The Convergence Rate for the Normal Approximation of Extreme Sums

Transcription

1 The Convergence Rate for the Normal Approximation of Extreme Sums Yongcheng Qi University of Minnesota Duluth WCNA 2008, Orlando, July 2-9, 2008

2 This talk is based on a joint work with Professor Shihong Cheng

3 Outline Outline Introduction Main Results Sketch of Proofs

4 Outline Outline Introduction Main Results Sketch of Proofs

5 Outline Outline Introduction Main Results Sketch of Proofs

6 Introduction Extreme Sum Let {X, X 1, X 2, } be a sequence of independent and identically distributed random variables with distribution F For each n 1 let X n,1 X n,n denote the order statistics of X 1,, X n Define S n,k = k j=1 X n,n j 1 as an extreme sum, where k = k n satisfies lim k n = and n lim n k n n = 0 (1)

7 Introduction Extreme Sum Let {X, X 1, X 2, } be a sequence of independent and identically distributed random variables with distribution F For each n 1 let X n,1 X n,n denote the order statistics of X 1,, X n Define S n,k = k j=1 X n,n j 1 as an extreme sum, where k = k n satisfies lim k n = and n lim n k n n = 0 (1)

8 Introduction Extreme Sum Let {X, X 1, X 2, } be a sequence of independent and identically distributed random variables with distribution F For each n 1 let X n,1 X n,n denote the order statistics of X 1,, X n Define S n,k = k j=1 X n,n j 1 as an extreme sum, where k = k n satisfies lim k n = and n lim n k n n = 0 (1)

9 Introduction Motivation Why extreme sums? Influence of extremes sums in the partial sums n j=1 X j Trimmed sums or extreme sums Estimation of the tail of a distribution in extreme-value statistics Hill estimator and moment estimator for tail-index

10 Introduction Motivation Why extreme sums? Influence of extremes sums in the partial sums n j=1 X j Trimmed sums or extreme sums Estimation of the tail of a distribution in extreme-value statistics Hill estimator and moment estimator for tail-index

11 Introduction Motivation Why extreme sums? Influence of extremes sums in the partial sums n j=1 X j Trimmed sums or extreme sums Estimation of the tail of a distribution in extreme-value statistics Hill estimator and moment estimator for tail-index

12 Introduction Domain of Attraction of An EV Distribution Assume that F belongs to the domain of attraction of an extreme value distribution, i.e. there exist constants c n > 0 and d n R such that for some γ R. X n,n d n c n d exp{ (1 + γx) 1/γ } =: G γ (x) (2) G γ : extreme-value distribution γ > 0: Fréchet γ = 0: Gumbel γ < 0: Weibull

13 Introduction Domain of Attraction of An EV Distribution Assume that F belongs to the domain of attraction of an extreme value distribution, i.e. there exist constants c n > 0 and d n R such that for some γ R. X n,n d n c n d exp{ (1 + γx) 1/γ } =: G γ (x) (2) G γ : extreme-value distribution γ > 0: Fréchet γ = 0: Gumbel γ < 0: Weibull

14 Introduction Some Distributions Heavy-tailed distributions (γ > 0) 1 F (x) = x 1/γ L(x), x > 0 where L(x) is a slowly varying function at infinity; Normal and exponential (γ = 0) Distributions with a finite right-endpoint ω F (γ < 0) 1 F(x) c(ω F x) 1/γ as x ω F.

15 Introduction Some Distributions Heavy-tailed distributions (γ > 0) 1 F (x) = x 1/γ L(x), x > 0 where L(x) is a slowly varying function at infinity; Normal and exponential (γ = 0) Distributions with a finite right-endpoint ω F (γ < 0) 1 F(x) c(ω F x) 1/γ as x ω F.

16 Introduction Some Distributions Heavy-tailed distributions (γ > 0) 1 F (x) = x 1/γ L(x), x > 0 where L(x) is a slowly varying function at infinity; Normal and exponential (γ = 0) Distributions with a finite right-endpoint ω F (γ < 0) 1 F(x) c(ω F x) 1/γ as x ω F.

