THIRD ORDER EXTENDED REGULAR VARIATION. Isabel Fraga Alves, Laurens de Haan, and Tao Lin

Transcription

1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 8094) 2006), DOI:02298/PIM069409F THIRD ORDER EXTENDED REGULAR VARIATION Isabel Fraga Alves, Laurens de Haan, and Tao Lin Abstract A theory of third order extended regular variation is developed, somewhat similar to the corresponding theories of first and second order Introduction The theory of regularly varying functions initiated by J Karamata 930, 933) has turned out to be very useful and in fact indispensable in the probabilistic and statistical theory of extreme values An extended form of regular variation serves as necessary and sufficient condition for the domain of attraction of extreme value distributions The same conditions are sufficient for the consistency of the estimators of the extreme value index Second order extended regular variation has proven very useful almost indispensable) for establishing asymptotic normality of estimators of, in particular for calculating the asymptotic variance and bias In second order extended regular variation an extra parameter shows up that we call ρ Then asymptotic bias depends on this parameter One way of reducing the asymptotic) bias of an estimator is subtracting the estimated bias from the estimator In order to do so one needs an estimator for the second order parameter ρ Such estimators have been developed They are generally consistent under second order extended regular variation But for proving asymptotic normality of a ρ-estimator third order extended regular variation comes into play It has been proved Fraga Alves, de Haan, and Lin, 2003) that indeed third order extended regular variation is sufficient for the asymptotic normality of ρ- estimators The cited paper also gives a sketch of the third order theory but this sketch had to be so short that it is extremely difficult to understand This paper offers full proofs of the main results for third order extended regular variation 2000 Mathematics Subject Classification: Primary 26A2, 62G32 Key words and phrases: Regular variation; third-order condition Partially supported by ERAS project FCT / POCI / FEDER 09

2 0 FRAGA ALVES, DE HAAN, AND LIN 2 The results A measurable function f is said to satisfy the extended regular variation property if there is a positive function a such that for x>0, 2) ftx) ft) at) = x, where is a real parameter Notation f ERV or f ERV ) The speed of convergence in this it relation can be captured by a relation of second order The measurable function f is said to satisfy the second extended regular variation property if there exist functions a>0anda, positive or negative, with At) = 0, such that for all x>0 ) ftx) ft) x x y = y u ρ du dy At) at) 22) = ρ x +ρ + ρ ) x, where ρ 0 is the second order parameter Notation f 2ERV or f 2ERV,ρ ) Now the third order relation becomes obvious The function f satisfies the third order extended regular variation property if it satisfies 22) and there exists a positive or negative function B, with Bt) = 0, such that for all x>0 [ ) ftx) ft) 23) x x y ] y u ρ du dy Bt) At) at) x y u = y u ρ s η ds du dy, where η 0 is the third order parameter Notation f 3ERV or f 3ERV,ρ,η ) One of the results established in the Theorem is that only in trivial cases the it function in 23) is not of the stated form Theorem 2 Write x D x) := y dy = x and x y H,ρ x) := y u ρ du dy = x +ρ ) x ρ + ρ Suppose f is a measurable function and there exist functions a 0 positive) and a and a 2 positive or negative) with a i t) =0for i =, 2 such that for all x>0 { 24) ftx) ft) a0 t)d x) a t)h,ρ x) } / =:Rx), exists t ), then for a judicious choice of a 0, a and a 2 and given that D, H and R are not linearly dependent, the it function R will be of the form x y u 25) R,ρ,η x) := y u ρ s η ds du dy,

