p o wer of n or n 1 = log n . (Here log is to the base e .) Pic k ands (1968) explored the conditions under wh ic h v arious momen ts of ( X n 1 − b n ) /a n con v erge to the corresp onding moments of the extreme v alue distr ib ution. It was p r o v ed that this is indeed tru e for all F in the domain of attract ion of an extreme v alue distribu tion pro vided that the moments are finite for sufficien tly large n . F or other w ork, we refer the readers to P olfeldt (1970 ), Ramac handran (1984) and Resnic k (1987). 1 The asymptotics of the quan tiles of X nr ha v e also b een stu died. Note that U nr = F ( X nr ) is the r th ord er statistics from U (0 , 1). F or 1 ≤ r 1 < r 2 < · · · < r k ≤ n set U n, r = { U nr i , 1 ≤ i ≤ k } . By Section 14.2 of Stuart and O rd (198 7), U n has the multiv ariate b eta densit y U n, r ∼ B ( u : r ) = k Y i =0 ( u i +1 − u i ) r i +1 − r i − 1 /B n ( r ) (1.1) on 0 < u 1 < · · · < u k < 1, where u 0 = 0, u k +1 = 1, r 0 = 0, r k +1 = n + 1 and B n ( r ) = k Y i =1 B ( r i , r i +1 − r i ) . (1.2) Da vid and John son (1954) expanded X nr i = F − 1 ( U nr i ) ab out u ni = E U nr i = r i / ( n + 1): X nr i = P ∞ j =0 G ( j ) ( u ni )( U ni − u ni ) j /j !, where G ( u ) = F − 1 ( u ), and using the p rop erties of (1 .1) sho wed th at if r dep en ds on n in suc h a wa ys that r /n → p ∈ ( 0 , 1 ) as n → ∞ then the m th order cumulan ts of X n, r = { X nr i , 1 ≤ i ≤ k } ha ve m agnitude O ( n 1 − m ) – at least for n ≤ 4, so that the distribution of X n, r has a multiv ariate Edgewo rth expansion in p o wers of n − 1 / 2 . (Alternativ ely one can use James and Ma yne (19 62) to derive the cumulan ts of X n, r from those of U n, r .) T he metho d requ ir es the deriv ativ es of F at { F − 1 ( p i ) , 1 ≤ i ≤ k } s o breaks down if p i = 0 or p k = 1 – whic h is the situation w e study here. F or defin iteness, we confine ourselv es to F − 1 ( u ) h a ving a p o w er singularit y at 1, sa y F − 1 ( u ) ∼ (1 − u ) − 1 /α as u → 1, where α > 0 that is, 1 − F ( x ) ∼ x − α (1.3) as x → ∞ . F or a nonparametric estimate of α see No v ak and Utev (1990). Distributions satisfying (1.3) are known as P areto t yp e distributions. These distributions arise in m an y areas of the sciences, engineering and medicine. Some of these areas – where publications inv olving P areto t yp e distrib utions ha ve app eared – are: h ydr ology , physics, wind engineering and indu strial aero d y n amics, computer science, w ater r esources, insurance math- ematics and economics, structural safet y , material science, p erf ormance ev aluation, queu eing systems, geoph ysical research, ironmaking and steelmaking, banking and finance, atmospheric en vironment, civil engineering, comm un ications, inform ation pro cessing and managemen t, h igh sp eed n etw orks, light wa v e tec hnology , solar energy engineering, su p ercomputin g, natural haz- ards and earth sys tem sciences, ocean enginee rin g, optics co mmunications, r eliabilit y engineer- ing, signal pro cessing and urban studies. In Withers and Nadara jah (2007a) w e sho wed that for fixed r when (1.3) holds the d is- tribution of X n,n 1 − r (where 1 is the vect or of ones in ℜ k ), suitably normalized tends to a certain m ultiv ariate extreme v alue distribution as n → ∞ , and so ob tained the leading terms of the expansions of its moments in in v erse p o w ers of n . Here we sho w ho w