EXPANSIONS FOR QUANTILES AND MULTIVARIATE MOMENTS OF EXTREMES FOR HEAVY TAILED DISTRIBUTIONS

Transcription

1 REVSTAT Statistical Journal Volume 15, Number 1, January 2017, EXPANSIONS FOR QUANTILES AND MULTIVARIATE MOMENTS OF EXTREMES FOR HEAVY TAILED DISTRIBUTIONS Authors: Christopher Withers Industrial Research Limited, Lower Hutt, New Zealand Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, UK Received: August 2014 Revised: March 2015 Accepted: May 2015 Abstract: Let X n,r be the r-th largest of a random sample of size n from a distribution function Fx) = 1 c i x α iβ for α > 0 and β > 0. An inversion theorem is proved and used to derive an expansion for the quantile F 1 u) and powers of it. From this an expansion in powers of n 1,n β/α) is given for the multivariate moments of the extremes {X n,n si, 1 i k}/n 1/α for fixed s = s 1,...,s k ), where k 1. Examples include the Cauchy, Student s t, F, second extreme distributions and stable laws of index α < 1. Key-Words: Bell polynomials; extremes; inversion theorem; moments; quantiles. AMS Subject Classification: 62E15, 62E17.

2 26 C.S. Withers and S. Nadarajah

3 Extremes for Quantiles and Multivariate Moments INTRODUCTION For 1 r n, let X n,r be the r-th largest of a random sample of size n from a continuous distribution function F on R, the real numbers. Let f denote the density function of F when it exists. The study of the asymptotes of the moments of X n,r has been of considerable interest. McCord [12] gave a first approximation to the moments of X n,1 for three classes. This showed that a moment of X n,1 can behave like any positive power of n or n 1 = log n. Here, log is to the base e.) Pickands [15] explored the conditions under which various moments of X n,1 b n )/a n converge to the corresponding moments of the extreme value distribution. It was proved that this is indeed true for all F in the domain of attraction of an extreme value distribution provided that the moments are finite for sufficiently large n. Nair [13] investigated the limiting behavior of the distribution and the moments of X n,1 for large n when F is the standard normal distribution function. The results provided rates of convergence of the distribution and the moments of X n,1. Downey [4] derived explicit bounds for E[X n,1 ] in terms of the moments associated with F. The bounds were given up to the order o n 1/ρ), where x ρ dfx) is defined, so E[X n,1 ] grows slowly with the sample size. For other work, we refer the readers to Ramachandran [16], Hill and Spruill [9] and Hüsler et al. [10]. The main aim of this paper is to study multivariate moments of {X n,n si, 1 i k} for fixed s = s 1,..., s k ), where k 1. We suppose F is heavy tailed, i.e., 1.1) 1 Fx) Cx α as x for some C > 0 and α > 0. For a nonparametric estimate of α, see Novak and Utev [14]. There are many practical examples giving rise to {X n,n si, 1 i k} for heavy tailed F. Perhaps the most prominent example is the Hill s estimator Hill [8]) for the extremal index given by k log X n,n k + k 1 log X n,n i+1. Clearly, this is a function of X n,n si, 1 i k. Real life applications of the Hill s estimator are far too many to list. Since Hill [8], many other estimators have been proposed for the extremal index, see Gomes and Guillou [6] for an excellent review of such estimators. Each of these estimators is a function of X n,n si, 1 i k. No doubt that many more