3 THIRD ORDER EXTENDED REGULAR VARIATION with ρ 0, η 0 Moreover, a 2 tx) 26) = x+ρ+η, 27) a tx) a t)x +ρ = x +ρ xη, η a 0 tx) a 0 t)x a t)x x ρ )/ρ 28) = x H ρ,η x) and for each ɛ>0 there exists t 0 such that for t t 0, tx t 0 29) e ɛ log x x ρ η ftx) ft) a 0 t)d x) a t)h,ρ x) R,ρ,η x) ɛ Proof We write for t, x, y > 0 ftxy) ft) =ftxy) ftx)) + ftx) ft)) and we connect the three parts to 24) as follows { ftxy) ft) a0 t)d xy) a t)h,ρ xy) } / = [{ ftxy) ftx) a 0 tx)d y) a tx)h,ρ y) } /a 2 tx) ] a 2 tx)/ 20) + {ftx) ft) a 0 t)d x) a t)h,ρ x)} / + { a 0 t)d xy) D x)) + a 0 tx)d y)} / + { a t)h,ρ xy) H,ρ x)) + a tx)h,ρ y)} / Note that a 0 t)d xy) D x)) + a 0 tx)d y) =a 0 tx) x a 0 t)) D y) and a t)h,ρ xy) H,ρ x)) + a tx)h,ρ y) = a tx) x +ρ a t) ) H,ρ y) a t) x xρ y ρ so that the sum of the last two terms of 20) is 2) a tx) a t)x +ρ H,ρ y)+ a 0tx) a 0 t)x + a t)x x ρ )/ρ D y) Let t in both sides of the equation 20) with the last two terms substituted by 2) We get for x, y > 0 [ Rxy) Rx) = Ry) + o)) a 2tx) + H,ρy) a tx) a t) x +ρ 22) + D y) a 0tx) a 0 t)x a t)x x ρ )/ρ We have required that there do not exist constants c 2 and c 3 such that Ry) c 2 H,ρ y)+c 3 D y) ]

4 2 FRAGA ALVES, DE HAAN, AND LIN Consequently, the set of vectors {Ry),Hy),Dy))} y>0 is not contained in a plane, hence there are y,y 2,y 3 such that the matrix Ry ) Hy ) Dy ) Ry 2 ) Hy 2 ) Dy 2 ) Ry 3 ) Hy 3 ) Dy 3 ) has rank 3 Then also the transposed matrix has rank 3 So there are no z,z 2,z 3, not all of them zero, such that z i Ry i )= z i Hy i )= z i Dy i )=0 i i i Now take z,z 2,z 3 such that 23) z i Hy i )= z i Dy i )=0; i i then we must have z i Ry i ) 0 i Now multiply the two sides of 22) by z i, i =, 2, 3, and add the three equations The resulting equation use 23)) shows that a 2 tx)/ exists for x>0 The it could be zero but that is not possible here) or else a power of x, sayx +ρ+η This is the definition of η Similarly repeating the above reasoning for this case) a tx) a t) x +ρ 24) and 25) a 0 tx) a 0 t) x a t) x x ρ )/ρ must exist The its in 24) and 25) must be [de Haan and Stadtmüller 996, Theorem ] c x +ρ x η )/η and x c H ρ,η x)+c 2 D ρ x)) By replacing a 2 with c a 2 and a with a + c 2 a 2 in 24) the its become x +ρ x η )/η and x H ρ,η x), respectively Now 22) leads to the functional equation x, y > 0) 26) x +ρ+η Ry) =Rxy) Rx) x H ρ,η x) y x +ρ H,ρ y) xη η The reader may want to check that the function R,ρ,η is a solution of 26) Let R be any other solution and define V := R,ρ,η R Then V satisfies the equation hence as in de Haan 970, section 4), x +ρ+η V y) =V xy) V x), V x) =c x+ρ+η + ρ + η