to extend those expansions when F − 1 ( u ) = ∞ X i =0 b i (1 − u ) α i (1.4) 2 with α 0 < α 1 < · · · , that is, { 1 − F ( x ) } x − 1 /α 0 has a p o wer series in { x − δ i : δ i = ( α i − α 0 ) /α 0 } . Hall (1978) considered (1.4) with α i = i − 1 /α , b ut did not giv e the corresp onding expansion for F ( x ) or expans ions in inv ers e p o wers of n . He applied it to the Cauc hy . In Section 2, we demonstrate the metho d when 1 − F ( x ) = x − α ∞ X i =0 c i x − iβ , (1.5) where α > 0 and β > 0. In this case, (1.4) holds with α i = ( iβ − 1) / α . In Section 3, w e apply it to the Student t , F a nd second extreme v alue d istribution and to stable la ws of exp onent α < 1. App endix A giv es the in v erse theorem needed to pass from (1.5) to (1.4), and expansions for p o w ers and logs of series. W e use the follo wing notation and termin ology . Let ( x ) i = Γ( x + i ) / Γ( x ) and < x > i = Γ( x + 1) / Γ( x − i + 1). An inequalit y in ℜ k consists of k inequalities. F or example, for x in C k , where C is the set of complex num b ers, R e ( x ) < 0 means that Re ( x i ) < 0 for 1 ≤ i ≤ k . Also I ( A ) = 1 or 0 for A true or false and δ ij = I ( i = j ). F or θ ∈ C k let ¯ θ den ote the v ector with ¯ θ i = P k j =1 θ j . 2 Main Results F or 1 ≤ r 1 < · · · < r k ≤ n set s i = n − r i . Here, we sh o w ho w to obtain expansions in inv ers e p o wers of n for the momen ts of the X n, s for fixed r when (1.4) holds , an d in particular when the upp er tail of F s atisfies (1.5). Theorem 2.1 Supp ose (1.5) holds with c 0 , α , β > 0 . Then F − 1 ( u ) is given by (1.4) with α i = ia − 1 /α , a = β /α and b i = C i, 1 /α , wher e C iψ = c ψ 0 b C i ( − ψ , c 0 , x ∗ of (3.31) and x ∗ i = x ∗ i ( a, 1 , c ) of (3.32 ): C 0 ψ = c ψ 0 , C 1 ψ = ψc ψ − a − 1 0 c 1 , C 2 ψ = ψc ψ − 2 a − 2 0  c 0 c 2 + ( ψ − 2 a − 1) c 2 1 / 2  , C 3 ψ = ψc ψ − 3 a − 3 0  c 2 0 c 2 + ( ψ − 3 a − 1) c 0 c 1 c 2 + { ( ψ + 1) 2 / 6( ψ + 3 a/ 2)( a + 1) } c 3 1  , and so on. Also for any θ in ℜ ,  F − 1 ( u )  θ = ∞ X i =0 (1 − u ) ia − ψ C iψ (2.6) at ψ = θ /α . Note 2.1 O n those r ate o c c asions wher e the c o effic i ents d i = C i, 1 /α in F − 1 ( u ) = P ∞ i =0 (1 − u ) ia − 1 /α d i ar e known fr om some alternative f ormula then one c an use C iψ = d θ 0 b C i ( θ , 1 /d 0 , d ) of (3.31). 3 Pro of of Theorem 2.1 By Theorem A.1 with k = 1, u = x − α , x = c , w e ha ve x − α = P ∞ i =0 x ∗ i (1 − u ) 1+ ia at u = F ( x ), wher e x ∗ 0 = c − 1 0 , x ∗ 1 = c − a − 2 0 c 1 , x ∗ 2 = c − 2 a − 3 0  − c 0 c 2 + ( a + 1) c 2 1  , x ∗ 3 = c − 3 a − 4 0  − c 2 0 c 3 + (2 + 3 a ) c 0 c 1 c 2 − (2 + 3 a )(1 + a ) c 2 1 / 2  , and so on. So, for S of (3.29), x − α = c − 1 0 v (1 + c 0 S ( v a , x ∗ )) at v = 1 − u . No w app ly (3.30).  Lemma 2.1 F or θ in C k , E k Y i =1 (1 − U n,r i ) θ i = b n  r : ¯ θ  , (2.7) wher e b n  r : ¯ θ  = k Y i =1 b  r i − r i − 1 , n − r i + 1 : ¯ θ i  (2.8) and b ( α, β : θ = B ( α, β + θ ) /B ( α, β ) . Also in (1.2), B n ( r ) = k Y i =1 B ( r i − r i − 1 , n − r i + 1) . (2.9) Note 2.2 Sinc e B ( α, β ) = ∞ for Reβ ≤ 0 , for (2.7) to b e finite we ne e d n − r i + 1 + Re ¯ θ > 0 for 1 ≤ i ≤ k . Pro of of Lemma 2.1 S et I k = LH S (2.7) = R B n ( u : r ) Q k i =1 (1 − u i ) θ i du 1 · · · du k in tegrated o v er 0 < u 1 < · · · < u k < 1 by (1.1). So, (2.7), (2.9) hold for k = 1. S et s i = ( u i − u i − 1 ) / (1 − u i − 1 ). Th en I 2 = Z 1 0 u r 1 − 1 1 (1 − u 1 ) θ 1 Z 1 u 1 ( u 2 − u 1 ) r 2 − r 1 − 1 (1 − u 2 ) r 3 − r 2 − 1+ θ 2 du 2 /B n ( r ) , whic h is the RHS (2.7) with denominator replaced b y the RHS (2.8). Putting θ = 0 giv es (2.7), (2.9) for k = 2. No w use induction.  