4 28 C.S. Withers and S. Nadarajah estimators taking the form of a function of X n,n si, 1 i k will be proposed in the future. A possible application of the results in this paper is to assess optimality of these estimators. Suppose we can write the general form of the estimators as 1.2) ω = ω X n,n s1, X n,n s2,..., X n,n sk ; µ ), where µ contains some parameters, which include k itself. The optimum values of µ can be based on criteria like bias and mean squared error. For example, µ could be chosen as the value minimizing the bias of ω or the value minimizing the mean squared error of ω. If 1.2) can be expanded as ω = θ 1,θ 2,...,θ k a θ1, θ 2,..., θ k ; µ ) k X θ i n,n s i then the bias and mean squared error of ω can be expressed as and MSEω) = Biasω) = θ 1,θ 2,...,θ k a a θ 1,θ 2,...,θ k ϑ 1,ϑ 2,...,ϑ k θ 1,θ 2,...,θ k a θ1, θ 2,..., θ k ; µ ) E[ k X θ i n,n s i ] ω θ1, θ 2,..., θ k ; µ ) a ϑ 1, ϑ 2,..., ϑ k ; µ ) E[ k θ1, θ 2,..., θ k ; µ ) E[ k 2 X θ i n,n s i ] + [ Biasω) ] 2, ] X θ i+ϑ i n,n s i respectively. Both involve multivariate moments of X n,n si, 1 i k. Expressions for the latter are given in Section 2, in particular, Theorem 2.2. Hence, general estimators can be developed for µ which minimize bias, mean squared error, etc. Such developments could apply to any future estimator also to any past estimator) of the extremal index taking the form of 1.2). Note that U n,r = F X n,r ) is the r-th order statistics from U0, 1). For 1 r 1 < r 2 < < r k n set U n,r = {U n,ri, 1 i k}. By Section 14.2 of Stuart and Ord [17], U n,r has the multivariate beta density function 1.3) U n,r B u : r) = k u i+1 u i ) r i+1 r i 1 / B n r) on 0 < u 1 < < u k < 1, where u 0 = 0, u k+1 = 1, r 0 = 0, r k+1 = n + 1 and 1.4) B n r) = k B r i, r i+1 r i ).

5 Extremes for Quantiles and Multivariate Moments 29 David and Johnson [3] expanded X n,ri = F 1 U n,ri ) about u n,i = E[U n,ri ] = r i /n + 1): X n,ri = G j) u n,i )U n,i u n,i ) j /j!, where Gu) = F 1 u) and j=0 G j) u) = d j Gu)/du j, and using the properties of 1.3) showed that if r depends on n in such a ways that r/n p 0,1) as n then the m-th order cumulants of X n,r = {X n,ri, 1 i k} have magnitude O n 1 m) at least for n 4, so that the distribution function of X n,r has a multivariate Edgeworth expansion in powers of n 1/2. Alternatively one can use James and Mayne [11] to derive the cumulants of X n,r from those of U n,r.) The method requires the derivatives of F at { F 1 p i ), 1 i k } so breaks down if p i = 0 or p k = 1 which is the situation we study here. In Withers and Nadarajah [18], we showed that for fixed r when 1.1) holds the distribution of X n,n1 r where 1 is the vector of ones in R k ), suitably normalized tends to a certain multivariate extreme value distribution as n, and so obtained the leading terms of the expansions of its moments in inverse powers of n. Here, we show how to extend those expansions when 1.5) F 1 u) = b i 1 u) α i with α 0 < α 1 <, that is, {1 Fx)} x 1/α 0 has a power series in {x δ i : δ i = α i α 0 ) /α 0 }. Hall [7] considered 1.5) with α i = i 1/α, but did not give the corresponding expansion for Fx) or expansions in inverse powers of n. He applied it to the Cauchy. In Section 2, we demonstrate the method when 1.6) 1 Fx) = x α c i x iβ, where α > 0 and β > 0. In this case, 1.5) holds with α i = iβ 1)/α. In Section 3, we apply it to the Student s t, F and second extreme value distributions and to stable laws of exponent α < 1. The appendix gives the inverse theorem needed to pass from 1.6) to 1.5), and expansions for powers and logs of series. We use the following notation and terminology. Let x) i = Γx + i)/γx) and x i = Γx + 1)/Γx i + 1). An inequality in R k consists of k inequalities. For example, for x in C k, where C is the set of complex numbers, Rex) < 0 means that Re x i ) < 0 for 1 i k. Also let IA) = 1 if A is true and IA) = 0 i if A is false. For θ C k let θ denote the vector with θ i = θ j. j=1