17 Introduction Limiting Distribution of S n,kn under (2) Csörgö, Horváth and Mason (1986), Csörgö and Mason (1986), Lo (1989), Haeusler and Mason (1987), and Csörgö, Haeusler and Mason (1991). In particular, Csörgö, Haeusler and Mason (1991) proved that the limiting distribution of S n,kn is normal if and only if γ 1/2.

18 Introduction Limiting Distribution of S n,kn under (2) Csörgö, Horváth and Mason (1986), Csörgö and Mason (1986), Lo (1989), Haeusler and Mason (1987), and Csörgö, Haeusler and Mason (1991). In particular, Csörgö, Haeusler and Mason (1991) proved that the limiting distribution of S n,kn is normal if and only if γ 1/2.

19 Introduction Distributional Convergence Rate Mijnheer (1988) showed that if sup P(n 1/α S n,kn x) G(x; α) = O(max(k 1 1/α n, n 1 )) x 1 F(x) = x α + O(x r ) x for some r > α, 0 < α < 1, where G(x; α) is an asymmetric stable law. The above condition implies (2) with γ = 1/α > 1.

20 Introduction Distributional Convergence Rate Mijnheer (1988) showed that if sup P(n 1/α S n,kn x) G(x; α) = O(max(k 1 1/α n, n 1 )) x 1 F(x) = x α + O(x r ) x for some r > α, 0 < α < 1, where G(x; α) is an asymmetric stable law. The above condition implies (2) with γ = 1/α > 1.

21 Main Results Our interest is To estimate rates of convergence to normality for the extreme sums under conditions: (2) with γ < 1/2, and Von-Mises condition

22 Main Results Our interest is To estimate rates of convergence to normality for the extreme sums under conditions: (2) with γ < 1/2, and Von-Mises condition

23 Main Results Our interest is To estimate rates of convergence to normality for the extreme sums under conditions: (2) with γ < 1/2, and Von-Mises condition

24 Main Results Von-Mises Conditions Denote ω F = sup{x : F(x) < 1}. The following von-mises conditions are sufficient for (2): F has the derivative function F satisfying γ > 0 : γ < 0 : γ = 0 : xf (x) ω F =, lim x 1 F(x) = 1 γ ; (ω ω F <, lim F x)f (x) = 1 x 1 F(x) γ ; F (x) ωf lim x ω F [1 F(x)] 2 [1 F(u)]du = 1. x

25 Main Results Von-Mises Conditions Denote ω F = sup{x : F(x) < 1}. The following von-mises conditions are sufficient for (2): F has the derivative function F satisfying γ > 0 : γ < 0 : γ = 0 : xf (x) ω F =, lim x 1 F(x) = 1 γ ; (ω ω F <, lim F x)f (x) = 1 x 1 F(x) γ ; F (x) ωf lim x ω F [1 F(x)] 2 [1 F(u)]du = 1. x

26 Main Results Von-Mises Conditions Let V (t) = inf{x : [1 F(x)] 1 t} for t 1 These von-mises conditions are equivalent to the assumption that V has a derivative function V Rv(γ 1), where Rv(γ 1) denotes the class of all regularly varying functions with index γ 1, i.e. V (tx) lim t V (t) = x γ 1, x > 0. (3) See Cheng, de Haan and Huang (1997).

27 Main Results Von-Mises Conditions Let V (t) = inf{x : [1 F(x)] 1 t} for t 1 These von-mises conditions are equivalent to the assumption that V has a derivative function V Rv(γ 1), where Rv(γ 1) denotes the class of all regularly varying functions with index γ 1, i.e. V (tx) lim t V (t) = x γ 1, x > 0. (3) See Cheng, de Haan and Huang (1997).