5 THIRD ORDER EXTENDED REGULAR VARIATION 3 Next we get rid of V Suppose the it in 24) is R := R,ρ,η + V Note that by changing the functions a 0, a and a 2 in 24) into ã 2 := c 2 a 2 27) ã := a + c ã 2, ã 0 := a 0 + c 0 ã 2 the it function R changes into c 2 R c 0 D c H,ρ Also note that if both η 0 and η + ρ 0, R,ρ,η x) = ie, x +ρ+η + ρ + η If η = 0, then x +ρ+η ηη + ρ) + ρ + η η H,ρx) ηη + ρ) D x), { = ηη + ρ) R,ρ,η x)+ η H,ρx)+ } ηη + ρ) D x) x +ρ+η + ρ + η = ρh,ρx)+d x) and if η + ρ = 0, then x +ρ+η + ρ + η = D x) Hence by the device 27) we can make sure that the it function in 24) is exactly R,ρ,η In order to prove 29) we distinguish various cases = ρ = η = 0 The proof is similar to Omey and Willekens 988) For x, y > 0 ftxy) fty) ftx) ft) a t)log x)log y) = ftxy) ft) a 0 t) logxy) a t) 2 logxy))2 2 {fty) ft) a 0 t) log y a t) logy))2} { ftx) ft) a 0 t) log x a t) 2 logx))2} Hence by 24) and 25) for the function g x t) :=ftx) ft), we have g x ty) g x t) a t) log x log y = a 0 t) 6 logxy))3 6 log y)3 log x)3 6 for all x, y > 0, ie, = 2 log x)2 log y + 2 log y)2 log x g x ty) g x t) { a t)log x) a 0 t) 2 log x)2} log y = a 0 t) log x 2 log y)2

6 4 FRAGA ALVES, DE HAAN, AND LIN Hence g x is second order Π-varying for each x>0 It follows from Omey and Willekens 988), that the function h x t) :=g x t) t t 0 g xs is in the class Π with auxiliary function a 0 t) log x, ie, for y>0 h x ty) h x t) = log y a 0 t) log x The theory of Π-variation applies, hence by Geluk and de Haan 987) 28) h x t) t ) h x s)ds = a 0 t) log x t 0 Now note that h x t) =h ) tx) h ) t) with h ) t) :=ft) t Consequently, it follows from 28) that h ) tx) h ) t) a 0 t) log x tx ie, tx 0 h ) s + t t t 0 0 fu) du ) h ) s =, h 2) tx) h 2) t) 29) = a 0 t) log x with h 2) t) :=h ) t) t h ) s)ds t 0 Again by the theory of Π-variation, relation 29) implies h 2) tx) h 2) t) = log x h 3) t) with h 3) t) :=h 2) t) t h 2) s)ds t 0 Then by Drees inequalities 998), for any ɛ,ɛ 2 > 0, there exists t 0 = t 0 ɛ,ɛ 2 ) such that for t t 0,tx t 0 220) e ɛ2 log x h 2) tx) h 2) t) log x h 3) t) <ɛ It is easily checked using the definition of h 2) that t h ) t) =h 2) t)+ h 2) s 0 s hence x h ) tx) h ) t) =h 2) tx) h 2) t)+ h 2) ts s,

7 THIRD ORDER EXTENDED REGULAR VARIATION 5 ie, 22) h ) tx) h ) t) h 2) t) log x = h 2) tx) h 2) t)+ and consequently h ) tx) h ) t) h 2) t)+h 3) t) ) log x log x)2 h 3) t) 2 = h2) tx) h 2) t) x h 2) ts) h 2) t) log x + h 3) t) h 3) t) Integrating 22) with respect to x we get x ) ds x 222) h ) ts) h ) t) s h2) t) log s s x ) ds x u = h 2) ts) h 2) t) s + Also we have the following obvious analogue of 22): 223) ) x ftx) ft) h ) t) log x = h ) tx) h ) t) + Combining 222) and 223) we get ftx) ft) =h ) t) log x + h 2) t)log x) 2 /2 x u + h 2) ts) h 2) t) s ) + h ) tx) h ) t), du u + x h 2) ts) h 2) t) s h 2) ts) h 2) t) x log s s s du u h ) ts) h ) t) s h 2) ts) h 2) t) s which by 22) equals ) x u h ) t)+h 2) t) log x + h 2) t)log x) 2 /2+ h 2) ts) h 2) du t) s u x +2 h 2) ts) h 2) t) s + h2) tx) h 2) t) Hence ftx) ft) h ) t)+h 2) t)+h 3) t) ) log x h 2) t)+2h 3) t) ) log x) 2 /2 h 3) t) log x u x)3 h 2) ts) h 2) ) t du = log s 6 h 3) t) s u x h 2) ts) h 2) ) t +2 log s h 3) t) s + h2) tx) h 2) t) log x h 3) t)