6 30 C.S. Withers and S. Nadarajah 2. MAIN RESULTS For 1 r 1 < < r k n set s i = n r i. Here, we show how to obtain expansions in inverse powers of n for the moments of the X n,s for fixed r when 1.5) holds, and in particular when the upper tail of F satisfies 1.6). Theorem 2.1. Suppose 1.6) holds with c 0, α, β > 0. Then F 1 u) is given by 1.5) with α i = ia 1/α, a = β/α and b i = C i,1/α, where C i,ψ = c ψ 0 Ĉi ψ, c 0,x ) of A.3) and x i = x i a,1,c) of A.4). In particular, C 0,ψ = c ψ 0, C 1,ψ = ψ c ψ a 1 0 c 1, C 2,ψ = ψ c ψ 2a 2 { 0 c0 c 2 + ψ 2a 1)c 2 1/2 }, [ ] C 3,ψ = ψ c ψ 3a 3 0 c 2 0c 2 + ψ 3a 1)c 0 c 1 c 2 + {ψ +1) 2 /6ψ + 3a/2)a+1)} c 3 1, and so on. Also for any θ in R, 2.1) { F 1 u) } θ = 1 u) ia ψ C i,ψ at ψ = θ/α. On those rate occasions, where the coefficients d i = C i,1/α in F 1 u) = 1 u) ia 1/α d i are known from some alternative formula then one can use C i,ψ = d θ 0Ĉi θ, 1/d 0,d) of A.3). Proof of Theorem 2.1: By Theorem A.1 with k = 1, we have x α = x i 1 u) 1+ia at u = Fx), where x 0 = c 1 0, x 1 = c0 a 2 c 1, x 2 = c0 2a 3 { c0 c 2 + a + 1)c 2 } 1, x 3 = c0 3a 4 { c 2 0 c a)c 0 c 1 c a)1 + a)c 2 1/2 }, and so on. So, for S of A.1), x α = c 1 0 v [1 + c 0S v a,x )] at v = 1 u. Now apply A.2).

7 Extremes for Quantiles and Multivariate Moments 31 Lemma 2.1. For θ in C k, [ k ] E 1 U n,ri ) θ ) 2.2) i = b n r : θ, where ) k b n r : θ = b ) 2.3) r i r i 1, n r i + 1 : θ i and b α, β : θ) = Bα, β + θ)/bα, β). Also in 1.4), 2.4) B n r) = k B r i r i 1, n r i + 1). Since Bα, β) = for Reβ 0, for 2.2) to be finite we need n r i Re θ i > 0 for 1 i k. Proof of Lemma 2.1: Let I k denote the left hand side of 2.2). Then k I k = B n u : r) 1 u i ) θ i du 1 du k integrated over 0 < u 1 < < u k < 1 by 1.3). So, 2.2), 2.4) hold for k = 1. Set s i = u i u i 1 ) / 1 u i 1 ). Then 1 1 I 2 = u r u 1 ) θ 1 u 2 u 1 ) r 2 r u 2 ) r 3 r 2 1+θ 2 / du 2 Bn r), u 1 0 which is the the right hand side of 2.2) with denominator replaced by the right hand side of 2.3). Putting θ = 0 gives 2.2), 2.4) for k = 2. Now use induction. Lemma 2.2. In Lemma 2.1, the restriction 2.5) 1 r 1 < < r k n may be relaxed to 1 r 1 r k n. Proof: For k = 2, the second factor in the right hand side of 2.3) is br 2 r 1, n r : θ 2 ) = f ) ) ) θ 2 /f0), where f θ2 = Γ n r θ 2 / Γ ) n r θ 2 = 1 if r2 = r 1 and the first factor is b ) [ ] r 1, n r : θ 1 = E 1 U n,r1 ) θ 1. Similarly, if r i = r i 1, the i-th factor is 1 and the product of the k ) others is E θ 1 j Un,rj, where θj = θ j for j i 1 and θj = θ i 1 + θ i for j = i 1. j=1,j i Corollary 2.1. In any formulas for E[g X n,r )] for some function g, 2.5) holds. In particular it holds for the moments and cumulants of X n,r.

8 32 C.S. Withers and S. Nadarajah This result is very important as it means we can dispense with treating the 2 k 1 cases r i < r i+1 or r i = r i+1, 1 i k 1 separately. For example, Hall [7] treats the two cases for cos X n,r, X n,s ) separately and David and Johnson [3] treat the 2 k 1 cases for the k-th order cumulants of X n,r separately for k 4. Theorem 2.2. Under the conditions of Theorem 2.1, 2.6) E [ k X θ i n,r i ] = i 1,...,i k =0 C i1,ψ 1 C ik,ψ k b n r : ia θ/α ) with b n as in 2.3), where ψ = θ/α. All terms are finite if Re θ < s + 1)α, where s i = n r i. Lemma 2.3. For α, β positive integers and θ in C, 2.7) α+β 1 bα, β : θ) = 1 + θ/j) 1. j=β So, for θ in C k, 2.8) b n r : θ ) = k s i 1 j=s i θi /j ) 1, where s i = n r i and r 0 = 0. Proof: The left hand side of 2.7) is equal to Γβ + θ)γα + β)/ {Γβ + θ + α)γβ)}. But Γα + x)/γx) = x) α, so 2.7) holds, and hence 2.8). From 2.3) we have, interpreting Lemma 2.4. For s i = n r i, k 1 i=2 b i as 1, 2.9) b n r : θ ) = B s : θ ) n! / Γ n θ1 ), where B s : θ ) = Γ ) k s θ 1 s1!) 1 b ) s i 1 s i, s i + 1 : θ i does not depend on n for fixed s. i=2