28 Main Results Notation Φ, φ the standard normal distribution function and its density D 2 X the variance of a random variable X; Z a random variable with distribution function F Z (x) = 1 x 1, x 1. Set a n = n k n + 1 n + 1 l n = 1 1 a n ; τ n = an (1 a n ) ; b n = ; n + 1 ( an EV (ln Z ) V (l n ) ) DV (l n Z )

29 Main Results Notation Φ, φ the standard normal distribution function and its density D 2 X the variance of a random variable X; Z a random variable with distribution function F Z (x) = 1 x 1, x 1. Set a n = n k n + 1 n + 1 l n = 1 1 a n ; τ n = an (1 a n ) ; b n = ; n + 1 ( an EV (ln Z ) V (l n ) ) DV (l n Z )

30 Main Results Notation Φ, φ the standard normal distribution function and its density D 2 X the variance of a random variable X; Z a random variable with distribution function F Z (x) = 1 x 1, x 1. Set a n = n k n + 1 n + 1 l n = 1 1 a n ; τ n = an (1 a n ) ; b n = ; n + 1 ( an EV (ln Z ) V (l n ) ) DV (l n Z )

31 Main Results We are going to estimate the convergence rate ( Sn,kn k n EV (l n Z ) ) n (x) = P kn (1 + τn 2 )D 2 V (l n Z ) x Φ(x)

32 Main Results Theorem 1 Theorem Under (1), if (3) holds for some γ [1/3, 1/2), then sup n (x) = o(kn ɛ ) (4) x R holds for any ɛ (0, (2γ) 1 1). (5)

33 Main Results Theorem 2 Theorem Under (1), if (3) holds for γ < 1/3, then n (x) = 1 2(1 2γ) 1/2 kn 6 ( 2(1 γ) ) 3/2 (1 3γ) ) ( 1 ) +Φ γ (x) + o kn holds uniformly on x R, where ( ( ξ 1 (γ) + ξ 2 (γ)x 2) φ(x) ξ 1 (γ) = 7 8γ + 9γ 2, ξ 2 (γ) = 1 16γ 15γ γ 3 Φ γ (x) = 2(1 2γ)(1 3γ) x (3u u 3 )φ(u)du. (6)

34 Sketch of the proofs Sketch of the proofs Express the extreme sum as a sum of k n conditionally iid RVs Estimate the moments of these RVs by using the regularity of V and V Find normal approximate error for the distribution of the sum of these conditionally iid RVs

35 Sketch of the proofs Sketch of the proofs Express the extreme sum as a sum of k n conditionally iid RVs Estimate the moments of these RVs by using the regularity of V and V Find normal approximate error for the distribution of the sum of these conditionally iid RVs

36 Sketch of the proofs Sketch of the proofs Express the extreme sum as a sum of k n conditionally iid RVs Estimate the moments of these RVs by using the regularity of V and V Find normal approximate error for the distribution of the sum of these conditionally iid RVs

37 Sketch of the proofs Sketch of the proofs Express the extreme sum as a sum of k n conditionally iid RVs Estimate the moments of these RVs by using the regularity of V and V Find normal approximate error for the distribution of the sum of these conditionally iid RVs

38 Sketch of the proofs Some Notation Let Z 1, Z 2, be independent random variables having the same distribution function as Z and let Z n,1 Z n,n be the order statistics of Z 1,, Z n. Then (X n,1,, X n,n ) d = (V (Z n,1 ),, V (Z n,n )). For every u R set a n (u) = {(a n + b n u) 1} 0; l n (u) = ( 1 a n (u) ) 1 ; k n S n,kn (u) = V (l n (u)z j ); φ n (u) = φ(u) ( 1 u3 3u 3 ). k n j=1

39 Sketch of the proofs Some Notation Let Z 1, Z 2, be independent random variables having the same distribution function as Z and let Z n,1 Z n,n be the order statistics of Z 1,, Z n. Then (X n,1,, X n,n ) d = (V (Z n,1 ),, V (Z n,n )). For every u R set a n (u) = {(a n + b n u) 1} 0; l n (u) = ( 1 a n (u) ) 1 ; k n S n,kn (u) = V (l n (u)z j ); φ n (u) = φ(u) ( 1 u3 3u 3 ). k n j=1

40 Sketch of the proofs Fact 1: Under (1), we have sup P(S n,kn x) x R P(S n,kn (u) x)φ n (u)du ( 1 = o ). kn (7)

41 Sketch of the proofs Fact 2: If (3) holds for γ R, then V (tx) V (t) lim t tv (t) = x γ 1, x > 0 (8) γ with convention x γ 1 γ = log x for x > 0. Furthermore, for γ=0 any δ > 0, there exists a t δ > 0 such that (1 δ) x γ δ 1 γ δ V (tx) V (t) tv (t) (1 + δ) x γ+δ 1 γ + δ (9) holds for all x 1 and t t δ.