8 6 FRAGA ALVES, DE HAAN, AND LIN Hence by 220) for any ɛ,ɛ 2 > 0, t t 0 ɛ,ɛ 2 ), tx t 0 ɛ,ɛ 2 ) ftx) ft) h ) t)+h 2) t)+h 3) t) ) log x h 2) t)+2h 3) t) ) log x) 2 /2 h 3) t) log x)3 6 x u ɛ2 log s ds du x <ɛ e s u +2ɛ ɛ2 log s ds e s + ɛ ɛ2 log x e <ɛ + + ) ɛ 2 ɛ 2 e ɛ2 log x 2 2 η<0: By Theorem 0 of Geluk and de Haan 987) relation 27), that is, tx) +ρ) a tx) t +ρ) a t) = xη t +ρ) η with η<0, implies that for some constant c>0 c t +ρ) a t) η t +ρ), that is, a t) =ct +ρ + η + o)) as t Hence we can replace a in 28) with a defined by a t) :=ct +ρ + η By inserting this choice in 28) and rearranging we get for x>0 a 0 tx) a 0 t)x ct +ρ x x ρ )/ρ = x H ρ,η x) η x xρ ρ which is the same as tx) a 0 tx) t a 0 t) ct ρ x ρ )/ρ t = η xρ+η ρ + η We can rewrite this as tx) a 0 tx) ctx) ρ )/ρ ) t a 0 t) ct ρ )/ρ ) t = η xρ+η ρ + η, where ρ + η<0 Again by Theorem 0 of Geluk and de Haan 987) this implies that for some constant c, c t a 0 t) c tρ ρ ηρ + η) t, t, hence in 24) we can replace a 0 with a 0 defined by a 0t) :=c t + c ρ t+ρ ηρ + η) a 2t)

9 THIRD ORDER EXTENDED REGULAR VARIATION 7 Inserting the special choices a and a 0 in 24) we get [ tx) ftx) c c tx) +ρ 224) η + ρ t ft) c c t +ρ ) c t +ρ x ] η + ρ η η = R,ρ,η x)+ η H x,ρx) ηρ + η) = H,ρ+η x) x ) = x +ρ+η η ρ + η η + ρ + η Relation 224) is a second order relation for which uniform bounds are well known Drees, 998) Hence for some functions â and â 2 [ e ɛ log x x ρ η tx) ftx) c c tx) +ρ ) η + ρ t ft) c c t +ρ ) ] â t) x η + ρ â 2 t) ɛ provided t t 0 ɛ) andtx t 0 ɛ) This gives 29) for η<0 3 ρ<0, η = 0 We start from 28), ie, tx) a 0 tx) t a 0 t) t a t) x ρ )/ρ t = H ρ,0 x) This relation is discussed in de Haan and Stadtmüller 996), Theorem 2,ii) Relation 2) of that paper gives { tx) a 0 tx) ρ tx) a tx) } { t a 0 t) ρ t a t) } t = ρ xρ ρ The latter relation is discussed in Proposition 2ii) of the same paper It follows that for some c>0 t a 0 t) ρ t a t) c t = ρ 2, ie, relation 24) holds with a replaced by a defined by a t) := cρt + ρa 0 t)+ ρ a 2t) After some rearranging we get [ ) ) ] ftx) c tx) ft) c t ct a 0 t)) x+ρ + ρ = R,ρ,0 x)+ ρ H,ρx) = ρ H +ρ,0x) This is a second order relation for which uniform bounds are well known Drees, 998) This gives 29) for η =0,ρ<0