9 Extremes for Quantiles and Multivariate Moments 33 Lemma 2.5. We have n!/ Γn θ) = n θ e i θ)n i, where e 0 θ) = 1, e 1 θ) = θ) 2 /2, e 2 θ) = θ) 3 3θ + 1)/24, e 3 θ) = θ) 4 θ) 2 / 4! 2), e 4 θ) = θ) 5 15θ θ 2 + 5θ 2 ) / 5! 48), e 5 θ) = θ) 6 θ) 2 3θ 2 + 7θ 2 ) / 6! 16), e 6 θ) = θ) 7 63θ θ θ 3 91θ 2 42θ + 16 ) / 7! 576), e 7 θ) = θ) 8 θ) 2 9θ θ θ 2 58θ + 16 ) / 8! 144). Proof: Apply equation ) of Abramowitz and Stegun [1]. So, 2.6), 2.9) yield the joint moments of X n,r n 1/α for fixed s as a power series in 1/n, n α ): Corollary 2.2. Under the conditions of Theorem 2.1, [ k ] E X θ i n,n s i = n! Γ ) ) n ja ψ 1 Cj s : ψ), j=0 where ψ = θ/α and C j s : ψ) = {C i1,ψ 1 C ik,ψ k B s : ia ψ ) } : i i k = j. So, if s, θ are fixed as n and Re θ ) < s + 1)α, then the left hand side of 2.10) is equal to 2.11) n ψ 1 n i ja ) e i ja ψ1 Cj s : ψ). i,j=0 If a is rational, say a = M/N then the left hand side of 2.10) is equal to 2.12) where n ψ 1 n m/n d m s : ψ), m=0 d m s : ψ) = ) } {e i ja ψ1 Cj s : ψ): in + jm = m = ) } {e m ja ja ψ1 Cj s : ψ): 0 j m/a if N = 1; so for d m to depend on c 1 and not just c 0 we need m M.

10 34 C.S. Withers and S. Nadarajah The leading term in 2.11) does not involve c 1 so may be deduced from the multivariate extreme value distribution that the law of X n,n si, suitably normalized, tends to. The same is true of the leading terms of its cumulants. See Withers and Nadarajah [18] for details. where The leading terms in 2.11) are n ψ 1[ {1 n 1 ψ 1 2 /2 } C 0 s : ψ) + n a C 1 s : ψ) + O n 2a 0 )], a 0 = mina,1), and I j,m = Im j). For k = 1, C 0 s : ψ) = c 0 B s : ψ ), k C 1 s : ψ) = c ψ 1 a 2 0 c 1 ψ j B s : ai j ψ ) j=1 C 0 s : ψ) = c ψ 0 s + 1) ψ = c ψ 0 / s ψ, C 1 s : ψ) = ψc ψ a 1 0 c 1 s + 1) a ψ = ψc ψ a 1 0 c 1 / s ψ a. Set π s λ) = b s 1 s 2, s : λ) = s 1 j=s / 1 + λ/j) for λ an integer. For example, π s 1) = s 2 + 1) / s 1 + 1) and π s 1) = s 1 /s 2. Then for k = 2, C 0 s : λ1) = c 2λ 0 s 1 1 2λ π s λ) = c 2 0 s 1 1) 1 s 2 for λ = 1 = c 2 0 s s for λ = 2 and C 1 s : λ1) = λ c0 2λ a 1 c 1 s 1 1 { 2λ a πs λ) + π s a λ) } = λ c 1 a 0 c 1 s 1 1 = λ c 3 a 0 c 1 s a 2 a{ s1 /s 2 + π s a 1) } for λ = 1 { s1 2 s π s a 2) } for λ = 2. Set λ = 1/α, Y n,s = X n,n s / nc 0 ) λ and E c = λ c a 1 0 c 1. Then for s > λ ) E[Y n,s ] = { 1 n 1 λ 2 /2 } s 1 λ and for s 1 > 2λ 1, s 2 > λ 1, s 1 s 2, + n a E c s 1 λ a + O n 2a 0 ) 2.14) E[Y n,s1 Y n,s2 ] = { 1 n 1 2λ 2 /2 } B 2,0 + n a E c D a + O n 2a 0 ),