42 Sketch of the proofs Fact 3: Under (1) and (3), we have lim t 1 [ V (tv) V (t) for all j = 1, 2,. tv (t) ] j dv v 2 = (v γ 1) j 1 γ j v 2 dv, γ < j 1 (10)

43 Sketch of the proofs Some estimates We conclude EV (l n Z ) V (l n ) lim n l n V = (l n ) 1 v γ 1 dv γ v 2 = 1 1 γ if γ < 1. (11) lim n DV (l n Z ) l n V (l n ) = 1 (1 γ)(1 2γ) 1/2 =: σ(γ) if γ < 1/2. (12)

44 Sketch of the proofs Some estimates We conclude EV (l n Z ) V (l n ) lim n l n V = (l n ) 1 v γ 1 dv γ v 2 = 1 1 γ if γ < 1. (11) lim n DV (l n Z ) l n V (l n ) = 1 (1 γ)(1 2γ) 1/2 =: σ(γ) if γ < 1/2. (12)

45 Sketch of the proofs Fact 4: If γ < 1/2, then under (1) and (3), the following equations hold uniformly on u [ k 1/6 n, k 1/6 n ]: V (l n (u)) V (l n ) = l nv (l n )u[1 + o(1)] kn ; (13) EV (l n (u)z ) EV (l n Z ) = uτ ndv (l n Z ) + (1 + γ)l nv (l n )u 2 [1 + o(1)] ; kn 2k n (1 γ) (14) DV (l n Z ) + o(1)] 1 = u[γ. (15) DV (l n (u)z ) kn

46 Sketch of the proofs Fact 5: Suppose that (1) and (3) hold. If γ < 1/2, then there exist two functions α n,1 (u) and α n,2 (u) such that η n (x, u) := DV (l nz )x k n [EV (l n (u)z ) EV (l n Z )] (16) DV (l n (u)z ) = x τ n u α n,1 (u)xu α n,2 (u)u 2 holds for all x R and u [ k 1/6 n, k 1/6 n ], where lim kn α n,1 (u) = γ and lim kn α n,2 (u) = [2σ(γ)] 1 n n uniformly on [ k 1/6 n, k 1/6 n ].

47 Sketch of the proofs For n = 1, 2, and x, u R, denote Ẑ n,j (u) = V (l n(u)z j ) EV (l n (u)z ), j = 1,, k n ; D 2 V (l n (u)z ) Ŝ n,kn (u) = 1 kn k n j=1 Ẑ n,j (u).

48 Sketch of the proofs Proof of Theorem 1. Denote n,1 = sup x R n,2 = sup x R n,3 = sup x R u >k 1/6 n k 1/6 n k 1/6 n k 1/6 n k 1/6 n [P(Ŝn,kn (u) η n (x, u)) Φ(x τ n u)]φ(u)du ; [P(Ŝn,kn (u) η n (x, u)) Φ(η n (x, u))]φ(u)du ; [Φ(η n (x, u)) Φ(x τ n u)]φ(u)du.

49 Sketch of the proofs Since ( x ) Φ = 1 + τ 2 n we have from (7) and (16) that ( x ) sup n (x) = sup n 1 + τ 2 n x R sup x R + C kn x R Φ(x ± τ n u)φ(u)du, (17) [P(Ŝn,k n (u) η n (x, u)) Φ(x τ n u)]φ(u)du n,1 + n,2 + n,3 + C kn.

