of (X n ) are available at certain points. Under assumption of weak dependency we proved the consistency of Hill s estimator of the tail

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1 Novi Sad J. Math. Vol. 38, No. 3, 2008, INCOMPLETE SAMPLES AND TAIL ESTIMATION FOR STATIONARY SEQUENCES 1 Ivaa Ilić 2, Pavle Mladeović 3 Abstract. Let (X ) be a strictly statioary sequece with a margial distributio fuctio F such that 1 F (x) = x α L(x), x > 0, where α > 0 ad L(x) is a slowly varyig fuctio. We assume that oly observatios of (X ) are available at certai poits. Uder assumptio of weak depedecy we proved the cosistecy of Hill s estimator of the tail idex α based o a icomplete sample from {X 1, X 2..., X }. This is a extesio of the results of Hsig [15] ad Mladeović ad Piterbarg [19]. Primary 60G70; Sec- AMS Mathematics Subject Classificatio (2000): odary 60G10 Key words ad phrases: Hill s estimator, idex of regular variatio, statioary sequeces, weak depedece, missig observatios 1. Itroductio Let F be a distributio fuctio with regularly varyig uper tail, that is (1.1) 1 F (x) = x α L(x), x > 0, where α > 0 ad L is slowly varyig at ifiity. Without loss of geerality we may assume that F (0) = 0. The problem of estimatig the tail idex α has attracted a great attetio amog statisticias. There is a huge umber of papers cocerig this problem i i.i.d. settigs, i.e. whe the estimator is defied usig a sample of idepedet radom variables X 1,..., X distributed accordig to F. Probably, the most popular is the Hill estimator defied as follows [see Hill (1975)]: Let X (1) X (2)... X () be a sequece of order statistics. Based o k + 1 largest of them, Hill s estimator is (1.2) H k, = 1 k k l X (i) l X (k+1). i=1 Asymptotic behavior of Hill s estimator was studied by may authors uder differet coditios. Here the umber k = k() should also icrease together 1 This work was supported by the Joit Research Project fiaced by The Miistry of Educatio ad Sciece of the Republic of Macedoia (MESRM) (Project No /1) ad The Scietific ad Techical Research Coucil of Turkey (TUBITAK) (Project No. TBGA- U-105T056). 2 Uiversity of Niš, Medical Faculty, Dept. of mathematics ad iformatics, Bul. dr Zoraa Djidjića 82, Nish, Serbia, ivaa@medfak.i.ac.yu 3 Uiversity of Belgrade, Faculty of Mathematics, Studetski trg 16, Belgrade, Serbia, paja@matf.bg.ac.yu

2 98 I. Ilić, P. Mladeović with. Maso [18] (1982) proved weak cosistecy uder coditios k ad k/ 0 as. Deheuvels, Haeusler ad Maso [7] (1988) proved strog cosistecy for ay sequece k = k() such that k/ l l ad k/ 0 as. For results cocerig asymptotic ormality of Hill s estimator see, for example, Davis ad Resick [4] (1984), Csörgó ad Maso [3] (1985), Haeusler ad Teugels [12] (1985), Goldie ad Smith [9] (1987), Hall [13] (1982), Hill [14](1975) ad Beirlat ad Teugels [1] (1989). Dekkers, Eimahl ad de Haa [5] (1989) exteded Hill s estimator for the idex of regular variatio to a estimate for the idex of a extreme-value distributio. Also see Dekkers ad de Haa [6] (1989), Gedeko [10](1943) ad de Haa [11] (1970). Some other estimators for extreme value idex i i.i.d. settigs were also proposed ad studied: see Pickads [20] (1975), Chörgó, Deheuvels ad Maso [3] (1985), Dekkers ad de Haa [5] (1989) ad Drees [8](1998). There are also relatively small umber of papers devoted to estimatio of the tail idex usig depedet data, see Hsig (1991) [15], Resick ad Starica [22, 23, 24] (1995, 1997, 1998), where asymptotic behavior of Hill s estimator was cosidered. Also see Seeta [25](1976) ad Smith [26](1987). 2. Some prelimiaries ad otatio Let (X ) 1 be a strictly statioary sequece of radom variables with short rage depedece, that is to say that the fiite dimesioal distributios of (X ) are ivariat uder shifts ad the depedece betwee observatios from (X ) becomes weaker as time separatio becomes larger. Moreover, we assume that oly observatios at certai poits are available. Deote observed radom variables amog {X 1,..., X } by X 1,..., XS. Here the radom variable S represets the umber of observed rv s amog the first terms of the sequece (S ). Icomplete sample ca be obtaied, for example, if every term of (X ) is observed with probability p, idepedetly of other terms, ad i this case S is biomial radom variable. But we shall assume that observed radom variables are determied by a geeral poit process, ad oly coditios o S will be imposed. See Leadbetter, Lidgre ad Rootzé [17] (1983) ad Resick [21] (1987). Assumptio. X 1, X 2,... does ot deped o S ad there exists a sequece of real umbers (γ ) such that: ad S γ p c 0 > 0 as + lim γ =. + Suppose β is a sequece of real umbers such that lim β =