11 Extremes for Quantiles and Multivariate Moments 35 where B 2,0 = s 1 1 2λ π s λ), D a = s 1 1 2λ a {π s λ) + π s a λ)} and 2.15) Cov Y n,s1, Y n,s2 ) = F 0 + F 1 /n + E c F 2 /n + O n 2a 0 ), where F 0 = B 2,0 s 1 1 λ s 2 1 λ, F 1 = λ 2 s 1 1 λ s 2 1 λ 2λ 2B 2,0 /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 may use 2.11) to approximate higher order cumulants. If a = 1 this gives E[Y n,s ] and Cov Y n,s1, Y n,s2 ) to O n 2). Example 2.1. Suppose α = 1. Then Y n,s = X n,n s / nc 0 ), E c = c0 a 1 c 1, B 2,0 = F 1 = s 1 1) 1 s 1 2, F 0 = s s 1 2, D a = s a G a, where G a = s 1 s 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 a s 1 2 s a. So, 2.16) E[Y n,s ] = s 1 + n a E c s 1 1 a + O n 2a 0 ) for s > 0 and 2.14) 2.15) hold if 2.17) s 1 > 1, s 2 > 0, s 1 s 2. A little calculation 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 [ k ] E Y n,si = { 1 + n 1 k 2 /2 } B k,0 + n a E c B k, + O n 2a ) 0 = m 0 s) + n 1 m 1 s) + n a m a s) + O n 2a 0 ) say for s i > k i, 1 i k and s 1 s k, where B k, = B k,0 = k B k,j, j=1 k 1/ s 1 k + i), j 1 k B k,j = s i k + a + i) 1 s j k + j + 1 a 1 s i k + i) 1, B k,k = k 1 s i k + a + i) 1 s k 1 1 a i=j+1 for s i > k i and 1 j < k. For example, B 1,0 = s 1, B 2m,0 = s 1 1) 1 s 1 2 and B 3,0 = s 1 2) 1 s 2 1) 1 s 1 3. So, κ ns) = κy n,s1,..., Y n,sk ), the joint cumulant of Y n,s1,..., Y n,sk ), is given by κ n s) = κ 0 s) + n 1 κ 1 s) + n a κ a s) +

12 36 C.S. Withers and S. Nadarajah O n 2a 0), where, for example, κ 0 s 1, s 2, s 3 ) = m 0 s 1, s 2, s 3 ) m 0 s 1 )m 0 s 2, s 3 ) m 0 s 2 ) m 0 s 1, s 3 ) 3 m 0 s 3 )m 0 s 1, s 2 ) + 2 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 ) m 0 s 1 )m 1 s 2, s 3 ) m 0 s 2 ) m 1 s 1, s 3 ) m 0 s 3 )m 1 s 1, s 2 ) = 2 { s 2 1 2s 1 ) + s 1 s 2 1} /D s1, s 2, s 3 ) since m 1 s 1 ) = 0, κ a s 1, s 2, s 3 ) = m a s 1, s 2, s 3 ) m 0 s 1 )m a s 2, s 3 ) m a s 1 ) m 0 s 2, s 3 ) m 0 s 2 )m a s 1, s 3 ) m a s 2 )m 0 s 1, s 3 ) m 0 s 3 )m a s 1, s 2 ) m a s 3 )m 0 s 1, s 2 ) + 2m 0 s 1 )m 0 s 2 )m a s 3 ) + 2m 0 s 3 )m 0 s 1 ) m a s 2 ) + 2m 0 s 2 )m 0 s 3 )m a s 1 ), where D s 1, s 2, s 3 ) = s 1 3 s 2 2 s 3. Consider the case 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 n 1 s 2 1 2s 1 ) + s 1 s 2 1) } / D s1, s 2, s 3 ) 2.18) Set s = k s j. Then j=1 + O n 2). B 1, = B 1,1 1, B 2,2 = 1/s 2, B 2,2 = 1/s 2, B 2,2 = s 1, B 2, = s s 1 2 = s 1 + s 2 ) / s 1 s 2 ), B 3,1 = s 2 1) 1 s 1 3, B 3,2 = s 1 1) 1 s 1 3, B 3,3 = s 1 1) 1 s 1 2, B 3, = { s 2 s 2) s 3 } s1 1) 1 s s 1 3, B 4,1 = s 2 2) 1 s 3 1) 1 s 1 4, B 4,2 = s 1 2) 1 s 3 1) 1 s 1 4, B 4,3 = s 1 2) 1 s 2 1) 1 s 1 4, B 4,4 = s 1 2) 1 s 2 1) 1 s 1 3, B 4, = { s s 3 s 2 2) + s 3 s 2 4s 2 + 4) s 2 s 4 } { s1 2) s s 3 2 s 4 } 1. Also E c = c 2 0 c 1, D a = s s 1 2, F 2 = 0, and 2.19) 2.20) 2.21) E[Y n,s ] = s 1 + n 1 E c + O n 2) for s > 0, E[Y n,s1 Y n,s2 ] = 1 n 1) B 2, + n 1 E c D a + O n 2) if 2.17) holds, Cov Y n,s1, Y n,s2 ) = s s 1 2 s2 n 1 ) s 1 + O n 2 ) if 2.17) holds. In the case a 2, 2.19) 2.21) hold with E c replaced by 0. In the case a 1, 2.14) 2.16) with a 0 = a give terms O n 2a) with the n 1 terms disposable if a 1/2.