50 Sketch of the proofs n,1 P( N > k 1/6 n ) = O(k 1/2 n ); From the Berry-Esseen theorem we get for any ɛ (0, 1/2] n,2 sup sup P(Ŝn,kn (u) η n (x, u)) Φ(η n (x, u)) u k 1/6 x R n Ck ɛ n n,3 = O(k 1/2 n ).

51 Sketch of the proofs n,1 P( N > k 1/6 n ) = O(k 1/2 n ); From the Berry-Esseen theorem we get for any ɛ (0, 1/2] n,2 sup sup P(Ŝn,kn (u) η n (x, u)) Φ(η n (x, u)) u k 1/6 x R n Ck ɛ n n,3 = O(k 1/2 n ).

52 Sketch of the proofs n,1 P( N > k 1/6 n ) = O(k 1/2 n ); From the Berry-Esseen theorem we get for any ɛ (0, 1/2] n,2 sup sup P(Ŝn,kn (u) η n (x, u)) Φ(η n (x, u)) u k 1/6 x R n Ck ɛ n n,3 = O(k 1/2 n ).

53 Sketch of the proofs Proof of Theorem 2 For n = 1, 2, and t, x, u R, let ρ(γ) = 2(1 + γ) 1 2γ ; 1 3γ G(x) = Φ(x) ρ(γ)(x 2 1)φ(x) 3 k n ; F n (x, u) = P(Ŝn,k n (u) x); f n (t, u) = E exp{itẑn,1(u)} (the characteristic function of Ẑn,1(u)); ( t ) ψ n (t, u) = log f n, u + t2. kn 2k n

54 Sketch of the proofs Fact 6: Suppose (1) and (3) hold. Then lim n sup u k 1/6 n and for any positive p with γp < 1 3γ lim sup n sup u k 1/6 n EẐ n,1 3 (u) ρ(γ) = 0, (18) E Ẑn,1(u) 3+p <. (19)

55 Sketch of the proofs Fact 7: Under (1) and (3), we have sup u k 1/6 n ( 1 sup F n (x, u) G(x) = o ). (20) x R kn

56 Sketch of the proofs Proof of Theorem 2. We get ( x ) n 1 + τ 2 n = k 1/6 n k 1/6 n ρ(γ) 3 k n + k 1/6 n k 1/6 n [Φ(η n (x, u)) Φ(x τ n u)])φ n (u)du k 1/6 n k 1/6 n [η 2 n(x, u)) 1]φ(η n (x, u))φ n (u)du ( 1 ) Φ(x τ n u))[φ n (u) φ(u)]du + o kn ( 1 ) =: K n,1 (x) + K n,2 (x) + K n,3 (x) + o kn holds uniformly on x R.

57 Sketch of the proofs Thank you very much!

58 References References: Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987). Regular Variation. Cambridge University, New York. Cheng, S. (1997). Approximation to the expectation of a function of order statistics and its applications. Acta Math. Appl. Sinica (English Ser.) 13, Cheng, S., de Haan, L. and Huang, X. (1997). Rate of convergence of intermediate order statistics. J. Theoret. Probab. 10, Csörgö, S., Horváth, L. and Mason, D. M (1986). What portion of the sample makes a partial sum asymptotically stable or normal? Probab. Theory Rel. Fields Csörgö, S. and Mason, D. M (1986). The asymptotic distribution of sums of extreme values from a regularly varying distribution. Ann. Probab. 14, Csörgö, S., Haeusler, E. and Mason, D. M (1991). The asymptotic distribution of extreme sums. Ann. Probab. 19,

59 References Geluk, J. L. and de Haan, L. (1970). Regular Variation, Extension and Tauberian Theotems. CWI Tracts 40, Centre for Mathematics and Computer Science, Amsterdam. Haeusler, E. and Mason, D. M (1987). A law of the iterated logaritheorem for sums of extreme values from a distribution with a regularly varying upper tail. Ann. Probab Lo, G. S. (1989). A note on the asymptotic normality of sums of extreme values. J. Statist. Plann. Inference 22, Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Clarendon Press, Oxford. Reiss, R.-D. (1989). Approximate Distribution of Order Statistics with Applications to Nonparametric Statistics. Springer, New York.