Lemma 2.2 In L emma 2.1, the r estriction 1 ≤ r 1 < · · · < r k ≤ n may b e r elaxe d to 1 ≤ r 1 ≤ · · · ≤ r k ≤ n. (2.10) 4 Pro of F or k = 2, the second factor in RHS (2.8) is b ( r 2 − r 1 , n − r 2 + 1 : ¯ θ 2 ) = f ( ¯ θ 2 ) /f (0), where f ( ¯ θ 2 ) = Γ ( n − r 2 + 1 + ¯ θ 2 ) / Γ( n − r 1 + 1 + ¯ θ 2 ) = 1 if r 2 = r 1 and the first factor is b ( r 1 , n − r 1 + 1 : ¯ θ 1 ) = E (1 − U nr 1 ) ¯ θ 1 . Similarly , if r i = r i − 1 , the i th factor is 1 and the pro du ct of the others is E Q k j =1 ,j 6 = i (1 − U nr j ) θ ∗ j , wh ere θ ∗ j = θ j for j 6 = i − 1 and θ ∗ j = θ i − 1 + θ i for j = i − 1.  Corollary 2.1 In any formulas for E g ( X n, r ) for some function g , (2.10 ) holds. In p articular it holds for the moments and cumulants of X n, r . This result is very imp ortan t as it means we can disp ense with treating the 2 k − 1 cases ( r i < r i +1 or r i = r i +1 , 1 ≤ i ≤ k − 1 separately . F or example, Hall (1978 ) treats the t wo cases for cos( X n, r , X n, s ) separately and Da vid and Johns on (1954) treat the 2 k − 1 cases for th e k th order cum ulants of X n, r separately for k ≤ 4. Theorem 2.2 U nder the c onditions of The or em 2.1, E k Y i =1 X θ i n,r i = ∞ X i 1 ,...,i k =0 C i 1 ,ψ 1 · · · C i k ,ψ k b n  r : ¯ i a − ¯ θ /α  (2.11) with b n as in (2.8). Al l terms ar e finite if Re ¯ θ < ( s + 1) α , wher e s i = u − r i . Lemma 2.3 F or α , β p ositive inte gers θ in C , b ( α, β : θ ) = α + β − 1 Y j = β (1 + θ /j ) − 1 . (2.12) So, for θ in C k , b n ( r : ¯ θ ) = k Y i =1 s i − 1 Y j = s i +1  1 + ¯ θ /j  − 1 , (2.13) wher e s i = n − r i and r 0 = 0 . Pro of: LHS (2.12 ) = Γ( β + θ )Γ( α + β ) / { Γ( β + θ + α )Γ( β ) } . But Γ( α + x ) / Γ( x ) = ( x ) α , s o (2.12) holds, and hence (2.13 ).  F rom (2.8) we h a v e, in terpr eting Q k i =2 b i as 1 when k − 1, Lemma 2.4 F or s i = n − r i , b n ( r : ¯ θ ) = B ( s : ¯ θ ) n ! / Γ  n + 1 + ¯ θ 1  , (2.14) 5 wher e B ( s : ¯ θ ) = Γ  s 1 + 1 + ¯ θ 1  ( s 1 !) − 1 k Y i =2 b  s i − 1 − s i , s i + 1 : ¯ θ 1  do es not dep e nd on n for fixe d s . Lemma 2.5 W e have n ! / Γ( n + 1 + θ ) = n − θ ∞ X i =0 e i ( θ ) n − i , wher e e 0 ( θ ) = 1 , e 1 ( θ ) = − ( θ ) 2 / 2 , e 2 ( θ ) = ( θ ) 3 (3 θ + 1) / 24 , e 3 ( θ ) = − ( θ ) 4 ( θ ) 2 / (4!2) , e 4 ( θ ) = ( θ ) 5 (15 θ 3 + 30 θ 2 + 5 θ − 2) / (5!48) , e 5 ( θ ) = − ( θ ) 6 ( θ ) 2 (3 θ 2 + 7 θ − 2) / (6!16 ) , e 6 ( θ ) = ( θ ) 7 (63 θ 5 + 315 θ 4 + 315 θ 3 − 91 θ 2 − 42 θ + 16) / (7!5 76) , e 7 ( θ ) = − ( θ ) 8 ( θ ) 2 (9 θ 4 + 54 θ 3 + 51 θ 2 − 58 θ + 16) / (8!1 44) . Pro of: App ly equation (6.1.4 7) of Abr amo witz and S tegun (1964) for i ≤ 2 and Withers and Nadara j ah (2007b) for i ≤ 7.  So, (2.11), (2.1 4 ) yield the join t momen ts of X n, r n − 1 /α for fixed s as a p ow er series in (1 /n, n − α ): Corollary 2.2 We have E k Y i =1 X θ i n,n − s i = ∞ X j =0 n !