13 Extremes for Quantiles and Multivariate Moments 37 We now investigate what extra terms are needed to make 2.19) 2.21) depend on c when a = 1 or 2. Example 2.2. α = β = 1. Here, we find the coefficients of n 2. By 2.12), d 2 s : ψ) = 2 ) e 2 j j ψ1 Cj s : ψ) j=0 = e 2 ψ1 ) C0 s : ψ) + e 1 1 ψ1 ) C1 s : ψ) + C 2 s : ψ) = C 2 s : ψ) if ψ 1 = 1 or 2. For k = 1, C 2 s : ψ) = C 2,ψ s+1) 2 ψ, where C 2,ψ = ψc ψ 4 { 0 c0 c 2 + ψ 3)c 2 1 /2}, so d 2 s : 1) = s + 1)F c, where F c = c 3 0 c0 c 2 c 2 ) 1, so in 2.19) we may replace O n 2) by n 2 s + 1)F c c O n 3). For k = 2, C 2 s : 1) = } {C i,1 C j,1 B s : 0, j 1): i + j = 2 = C 0,1 C 2,1 { B s : 0, 1) + B s : 0, 1) } + C 2 1,1 B s : 0, 0), where B s : 0, λ) = b s 1 s 2, s : λ) = π s λ), so d 2 s : 1) = C 2 s : 1) D 2,s H c + c 2 0 c2 1, where D 2,s = s 2 + 1) s 1 + 1) 1 + s 1 s 1 2, H c = c 2 0 c0 c 2 c 2 1) and in 2.20) we may replace O n 2) by n 2 d 2 s : 1)c O n 3). Upon simplifying this gives Cov Y n,s1, Y n,s2 ) = s s n 1 ) s 1 c 2 0 H cf 3,s n 2 + O n 2), where F 3,s = s 2 + 1) / s s 1 2. For k = 1, Example 2.3. α = 1, β = 2. So, a = 2, λ = 1, ψ = θ. By 2.12), d 2 s : ψ) = 1 ) e 2 2j 2j ψ1 Cj s : ψ) j=0 C 1 s : ψ) = ψ c ψ 3 0 c 1 s 1 ψ 2 = = e 2 ψ1 ) C0 s : ψ) + C 1 s : ψ) = C 1 s : ψ) if ψ 1 = 0, 1 or 2. { c 2 0 c 1s + 1), if ψ = 1, 2 c 1 0 c 1, if ψ = 2, so E[Y n,s ] = s 1 + c 3 0 c 1s + 1)n 2 + O n 3) for s > 0. For k = 2, C 1 s : 1) = c 1 0 c 1D 2,s for D 2,s above, so and E[Y n,s1 Y n,s2 ] = 1 n 1) s 1 1) 1 s n 2 c 3 0 c 1D 2,s + O n 3) Cov Y n,s1, Y n,s2 ) = s s n 1 s 1 ) n 2 c 3 0 c 1F 3,s + O n 3).

