Sibuya s Measure of Local Dependence

Transcription

1 Versão formatada segndo a reista Estadistica 5/dez/8 Sibya s Measre of Local Dependence SUMAIA ABDEL LATIF () PEDRO ALBERTO MORETTIN () () School of Arts, Science and Hmanities Uniersity of São Palo, Ra Arlindo Béttio,, CEP: 88-, Ermelino Matarazzo, São Palo, Brazil. Phone/Fax: salatif@sp.br () Institte of Mathematics and Statistics Uniersity of São Palo, Ra do Matão CEP: 558-9, Btantã, São Palo, Brazil. Phone: Fax: pam@ime.sp.br ABSTRACT Common measres of association qantify the global association between two ariables, howeer, the data may hae different behaiors of association or dependence if we consider sbsets of the data. The aim of this paper is to stdy the fnction of local dependence of Sibya (96) for two continos random ariables. We rewrite this fnction in terms of copla. We propose three nonparametric estimators, deriing their respectie properties. Simlations and applications to real data are also gien. RESUMEN Medidas de asociación comnes cantifican la asociación geral entre dos ariables, sin embargo, los datos peden tener diferentes comportamientos de asociación o dependencia si se tiene en centa sbconjntos de los datos. El objetio de este trabajo es estdiar la fnción de la dependencia local de Sibya (96) para dos ariables aleatorias continas. Estamos reescribir esta fnción en términos de la cópla. Proponemos tres estimadores no paramétrico, qe se derian de ss respectias propiedades. Simlaciones y aplicaciones a datos reales también se dan. Key words: local dependence, coplas, nonparametric estimation Introdction Some measres like Pearson s correlation coefficient, Kendall s ta and Spearman s rho, measre the global association between two ariables. Howeer, the data may hae different behaiors of association if we consider sbsets of this data. In order to detect local behaiors in the data, local measres were proposed for two ariables, sch as the fnction of local dependence of Holland and Wang (987), the cre of correlation of Bjere and Doksm (99) and the measre of local dependence of Bairamo et al. (). Also, other proposals to measre the local dependence hae been sggested by Nelsen (999) and Sibya (96), for which Kole et al. (7) deeloped additional properties. The measre of local dependence between two ariables e Y proposed by Bairamo et al. () is gien by the expression E [( E [ / Y y] ) ( Y E [ Y / x] ) ] H (, ( S, E [ ( E [ / Y y] ) ] E [ ( Y E [ Y / x] ) ] which refers to the known Pearson correlation coefficient where E [ ] and E [Y ] was replaced by E [ / Y y] and E [ Y / x], respectielly. Ths, this measre can be interpreted as a local dependence between and Y at the point ( x,. Here, S denote the spport of (, Y). An alternatie representation of the aboe expression is gien by

2 Versão formatada segndo a reista Estadistica 5/dez/8 where ρ ( H ( Y + ϕ ϕ Y, + ϕ ( + ϕ Y E[ ] E[ / Y y] E[ Y ] E[ Y / x] ϕ (, ϕ Y and Var[ ] Var[ Y ] ρ Co(, Y ) Y Var[ ] Var[ Y]. Bjere and Doksm (99) introdced the correlation cre between and Y which is gien by β σ ρ( x), x S, β σ + σ ε where β ( x ) E[ Y / x] and σ ε ( x ) Var[ Y / x], assming that σ x and σ ε ( x ) exist, x S, where S is the domain of. This local measre is a generalization of the Pearson correlation coefficient for sal linear models which can be written by / ρ βσ { β σ + σ ε }. Ths, this local measre is appropriate for nonlinear models becase it considers nonconstant mean and ariance. Howeer, it is not symmetric in and Y. The expression of the fnction of local dependence proposed by Holland and Wang (987) is f ( f ( γ ( log f ( f ( x y f ( f ( where f is the biariate density fnction of (,Y) sch that the mixed partial deriaties f ij ( exist and f is defined in a Cartesian prodct set. The motiation of these athors was the extension of an r c contingence table for two discrete ariables to the case of two continos ariables, taking it simply partitions in a thin rectanglar grid. Then, this measre are motiated from considering continos density fnctions as contingency tables. These athors obsere that this measre not considers the marginal distribtions. The aim of this paper is to stdy the fnction of Sibya (96) for two continos random ariables. To this end, we rewrite this fnction in terms of copla, discssing their properties. Also, we propose one empirical estimator and two kernel smoothed estimators, checking their conergence in probability and in distribtion. The behaior of this local measre and their smoothed estimators were ealated throgh simlations and applications to real data (in this case, we made a comparison with copla and copla densit. Aiming to extend the notion of extreme statistics from the niariate case to the biariate case, Sibya (96) proposed a fnction of dependence between two continos random ariables, which relates the joint distribtion fnction with their marginal distribtion fnctions. Let and Y be continos random ariables defined in a probability space (Ω,A,P). Let F( P[ Y y] be a joint distribtion of and Y, and F P[ x] and F ( P[ Y y] their respecties marginal distribtions. Then, the dependence fnction of Sibya, ( F, F ( ), is gien by F( ( F ( x), F ( ), ( x, R, () F F (