Γ  n + 1 + j a − ¯ ψ 1  − 1 C j ( s : ψ ) , (2.15) wher e ψ = θ /α and C j ( s : ψ ) = X  C i 1 ,ψ 1 · · · C i k ,ψ k B  s : ¯ i a − ¯ ψ  : i 1 + · · · + i k = j  . So, if s , θ are fixed as n → ∞ and Re ( ¯ θ ) < ( s + 1 ) α , LH S (2.15) = n ψ 1 ∞ X i,j =0 n − i − j a e i  j a − ¯ ψ 1  C j ( s : ψ ) . (2.16) If a is rational, sa y a = M / N th en LH S (2.15) = n ¯ ψ 1 ∞ X m =0 n − m/ N d m ( s : ψ ) , (2.17) 6 where d m ( s : ψ ) = X  e i  j a − ¯ ψ 1  C j ( s : ψ ) : iN + j M = m  = X  e m − j a  j a − ¯ ψ 1  C j ( s : ψ ) : 0‘ j ≤ m/a  if N = 1; so for d m to dep end on c 1 and not just c 0 w e need m ≤ M . Note 2.3 The fol lowing dimensional che cks c an b e u se d thr oughout. By (1.5), dimc i = ( dimX ) α + iβ . By (2.6), dimC iψ = ( dimX ) θ . Also dim ¯ x i = ( dimX ) − α and dimd m ( s : ψ ) = dimC j ( s : ψ ) = ( dimX ) ¯ θ 1 . Note 2.4 The le ading term in (2.16) do es not involve c 1 so may b e de duc e d fr om the multi- variate extr eme value distribution that the law of X n,n − s i , suitably normalize d, tends to. The same is true of the le ading terms of its cumulants. Se e Withers and Nadar ajah (2007a) for details. The leading terms in (2.16) are n ¯ ψ 1  1 − n − 1 < ¯ ψ 1 > 2 / 2  C 0 ( s : ψ ) + n − a C 0 ( s : ψ ) + O  n − 2 a 0  , where a 0 = min( a, 1) , C 0 ( s : ψ ) = c 0 B ( s : − ¯ ψ ) , C 1 ( s : ψ ) = c ¯ ψ 1 − a − 2 0 c 1 k X j =1 ψ j B  s : a I j − ¯ ψ  and for I j = ¯ i for i m = δ mj , that is I j m = I ( m ≤ j ). F or k = 1, C j ( s : ψ ) = C j ψ ( s + 1) j a − ψ , C 0 ( s : ψ ) = c ψ 0 ( s + 1) − ψ = c ψ 0 / < s > ψ , C 1 ( s : ψ ) = ψ c ψ − a − 1 0 c 1 ( s + 1) a − ψ = ψ c ψ − a − 1 0 c 1 / < s > ψ − a . Set π s ( λ ) = b ( s 1 − s 2 , s 2 + 1 : λ ) = Q s 1 j = s 2 +1 1 / (1 + λ/j ) for λ an in teger. F or example, π s (1) = ( s 2 + 1) / ( s 1 + 1) and π s ( − 1) = s 1 /s 2 . Th en for k = 2, C 0 ( s : λ 1 ) = c 2 λ 0 < s 1 > − 1 2 λ π s ( − λ ) = c 2 0 ( s 1 − 1) − 1 s 2 for λ = 1 = c 2 0 < s 2 − 2 > − 1 2 < s 2 > − 1 2 for λ = 2 and C 1 ( s : λ 1 ) = λc 2 λ − a − 1 0 c 1 < s 1 > − 1 2 λ − a { π s ( − λ ) + π s ( a − λ ) } = λc 1 − a 0 c 1 < s 1 > − 1 2 − a { s 1 /s 2 + π s ( a − 1) } f or λ = 1 = λc 3 − a 0 c 1 < s 1 > − 1 4 − a  < s 1 > 2 < s 2 > − 1 2 + π s ( a − 2)  for λ = 2. 7 Set λ = 1 /α , Y ns = X n,n − s / ( nc 0 ) λ and E c = λc − a − 1 0 c 1 . Th en for s > λ − 1 E Y ns =  1 − n − 1 < λ > 2 / 2  < s > − 1 λ + n − a E c < s > − 1 λ − a + O  n − 2 a 0  (2.18) and for s 1 > 2 λ − 1, s 2 > λ − 1, s 1 ≥ s 2 , E Y ns 1 Y ns 2 =  1 − n − 1 < 2 λ > 2 / 2  B 20 + n − a E c D a + O  n − 2 a 0  , (2.19) where B 20 = < s 1 > − 1 2 λ π s ( − λ ), D a = < s 1 > − 1 2 λ − a { π s ( − λ ) + π s ( a − λ ) } and C ov ar ( Y ns 1 , Y ns 2 ) = F 0 + F 1 /n + E c F 2 /n + O  n − 2 a 0  , (2.20) where F 0 = B 20 − < s 1 > − 1 λ < s 2 > − 1 λ , F 1 = < λ > 2 < s 1 > − 1 λ < s 2 > − 1 λ − < 2 λ > 2 B 20 / 2 and F 2 = D a − < s 1 > − 1 λ < s 2 > − 1 λ − a − < s 1 > − 1 λ − a < s 2 > − 1 λ . Similarly , we m a y use (2.16) to approximat e higher order cumulan ts. If a = 1 this giv es E Y ns and C ov ar ( Y ns 1 , Y ns 2 ) to O ( n − 2 ). Example 2.1 Su pp ose α = 1 . Then Y ns = X n,n − s / ( nc 0 ) , E c = c − a − 1 0 c 0 , B 20 = − F 1 = ( s 1 − 1) − 1 s − 1 2 , F 0 = < s 1 > − 1 2 s − 1 2 , D a = < s 1 > − 1 2 − a G a , wher e G a = s 1 s − 1 2 + π s ( a − 1) for s 1 ≥ s 2 , G a = 2 for s 1 = s 2 and F 2 = D a − s − 1 1 < s 2 > − 1 1 − a − s − 1 2 < s 1 > − 1 1 − a . So, E Y ns = s − 1 + n − a E c < s > − 1 1 − a + O ( n − 2 a 0 ) (2.21) for s > 0 and (2.19)-(2.20) hold if s 1 > 1 , s 2 > 0 , s 1 ≥ s 2 . (2.22) A little c alculation shows that C 0 ( s : 1 ) = c k 0 B k 0 , C 1 ( s : 1 ) = c k − a − 1 0 c 1 B k · , and E k Y i =1 Y n,s i =  