14 38 C.S. Withers and S. Nadarajah 3. EXAMPLES Example 3.1. For Student s t distribution, X = t N has density function 1 + x 2 /N ) γ gn = d i x 2γ 2i, { Nπ } ) γ where γ = N + 1)/2, g N = Γγ)/ ΓN/2) and d i = N γ+i g N. i So, 1.6) holds with α = N, β = 2 and c i = d i /N + 2i). In particular, c 0 = N γ 1 g N, c 1 = γn 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 2.12) gives an expression in powers of n a/2 if N is odd or n a if N is even. The first term in 2.12) to involve c 1, not just c 0, is the coefficient of n a. Putting N = 1 we obtain Example 3.2. For the Cauchy distribution, 1.6) holds with α = 1, β = 2 and c i = 1) i 2i + 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 {1/105 2ψ/15 + ψ + 1) 2 /162}. By Example 2.3, Y n,s = π/n)x n,n s satisfies 3.1) E[Y n,s ] = s 1 n 2 π 2 s + 1) + O n 3) for s > 0 and when 2.17) holds 3.2) E[Y n,s1 Y n,s2 ] = 1 n 1) s 1 1) 1 s 1 2 n 2 π 2 D 2,s /3 + O n 3) for D 2,s = s 2 + 1) / s 1 + 1) + s 1 /s 2 and Cov Y n,s1, Y n,s2 ) = s s n 1 ) s 1 + n 2 π 2 F 3,s /3 + O n 3) for F 3,s = s 2 + 1)/ s s 1 2. Page 274 of Hall [7] gave the first term in 3.1) and 3.2) when s 1 = s 2 but his version of 3.2) for s 1 > s 2 replaces s 1 1) 1 s 1 2 and D 2,s by complicated expressions each with s 1 s 2 terms. The joint order of order three for {Y n,si, 1 i 3} is given by 2.18). Hall points out that F 1 u) = cotπ πu), so F 1 u) = 1 u) 2i 1 C i,1, where C i,1 = 4π 2) i π 1 B 2,i /2i)!.

15 Extremes for Quantiles and Multivariate Moments 39 Example 3.3. Consider the F distribution. For 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 function x M/2 1 + νx) γ g M,N = ν γ x N/2 1 + ν 1 x 1) γ gm,n = d i x N/2 i, ) γ where d i = h M,N ν i and h M,N = g M,N ν γ = ν N/2/ BM/2, N/2). So, for i N > 2, 2.1) holds with α = N/2 1, β = 1 and c i = d i /N/2 + i 1). If N = 4 then α = 1 and Examples apply. Otherwise 2.13) 2.15) give E[Y n,s ], E[Y n,s1 Y n,s2 ] and Cov Y n,s1, Y n,s2 ) to O n 2a 0), where Yn,s = X n,n s / nc 0 )λ, λ = 1/α, a = 2/N 2), a 0 = mina,1) = a if N 4 and a 0 = mina,1) = 1 if N < 4. Example 3.4. Consider the stable laws. Page 549 of Feller [5] proves that the general stable law of index α 0, 1) has density function x 1 αk a k α, γ), k=1 where a k α, γ) = 1/π)Γkα + 1) { 1) k /k! } sin{kπγ α)/2} and γ α. So, for x > 0 its distribution function F satisfies 2.1) with β = α and c i = a i+1 α, γ)γ 1 i+1) 1. Since a =1 the first two moments of Y n,s = X n,n s / nc 0 ) λ, where λ = 1/α are O n 2) by 2.13) 2.15). Example 3.5. Finally, consider the second extreme value distribution. Suppose Fx) = exp x α ) for x > 0, where α > 0. Then 1.6) holds with β = α and c i = 1) i /i + 1)!. Since a = 1 the first two moments of Y n,s = X n,n s /n 1/α are given to O n 2) by 2.13) 2.15).