3 Icomplete samples ad tail estimatio for statioary sequeces 99 ad Let = [ S β ] β lim = 0 γ { 0, S = 0 ad B = S, S 1 We are iterested i estimatio of α, usig some portio of a sample. Let X 1,S X 2,S... X S,S be the order statistics defied by S observed variables. Defie F 1 (y) = if{x : F (x) y}, 0 < y < 1. Let us also deote, for every x R, x + is max(x, 0). Hill s estimator is give by: M 1 H S = l X j,s l X, S M+1,S β j=1 0, S < β or: Let us also defie: M 1 H S = l X j,s l F 1 (1 B ), S β, j=1 0, S < β, H + S = 1 S (l X j l F 1 (1 B )) +, S β, j=1 0, S < β. Suppose Ỹi (Y i ) is a fuctioal of X i (X i ), for example, Ỹi may be: (l X i l F 1 (1 B )) +, I{l X i > lf 1 (1 B ) + ɛ}. Let F b a{y i } be the σ-field; σ{y i : a i b} ad for 1 l 1 let: β(l, {Y i }) = sup{ P (A B) P (A)P (B) : 3. Results A F k 1 {Y i }, B F k+l {Y i}, 1 k l}. Theorem 3.1. Suppose (r ) is a sequece of positive itegers ad r γ 0, whe. Let Sk be a radom variable measurable with respect to F kr (k 1)r +1 {Ỹi}, where Ỹi is a fuctioal of Xi ad 1 k K, where K = [ S r ]. Assume that:

4 100 I. Ilić, P. Mladeović (a) r β(r, {Y i }) 0, (b) I{S β } 1 K E S k I{ S k > } p 0, k=1 (c) I{S β } 1 M 2 The: K E( S k ) 2 I{ S k } p 0. k=1 Proof. Write: I{S β } 1 K ( S k E S k ) p 0. k=1 I{S β } 1 ( S k E S k ) = I{S β } 1 k=1 +I{S β } 1 k=2, k eve k=1, k odd ( S k E S k ). ( S k E S k ) Let us deote the set of all odd umbers i {1, 2,..., K } by O S. It was prove i Hsig [15], usig results of Ibragimov ad Liik [16], ad coditio (a) that, i the case whe all the data are preset, the variables S k for the set of all odd k {1, 2...} were treated as idepedet. From above, we ca proceed by assumig that S k are idepedet for every k O S. Defie: S k = S k I( S k ), 1 k K. For every ɛ > 0,

5 Icomplete samples ad tail estimatio for statioary sequeces 101 { 1 I{S β }P } ( S k ESk) > ɛ k O s I{S β }P S k Sk, for some k O S } { 1 } +I{S β }P (Sk ESk) > ɛ k O S I{S β } P { S k > } + I{S β } 1 M k O 2 S I{S β } 1 1 ɛ 2 I{S β } 1 I{S β } 1 k O S k O S E S k + I{S β } 1 M 2 E S k + I{S β } 1 1 M 2 ɛ 2 k O S E S k +I{S β } 1 1 M 2 ɛ 2 k O S 1 ɛ 2 V ar( k O S k O S E( S k ) 2 I( S k ) p 0. We used the fact that I 2 { S k } = I{ S k }. Sice the followig equality holds we have that Fially, we have that S k = S k I{ S k > } + S k I{ S k } = S k I{ S k > } + S k k O S S k)) V ar(s k) E(S k) 2 I{S β } 1 (E S k ESk) k O S = I{S β } 1 E S k I{ S k > } k O S I{S β } 1 E M S k I{ S k > } p 0. I{S β } 1 k O S k O S ( S k ES k (E S k ES k)) = I{S β } 1 ( M S k E S k ) p 0. k O S

6 102 I. Ilić, P. Mladeović We have similar deductio for eve umbers k from the set {1, 2,..., K }. Theorem 3.2. All three quatities H S, H + S, ad H S coverge to α 1 i probability uder the followig coditios. (i) There exists a sequece (r ) of positive itegers such that r γ 0, ad that r β(r, {Ỹi}) 0. Let us deote Ỹi = (l X i l F 1 (1 B )) + ad suppose that (b) ad (c) from Theorem 3.1 hold for S k = kr i=(k 1)r +1 Ỹi. (ii) For each ε R ad ρ i some iterval cotaiig 1 there exists a sequece (r ) of positive costats for which r γ 0, such that r β(r, {Ĩi}) 0, where Ĩi = I{l X i > l F 1 (1 ρb ) + ε}. Suppose that (b) ad (c) from Theorem 3.1 hold for S k = [ kr i=(k 1)r Ĩi, S +1 where K = r ]. ( ) (iii) r E I{S β} 1 0. Proof. It follows from Theorem 3.1 ad the coditio (i) that the followig relatios hold: I{S β } 1 I{S β } 1 kr k=1 i=(k 1)r +1 r i=(k 1)r +1 It follows from (iii) that the positive quatity I{S β } 1 (Ỹi EỸi) p 0, (Ỹi EỸi) p 0. S i=k r +1 has the expectatio tedig to 0. Cosequetly we coclude that: I{S β } 1 Coditio (ii) implies that: I{S β } 1 S i=1 S i=1 Ỹ i (Ỹi EỸi) p 0. (I i EI i ) p 0. Fially, the coclusio of the theorem is a cosequece of Theorem 1 from Mladeović ad Piterbarg [19].