1 + n − 1 < k > 2 / 2  B k 0 + n − a E c B k + O ( n − 2 a 0 ) = m 0 ( s ) + n − 1 m 1 ( s ) + n − a m a ( s ) + O ( n − 2 a 0 ) say for s i > k − i , 1 ≤ i ≤ k and s 1 ≥ · · · ≥ s k , wher e B k · = k X j =1 B k j , B k 0 = k Y i =1 1 / ( s 1 − k + 1) , B k j = j − 1 Y i =1 ( s i − k + a + i ) − 1 < s j − k + j + 1 > a − 1 k Y i = j +1 ( s i − k + i ) − 1 , B k k = k − 1 Y i =1 ( s i − k + a + i ) − 1 < s k > − 1 1 − a 8 for s i > k − i and 1 ≤ j < k . F or example, B 10 = s 1 , B 20 = ( s 1 − 1) − 1 s − 1 2 and B 30 = ( s 1 − 2) − 1 ( s 2 − 1) − 1 s − 1 3 . So, κ n ( s ) = κ ( Y ns 1 , . . . , Y ns k ) is given by κ n ( s = κ 0 ( s ) + n − 1 κ 1 ( s ) + n − a κ a ( s )+ O ( n − 2 a 0 ) , wher e , for example, writing Σ 3 a ( s 1 ) b ( s 2 s 3 ) = a ( s 1 ) b ( s 2 s 3 )+ a ( s 2 ) b ( s 3 s 1 )+ a ( s 3 ) b ( s 1 s 2 ) , κ 0 ( s 1 s 2 s 3 ) = m 0 ( s 1 s 2 s 3 ) − 3 X m 0 ( s 1 ) m 0 ( s 2 s 3 ) + 2 3 Y i =1 m 0 ( s i ) = 2 ( s 1 + s 2 − 2) D ( s 1 s 2 s 3 ) , κ 1 ( s 1 s 2 s 3 ) = m 1 ( s 1 s 2 s 3 ) − 3 X m 0 ( s 1 ) m 1 ( s 2 s 3 ) = 2  s 2 (1 − 2 s 1 ) + s 1 − s 2 1  /D ( s 1 s 2 s 3 ) sinc e m 1 ( s 1 ) = 0 , κ a ( s 1 s 2 s 3 ) = m a ( s 1 s 2 s 3 ) − 3 X { m 0 ( s 1 ) m a ( s 2 s 3 ) + m a ( s 1 ) m 0 ( s 2 s 3 ) } +2 3 X m 0 ( s 1 ) m 0 ( s 2 ) m a ( s 3 ) , wher e D ( s 1 s 2 s 3 ) = < s 1 > 3 < s 2 > 2 s 3 . Consider the c ase a = 1 . Then κ a ( s 1 s 2 s 3 ) = 0 so κ n ( s 1 s 2 s 3 ) = 2  s 1 + s 2 − 2 + n − 1  s 2 (1 − 2 s 1 ) + s 1 − s 2 1  /D ( s 1 s 2 s 3 ) + O  n − 2  . (2.23) Set s · = P k j =1 s j . Then B 1 · = B 11 − 1 , B 22 = 1 /s 2 , B 22 = 1 /s 2 , B 22 = s 1 , B 2 · = s − 1 1 + s − 1 2 = ( s 1 + s 2 ) / ( s 1 s 2 ) , B 31 = ( s 2 − 1) − 1 s − 1 3 , B 32 = ( s 1 − 1) − 1 s − 1 3 , B 33 = ( s 1 − 1) − 1 s − 1 2 , B 3 · = { s 2 ( s · − 2) − s 3 } ( s 1 − 1) − 1 < s 2 > − 1 2 s − 1 3 , B 41 = ( s 2 − 2) − 1 ( s 3 − 1) − 1 s − 1 4 , B 42 = ( s 1 − 2) − 1 ( s 3 − 1) − 1 s − 1 4 , B 43 = ( s 1 − 2) − 1 ( s 2 − 1) − 1 s − 1 4 , B 44 = ( s 1 − 2) − 1 ( s 2 − 1) − 1 s − 1 3 , B 4 · = { s · s 3 ( s 2 − 2) + s 3 ( s 2 − 4 s 2 + 4) − s 2 s 4 } { ( s 1 − 2) < s 2 − 2 > 2 < s 3 > 2 s 4 } − 1 . Also E c = c − 2 0 c 1 , D a = s − 1 1 + s − 1 2 , F 2 = 0 , and E Y ns = s − 1 + n − 1 E c + O  n − 2  for s > 0 , (2.24) E Y n,s 1 Y n,s 2 =  1 − n − 1  B 2 + n − 1 E c D a + O  n − 2  if (2.22 ) holds, (2.25) C ov ar ( Y n,s 1 , Y n,s 2 ) = < s 1 > − 1 2 s − 1 2  s − n − 1 s 1  + O  n − 2  if (2.22) holds. (2.26) In the c ase a ≥ 2 , (2.24)-(2.26) hold with E c r eplac e d by 0 . In the c ase a ≤ 1 , (2.19)-(2.21) with a 0 = a give terms O ( n − 2 a ) with the n − 1 terms disp osable if a ≤ 1 / 2 . W e n o w in ve stigate what extra terms are needed to mak e (2.24)-(2.26 ) dep end on c when a = 1 or 2. 9 Example 2.2 α = β = 1 . Her e, we fine the c o efficients of n − 2 . By (2.17 ), d 2 ( s : ψ ) = 2 X j =0 e 2 − j  j − ¯ ψ 1  C j ( s : ψ ) + e 2  − ¯ ψ 1  C 0 ( s : ψ ) + ¯ e 1  1 − ¯ ψ 1  C 1 ( s : ψ ) + C 2 ( s : ψ ) = C 2 ( s : ψ ) if ¯ ψ 1 = 1 or 2 . F or k = 1 , C 2 ( s : ψ ) = C 2 ψ ( s + 1) 2 − ψ , wher e C 2 ψ = ψ c ψ − 4 0 { c 0 c 2 + ( ψ − 3) c 2 1 / 2 } , so d 2 ( s : 1) = ( s + 1) F c , wher e F c = c − 3 0 ( c 0 c 2 − c 2 1 ) , so in (2.24) we may r eplac e O ( n − 2 ) by n − 2 ( s + 1) F c c − 1 0 + O ( n − 3 ) . F or k = 2 , C 2 ( s : 1 ) = X { C i 1 C j 1 B ( s : 0 , j − 1) : i + j = 2 } = C 01 C 21 { B ( s : 0 , 1) + B ( s : 0 , − 