16 40 C.S. Withers and S. Nadarajah APPENDIX: AN INVERSION THEOREM Given x j = y j /j! for j 1 set A.1) S = Ŝt,x) = x j t j = St,y) = y j t j /j!. j=1 j=1 The partial ordinary and exponential Bell polynomials B r,i x) and B r,i y) are defined for r = 0, 1,... by S i = t r Br,i x) = i! t r B r,i y)/r!. r=i r=i So, Br,0 x) = B r,0 y) = Ir = 0), Br,i λx) = λ i Br,i x) and B r,i λy) = λ i B r,i y). They are tabled on pages of Comtet [2] for r 10 and 12. Note that A.2) where A.3) and 1 + λs) α = t r Ĉ r = r=0 Ĉ r = Ĉr α, λ,x) = C r = C r α, λ,y) = t r C r /r!, r=0 r ) α B r,i x) λ i i r B r,i y) α i λ i. So, Ĉ 0 = 1, Ĉ 1 = αλx 1, Ĉ 2 = αλx 2 + α 2 λ 2 x 2 1 /2, Ĉ 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. Similarly, and where log1 + λs) = t r Dr = t r D r /r! r=1 r=1 expλs) = 1 + t r Br = 1 + t r B r /r!, r=1 r=1 D r = D r r λ,x) = B r,i x) λ) i /i!, r D r = D r λ,y) = B r,i y) λ) i /i 1)!,

17 Extremes for Quantiles and Multivariate Moments 41 and B r = B r λ,x) = B r = B r λ,y) = r B r,i x)λ i /i! r B r,i y)λ i. Here, B r 1,x) and B r 1,y) are known as the complete ordinary and exponential Bell polynomials. If x j = y j = 0 for j even, then S = t 1 X j t 2j, where X j = x 2j 1, so S i = t i t 2r Br,i X) and expλs) = 1 + t k Bk, j=1 r=i k=1 where B k = { } Br,i X)λ i /i!: i = 2r k, k/2 < r k. The following derives from Lagrange s inversion formula. Theorem A.1. Let k be a positive integer and a any real number. Suppose v/u = x i u ia = y i v ia /i! with x 0 0. Then u/v) k = x i v ia = yi v ia /ia)!, where x i = x i a, k,x) and y i = y i a, k,y) are given by A.4) and A.5) x i = k n 1 Ĉ i n, 1/x 0,x) = k x n 0 y i = k n 1 C i n, 1/y 0,y) = k y n 0 i n + 1) j 1 Bi,j x) x 0 ) j /j! j=0 i n + 1) j 1 B i,j y) y 0 ) j, j=0 respectively, where n = k + ai. Proof: u/v has a power series in v a so that u/v) k does also. A little work shows that A.4) A.5) are correct for i = 0, 1, 2, 3 and so by induction that x i xia 0 and yi yia 0 are polynomials in a of degree i 1. Hence, A.4) A.5) will hold true for all a if they hold true for all positive integers a. Suppose then a is a positive

18 42 C.S. Withers and S. Nadarajah integer. Since v/u = x x 1 0 S) for S = Ŝ ua,x) = S u a,y), the coefficient of u ai in v/u) n is x n 0 Ĉ i n, 1/x 0,x) = y0 n C i n, 1/y 0,y)/n k)!. Now set n = k + ai and apply Theorem A in page 148 of Comtet [2] to v = fu) = x i u 1+ai. Theorem F in page 15 of Comtet [2] proves A.4) for the case k = 1 and a a positive integer. ACKNOWLEDGMENTS The authors would like to thank the Editor and the referee for careful reading and comments which greatly improved the paper. REFERENCES [1] Abramowitz, M. and Stegun, I.A. 1964). Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C. [2] Comtet, L. 1974). Advanced Combinatorics, Reidel, Dordrecht. [3] David, F. N. and Johnson, N. L. 1954). Statistical treatment of censored data. Part I: Fundamental formulae, Biometrika, 41, [4] Downey, P. J. 1990). Distribution-free bounds on the expectation of the maximum with scheduling applications, Operations Research Letters, 9, [5] Feller, W. 1966). An Introduction to Probability Theory and Its Applications, volume 2, John Wiley and Sons, New York. [6] Gomes, M. I. and Guillou, A. 2014). Extreme value theory and statistics of univariate extremes: A review, International Statistical Review, doi: /insr [7] Hall, P. 1978). Some asymptotic expansions of moments of order statistics, Stochastic Processes and Their Applications, 7, [8] Hill, B. 1975). A simple general approach to inference about the tail of a distribution, Annals of Statistics, 3, [9] Hill, T. P. and Spruill, M. C. 1994). On the relationship between convergence in distribution and convergence of expected extremes, Proceedings of the American Mathematical Society, 121,

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