7 Icomplete samples ad tail estimatio for statioary sequeces 103 Ackowledgemet The work is supported by the Miistry of Sciece ad Evirometal Protectio of the Republic of Serbia, Grat No ad Grat No Refereces [1] Beirlat, J., Teugels, J.L., Asymptotic ormality of Hill s estimator. I: Hüsler, J. ad Reiss, R.D. (Eds.), Lecture Notes i Statistics, vol. 51., Berli: Spriger, pp , (1989). [2] Bigham, N., Goldie, C., Teugels, J., Regular Variatio. Ecyclopedia of Mathematics ad its Applicatios, vol. 27., Cambridge, UK: Cambridge Uiversity Press, (1987). [3] Chörgó, S., Deheuvels, P., Maso, D.M.,: Kerel estimates of the tail idex of a distributio. A. Statist. 13 (1985), [4] Davis, R.A., Resick, S.T., Tail estimates motivated by extreme value theory. A. Statist. 12 (1984), [5] Dekkers, A.L.M., Eimahl J.H.J., de Haa, L., A momet estimator for the idex of a extreme-value distributio. A. Statist. 17 (1989), [6] Dekkers, A.L.M., de Haa, L., O the estimatio of the extreme value idex ad large quatile estimatio. A. Statist. 17 (1989), [7] Deheuvels, P., Haeusler, E., Maso, D.M., Almost sure covergece for the Hill estimator. Math. Proc. Cambridge Philos. Soc. 104, (1988), [8] Drees, H. (1998). A geeral class of estimators of the extreme value idex. Joural of Statistical Plaig ad Iferece 66, [9] Goldie, C.M. ad Smith, R.L. Slow variatio with remaider: Theory ad applicatios. Quart. J. Math. Oxford Ser. (2) 38 (1987) [10] Gedeko, B.V., Sur la distributio limite du terme maximum d ue série aléatoire. A. Math. 44 (1943), [11] de Haa, L., O Regular Variatio ad its Applicatio to the Weak Covergece of Sample Extremes. Mathematical Cetre Tracts 32, Amsterdam, [12] Haeusler, E., Teugels, J.L., O asymptotic ormality of Hill s estimator for the expoet of regular variatio. A. Statist. 13 (1985), [13] Hall, P., O some simple estimates of a expoet of regular variatio. J. Roy. Statist. Soc. Ser. B 44 (1982), [14] Hill, B.M., A simple geeral approach to iferece about the tail of a distributio. A. Statist. 3 (1975), [15] Hsig, T., O tail idex estimatio usig depedet data. A. Statist. 19 (1991), [16] Ibragimov, I.A. ad Liik, Y.V. Idepedet ad Statioary Sequeces of Radom Variables. Wolters-Noordhoff, Groige (1969). [17] Leadbetter, M.R., Lidgre, G., Rootzé, H., Extremes ad Related Properties of Radom Sequeces ad Processes. New York Heidelberg Berli: Spriger- Verlag, (1983).

8 104 I. Ilić, P. Mladeović [18] Maso, D.M.,Laws of large umbers for sums of extreme values. A. Probab. 10 (1982), [19] Mladeović, P., Piterbarg, V., O estimatio of the expoet of regular variatio usig a sample with missig observatios. Statist. Prob. Letters. 78 (2008), [20] Pickads, J., Statistical iferece usig extreme order statistics. A. Statist. 3 (1975), [21] Resick, S.I., Extreme Values, Regular Variatio ad Poit Processes. New York, Berli, Heidelberg, Lodo, Paris, Tokyo: Spriger, (1987). [22] Resick, S. Starica, C., Cosistecy of Hill s estimator for depedet data. J. Appl. Probab. 32 (1995), [23] Resick, S., Starica, C., Asymptotic behavior of Hill s estimator for autoregressive data. Stochastic Models 13 (1997), [24] Resick, S. ad Starica, C., Tail idex estimatio for depedet data. A. Appl. Probab. 8 (1998), [25] Seeta, E., Regularly varyig fuctios. I: Lecture Notes i Mathematics, vol. 508., New York: Spriger, [26] Smith, R.L., Estimatig tails of probability distributios. A. Statist. 15 (1987), Received by the editors October 1, 2008