1) } + C 2 11 B ( s : 0 ) , wher e B ( s : 0 , λ ) = b ( s 1 − s 2 , s 2 + 1 : λ ) = π s ( λ ) , so d 2 ( s : 1 ) = C 2 ( s : 1 ) − D 2 , s H c + c − 2 0 c 2 1 , wher e D 2 , s = ( s 2 + 1)( s 1 + 1) − 1 + s 1 s − 1 2 , H c = c − 2 0 ( c 0 c 2 − c 2 1 ) and in (2.25) we may r eplac e O ( n − 2 ) b y n − 2 d 2 ( s : 1 ) c − 2 0 + O ( n − 3 ) . Up on simplifying this g i ves C ov ar ( Y n,s 1 , Y n,s 2 ) = < s 1 > − 1 2 s − 1 2  1 − n − 1 s 1  − c − 2 0 H c F 3 , s n − 2 + O  n − 2  , wher e F 3 , s = ( s 2 + 1) / < s 1 > 2 + s − 1 2 . Example 2.3 α = 1 , β = 2 . So, a = 2 , λ = 1 , ψ = θ . By (2.17), d 2 ( s : ψ ) = 1 X j =0 e 2 − 2 j  2 j − ¯ ψ 1  C j ( s : ψ ) = e 2  − ¯ ψ 1  C 0 ( s : ψ ) + C 1 ( s : ψ ) = C 1 ( s : ψ ) if ¯ ψ 1 = 0 , 1 or 2 . F or k = 1 , C 1 ( s : ψ ) = ψ c ψ − 3 0 c 1 < s > − 1 ψ − 2 =  c − 2 0 c 1 ( s + 1) , if ψ = 1 , 2 c − 1 0 c 1 , if ψ = 2 , so E Y ns = s − 1 + c − 3 0 c 1 ( s + 1) n − 2 + O ( n − 3 ) for s > 0 . F or k = 2 , C 1 ( s : 1 ) = c − 1 0 c 1 D 2 , s for D 2 , s ab ove, so E Y n,s 1 Y n,s 2 =  1 − n − 1  ( s 1 − 1) − 1 s − 1 2 + n − 2 c − 3 0 c 1 D 2 , s + O  n − 3  and C ov ar ( Y n,s 1 , Y n,s 2 ) = < s 1 > − 1 2 s − 1 2  1 − n − 1 s 1  − n − 2 c − 3 0 c 1 F 3 , s + O  n − 3  . 10 3 Examples Example 3.1 F or Student’s t distribution, X = t N has density  1 + x 2 / N  − γ g N = ∞ X i =0 d i x − 2 γ − 2 i , wher e γ = ( N + 1) / 2 , g N = Γ( γ ) / { √ N π Γ( N / 2) } and d i =  − γ i  N γ + i g N . So, (1.5) holds with α = N , β = 2 and c i = d i / ( N + 2 i ) : c 0 = N γ − 1 g N , c 1 = − γ N γ +1 ( N + 2) − 1 g N = − N γ +1 ( N + 1)( N + 2) − 1 g N / 2 , c 2 = ( γ ) 2 N γ +2 ( N + 4) − 1 g N / 2 , c 3 = − ( γ ) 3 N γ +3 G N ( N + 6) − 1 / 6 , and so on. So, a = 2 / N and (2.17) gives an expr ession in p owers of n − a/ 2 if N is o dd or n − a if N is even. The first term in (2.17) to involve c 1 , not just c 0 , is the c o efficient of n − a . Putting N = 1 we get Example 3.2 F or the Cauchy distribution, (1.5) ho lds with α = 1 , β = 2 and c i = ( − 1) i (2 i + 1) − 1 π − 1 . So, a = 2 , ψ = θ , C 0 ψ = π − ψ , C 1 ψ = − ψ π 2 − ψ / 3 , C 2 ψ = ψ π 4 − ψ { 1 / 5 + ( ψ − 5) /a } and C 3 ψ = − ψ π 6 − ψ { 1 / 10 5 − 2 ψ / 15 + ( ψ + 1) 2 / 162 } . By Example 2.3, Y ns = ( π /n ) X n,n − s satisfies E Y ns = s − 1 − n − 2 π 2 ( s + 1) + O  n − 3  (3.27) for s > 0 and when (2.22) holds E Y n,s 1 Y n,s 2 =  1 − n − 1  ( s 1 − 1) − 1 s − 1 2 − n − 2 π 2 D 2 , s / 3 + O  n − 3  (3.28) for D 2 , s = ( s 2 + 1) / ( s 1 + 1) + s 1 /s 2 and C ov ar ( Y n,s 1 , Y n,s 2 ) = < s 1 > − 1 2 s − 1 2  1 − n − 1 s 1  + n − 2 π 2 F 3 , s / 3 + O  n − 3  for F 3 , s = ( s 2 + 1) / < s 1 > 2 + s − 1 2 . Hal l (1978, p age 274) gave the first term in (3.27) and (3.28 ) when s 1 = s 2 but his v ersion of (3 .28) f or s 1 > s 2 r eplac es ( s 1 − 1) − 1 s − 1 2 and D 2 , s by c omplic ate d e xpr essions e ach with s 1 − s 2 terms. The joint or der of or der thr e e for { Y n,s i , 1 ≤ i ≤ 3 } is given by (2.23). H al l p oints out that F − 1 ( u ) = co t ( π − π u ) , so F − 1 ( u ) = P ∞ i =0 (1 − u ) 2 i − 1 C i 1 , wher e C i 1 = ( − 4 π 2 ) i π − 1 B 2 i / (2 i )! . Note 2.1 c ould b e use d. W e have not done so. Example 3.3 Consider the F distribution. F or N , M ≥ 1 , set ν = M / N , γ = ( M + N ) / 2 and g M N = ν M / 2 /B ( M / 2 , N / 2) . Then X = F M ,N has density x M / 2 (1 + ν x ) − γ g M N = ν − γ x − N/ 2  1 + ν − 1 x − 1  − γ g