10 8 FRAGA ALVES, DE HAAN, AND LIN 4 In the case 0,ρ = η = 0 we have, as t : ftx) ft) a 0 t)x )/ a t)h,0 x) Hence Note that R,0,0 x) = x log 2 x 2 ftx) ft) a 0 t)x )/ a t) / ) H,0 x) We have [ ftx) ft) Define Then H,0 x) = x log x a 0 t) a t)+ 2 a 2t) â 0 t) :=a 0 t) a t)+ 2 a 2t), [ ftx) ft) â 0 t) x which can be rewritten as ftx) â 0 tx) ) ft) â 0 t) ) 225) Relations 26), 27) and 28) imply â 0 tx) â 0 t)x â t)x log x 226) From 225) and 226) we have 227) x 2 H,0x) x log 2 x 2 ) x a t) ) x ] a log x 2t) â t) :=a t) a 2t) â t) x log x ] x log 2 x, 2 x log 2 x 2 + â0tx) â 0 t)x â t)x log x x log 2 x 2 x log 2 x 2 ftx) â 0 tx)) ft) â 0 t)) 0 If <0, by Theorem 30 of Bingham, Goldie and Teugels 987, due to Ash, Erdös and Rubel, 974), the it c := ft) â 0 t) exists, finite, and c ft)+ â 0 t) =0,

11 THIRD ORDER EXTENDED REGULAR VARIATION 9 ie, â 0 t) = ft) c)+o), as t Combining this with 226) gives 228) ftx) ft)x â t)x log x x log 2 x, 2 ie, tx) ftx) t ft) â t)x log x)/ log2 x / 2 Note that this is just a second order condition for which uniform inequalities have been derived in Drees 998) They can be rewritten as follows: one can find functions â and â 2 such that for any ɛ>0, t, tx t 0 ɛ), 229) x exp{ log x ɛ} [ â 2 t) ftx) ft) 2 ft) x ] â t)x log x x log 2 x 2 <ɛ For >0, Theorem 32 of of Bingham, Goldie and Teugels 987, due to Bojanic and Karamata, 963), applied to 227) implies ft) â 0 t) 0, ie, â 0 t) =ft)+o), as t Combining this with 226) gives 228) once again and hence 229) is obtained References [] J M Ash, P Erdös, P and L A Rubel, Very slowly varying functions, Aeq Math 0 974), 9 [2] N Bingham, C Goldie, and J Teugels, Regular Variation, Cambridge University Press, New York, 987 [3] R Bojanić and J Karamata, On a Class of Functions of Regular Asymptotic Behaviour, Math Research Center Tech Report ), Madison, Wis [4] ALM Dekkers, JHJ Einmahl and L de Haan, A moment estimator for the index of an extreme-value distribution, Ann Statist 7 989), [5] H Drees, On smooth statistical tail functionals, Scandinavian J Stat ), [6] M I Fraga Alves, L de Haan and T Lin, Estimation of the parameter controlling the speed of convergence in extreme value theory, Math Methods of Statist ), [7] J Geluk and L de Haan, Regular Variation, Extensions and Tauberian Theorems, CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, Netherlands, 987 [8] L de Haan, On regular variation and its application to the weak convergence of sample extremes, Thesis, University of Amsterdam / Mathematical Centre tract 32, 970 [9] L de Haan and U Stadtmüller, Generalized regular variation of second order, J Austral Math Soc A) 6 996), [0] J Karamata, Sur un mode de croissance régulière des fonctions, Mathematica Cluj) 4 930), 38 53

12 20 FRAGA ALVES, DE HAAN, AND LIN [] J Karamata, Sur un mode de croissance régulière Théorèmes fondamentaux, Bull Soc Math France 6 933), [2] E Omey and E Willekens, π-variation with remainder, J London Math Soc ), 05 8 CEAUL and DEIO Faculty of Science Received ) University of Lisbon Portugal isabelalves@fculpt Department of Economics Faculty of Economics Erasmus University Rotterdam The Netherlands ldhaan@feweurnl Xiamen University Department of Mathematics Xiamen University China weiminghult@tomcom