M N = ∞ X i =0 d i x − N/ 2 − i , 11 wher e d i = h M N  − γ i  ν i and h M N = g M N ν − γ = ν − N/ 2 /B ( M / 2 , N / 2) . So, for N > 2 , (2.6) holds with α = N / 2 − 1 , β = 1 and c i = d i / ( N/ 2 + i − 1) . If N = 4 then α = 1 and Exam- ples 2.1-2.2 apply. Otherwise (2.18)-(2.20) give E Y n, s , E Y n,s 1 Y n,s 2 and C ov ar ( Y n,s 1 , Y n,s 2 ) to O ( n − 2 a 0 ) , wher e Y n,s = X n,n − s / ( nc 0 ) λ , λ = 1 / α , a = 2 / ( N − 2) , a 0 = min( a, 1) = a if N ≥ 4 and a 0 = min ( a, 1) = 1 if N < 4 . Example 3.4 Consider the stable laws. F el ler (1966, p age 549) pr oves that the gener al stable law of index α ∈ (0 , 1) has density ∞ X k =1 | x | − 1 − ak a k ( α, γ ) , wher e a k ( α, γ ) = (1 /π )Γ( k α + 1) { ( − 1) k /k ! } sin { k π ( γ − α ) / 2 } and | γ |≤ α . So, f or x > 0 its distribution F satisfies (2.6) with β = α and c i = a i +1 ( α, γ ) γ − 1 ( i + 1) − 1 . Sinc e a = 1 the first two moments of Y n,s = X n,n − s / ( nc 0 ) λ , wher e λ = 1 /α ar e given to O ( n − 2 ) b y (2.18)-(2.20). Example 3.5 Final ly, c onsider the se c ond extr e me value distribution. Supp ose F ( x ) = exp ( − x − α ) for x > 0 , wher e α > 0 . Then (1.5) holds with β = α and c i = ( − 1) i / ( i + 1)! . Sinc e a = 1 the first two moments of Y n,s = X n,n − s /n 1 /α ar e given to O ( n − 2 by (2.18)-(2.20). References [1] Abramo witz, M. and Stegun, I. A. (1964 ). Handb o ok of Mathematic al F unctions . National Bureau of Standards, W ashington DC. [2] Com tet, L. (1974). A dvanc e d Combinatorics . Reidel, Dordrec ht. [3] Da vid , F. N. and Johnson , N. L. (1954 ). Statistica l treatment of censored data. Part I: F un damen tal formulae. Biometrika , 41 , 225–231. [4] F eller, W. (196 6). An Intr o duction to Pr ob ability Thory and Its Applic ations , vol um e 2. John Wiley and Sons, New Y ork. [5] Hall, P . (1978). Some asymptotic expansions of m oments of order statistics. Sto chastic Pr o c esses and Their A pplic ations , 7 , 265– 275. [6] James, G. S. and Ma yne, A. J. (1962). C um ulants of functions of random v ariables. Sankhy¯ a , A, 24 , 47–54 . [7] McCord, J. R. (1964 ). On asymptotic moments of extreme statistics. Ann als of Mathe- matic al Statistics , 64 , 1738– 1745. [8] No v ak, S. Y. and Utev, S. A. (199 0). Asy mtotic s of the distribu tio of the r atio of su ms of random v ariables. Sib erian Mathematic al Journal , 31 , 781–788 . 12 [9] Pic k ands, J. (1968). Momen t conv ergence of sample extremes. Annal s of Mathematic al Statistics , 39 , 881– 889. [10] P olfeldt, T. (1970) . Th e order of the m in im um v ariance in a non-regular case. A nnals of Mathematic al Statistics , 41 , 667–67 2. [11] Ramac handran, G. (1984) . Appro ximate v alues for the momen ts of extreme order stat is- tics in large samples. S tatistica l extremes and app lications (Vimeiro, 1983), pp. 563–578, NA TO Ad v anced Science Institutes Series C: Mathemat ical and Ph ysical Sciences, 131, Reidel, Dordrec ht. [12] Resnic k, S. I. (1987). Extr eme V alues, R e gu lar V ar iation, and Point Pr o c esses . Springer– V erlag, New Y ork. [13] Stuart, A. and O rd, J. K. (1987). Kendal l’s A dvanc e d The ory of Statistics , 5th edition, V olume 1. Griffin, London. [14] Withers, C. S. and Nadara jah, S. (2007a). Asymptotic multiv ariate distr ibutions and mo- men ts of extremes. T e chnic al R ep ort , Applied Mathematics Group, In dustrial Research Ltd., Lo we r Hu tt, New Z ealand. [15] Withers, C. S. and Nadara jah, S. (2007b). Exp ansions f or the b eta fu nction and its in ve rse when on e paramete r is large. T e chnic al R ep ort , Applied Mathematics Group, Industr ial Researc h L td., Lo w er Hutt, New Zealand. 13 App endix A : An In v ersion Theorem Giv en x j = y j /j ! for j ≥ 1 set S = b S ( t, x ) = ∞ X j =1 x j t j = S ( t, y ) = ∞ X j =1 y j t j /j ! . (3.29) The partial ordinary and exp onent ial Bell p olynomials b B r i ( x ) and B r i ( y ) are defined for r = 0 , 1 , . . . by S i = ∞ X r = i t r b B r i ( x ) = i ! ∞ X r = i t r B r i ( y ) /r ! . So, b B r 0 ( x ) = B r 0 ( y ) = δ r 0 (1 or 0 as r = 0 or r 6 = 0), b B r i ( λx ) = λ i b B r i ( x ) and B r i ( λy ) = λ i B r i ( y ). T hey are tabled on p ages 307 –309 of Comtet (197 4) for r ≤ 10 and 12. Note that (1 + λS ) α = ∞ X r =0 t r b C r = ∞ X r =0 t r C r /r ! , (3.30) where b C r = b C r ( α, λ, x ) = r X i =0 b B r i ( x )  α i  λ i (3.31) and C r = C r ( α, λ, x ) = r X i =0 B r i ( y ) < α > i λ i . So, b C 0 = 1, b C 1 = αλx 1 , b C 2 = αλx 2 + < α > 2 λ 2 x 2 1 / 2, b C 3 = αλx 3 + < α > 2 λ 2 x 1 x 2 + < α > 3 λ 3 x 3 1 / 6 and C 0 = 1, C 1 = αλy 1 , C 2 = αλy 2 + < α > 2 λ 2 y 2 1 . Sim ilarly , log(1 + λS ) = ∞ X r =1 t r b D r = ∞ X r =1 t r D r /r ! and exp( λS ) = 1 + ∞ X r =1 t r b B r = 1 + ∞ X r =1 t r B r /r ! , where b D r = b D r ( λ, x ) = − r X i =1 b B r i ( x )( − λ ) i /i ! , D r = D r ( λ, y ) = − r X i =1 B r i ( y )( − λ ) i / ( i − 1)! , 14 b B r = b B r ( λ, x ) = r X i =1 b B r i ( x ) λ i /i ! and B r = B r ( λ, y ) = r X i =1 B r i ( y ) λ i . Here, b B r (1 , x ) and B r (1 , y ) are known as the c omplete ordin ary and exp onenti al Bell p olyno- mials. If x j = y j = 0 for j ev en, then S = t − 1 P ∞ j =1 X j t 2 j , where X j = x 2 j − 1 , so S i = t − i ∞ X r = i t 2 r b B r i ( X ) and exp( λS ) = 1 + ∞ X k =1 t k b B k , where b B k = X n b B r i ( X ) λ i /i ! : i = 2 r − k , k / 2 < r ≤ k o . The follo win g derive s from Lagrange’s inv ersion formula. Theorem A.1 L et k b e a p ositive inte ger and a any r e al numb er. Su pp ose v /u = ∞ X i =0 x i u ia = ∞ X i =0 y i v ia /i ! with x 0 6 = 0 . Then ( u/v ) k = ∞ X i =0 x ∗ i v ia = ∞ X i =0 y ∗ i v ia / ( ia )! , wher e x ∗ i = x ∗ i ( a, k , x ) and y ∗ i = y ∗ i ( a, k , y ) ar e given by x ∗ i = k n − 1 b C i ( − n, 1 /x 0 , x ) = k x − n 0 i X j =0 ( n + 1) j − 1 b B ij ( x ) ( − x 0 ) − j /j ! (3.32) and y ∗ i = k n − 1 C i ( − n, 1 /y 0 , y ) = k y − n 0 i X j =0 ( n + 1) j − 1 B ij ( y ) ( − y 0 ) − j , (3.33) r esp e ctiv ely, wher e n = k + ai . Pro of: u/v has a p o w er series in v a so that ( u/v ) k do es also. A little w ork shows that (3.32)- (3.33) are correct for i = 0 , 1 , 2 , 3 and so by induction that x ∗ i x ia 0 and y ∗ i y ia 0 are p olynomials in a of degree i − 1. Hence, (3.32)-(3.3 3 ) will hold true for all a if they hold true for all p ositiv e in tegers a . Sup p ose then a is a p ositiv e in teger. Since v/u = x 0 (1 + x − 1 0 S ) f or S = b S ( u a , x ) = S ( u a , y ), th e coefficien t of u ai in ( v /y ) − n is x − n 0 b C i ( − n, 1 /x 0 , x ) = y − n 0 C i ( − n, 1 /y 0 , y ) / ( n − k )!. No w set n = k + ai and apply Theorem A in Comte t (1974, page 148) to v = f ( u ) = P ∞ i =0 x i u 1+ ai .  Note A.1 Comtet (1978, p age 15 , The or em F) pr oves (3.32) for the c ase k = 1 and a a p ositive inte ger. 15