3 Versão formatada segndo a reista Estadistica 5/dez/8 where F > and F ( >. For F or F (, then ( F, F ( ) may be defined if the limit lim F( ( F F ( ) ( or lim ( F( F F ( ) F x) F ( exists, respectiely. The behaior of this measre of local dependence for a biariate random ector with standardized normal distribtion and correlation coefficient +,8, -,8, +, and, are shown in Figre. The graphics of this figre, whose scales are not always the same, hae biariate grid with 99% of the central distribtion of each marginal. Figre Graphics of perspectie and corresponding graphics of leel cres of the fnction of local dependence of Sibya gien by () for (,Y) with standardized normal distribtion and correlation coefficient ρ +,8 in (a) and (e), ρ -,8 in (b) and (f), ρ +, in (c) and (g), and ρ -, in (d) and (h)... F( F(x) F( F(x) F( F(x) F( F(x) (a) (b) (c) (d) F(. 8 6 F( F( F( F(x) F(x) F(x) F(x) (e) (f) (g) (h) The properties of defined by Sibya (96) are gien below. See also Kole et al. (7). (i) F ( ) + ( ), x F y máx ( F, F ( ) mín,, ( x, R ; F F ( F F ( (ii) ( F, F ( ), ( x, R, if and only if and Y are independent; (iii) If and Y are PQD (NQD) (), then ( F ( ), F ( Y)) ( ) almost shre; (i Consider the arbitrary fnction ϕ (.) and ψ (.) of and Y, respectiely, sch that ϕ (.) and ψ (.) exist. Under the notation F F, F ( F (, Fϕ ( ) P[ ϕ( ) x], Fψ ( Y ) ( P[ ψ ( Y ) y], ϕ( ) ψ ( Y ) ( ( Fϕ ( ), Fψ ( Y ) ( ) and ( ( ), ( )) ( ( ( )), ( Y ϕ x ψ y F ϕ x F ψ ( )), and S being the spport of (, Y ), then: (a) If ϕ (.) and ψ (.) are increasing fnctions on the spport of and Y, respectiely, then ϕ ( ) ψ ( Y )( Y ( ϕ, ψ ( ), ( S ; (b) If ϕ (.) is a decreasing fnction on the spport of and ψ (.) is an increasing fnction on the spport of Y, then

4 Versão formatada segndo a reista Estadistica 5/dez/8 F ( ( )) (, ) ϕ x ( ( ), ( ) ( ) x y x ϕ ψ Y ( ) F ( ( )) ( Y ϕ ψ, ( S ; ϕ x F ϕ ) (c) If ϕ (.) is an increasing fnction on the spport of, and ψ (.) is a decreasing fnction on the spport of Y, then F ( ( )) (, ) ψ y ( ( ), ( ) ( ) x y x ϕ ψ Y ( ) F ( ( )) ( Y ϕ ψ, ( S ; ψ y F ψ ( ) (d) If ϕ (.) and ψ (.) are decreasing fnctions on the spport of and Y, respectiely, then F ( ( )) ( ( )) ( ( )) ( ( )) (, ) ϕ x F ψ y F ϕ x F ψ y ( ( ), ( ) ( ) x y x ϕ ψ Y + ( ) F ( ( )) ( ( )) ( ( )) ( Y ϕ ψ, ϕ x F ψ y F ϕ x F ψ ( ) ( S ; ( If ρ Y ( >, < ) then ( F ( ), F ( Y )) ( >, < ), in biariate normal case. By property (i, we see that at the point ( x,, is not inariant nder strictly monotone transformations since it depend of the fnctions ϕ (.) and ψ (.), among possible others. Also, Kole et al. (7) established to : exact form of the cmlatie distribtion of for the cases where and Y are contermonotonic, independent or comonotonic, and also deried the relation between these distribtions; lower bonds for expectation of. We obsere that the Sibya s dependence fnction gien by () can also be write throgh copla. Coplas are fnctions that cople the joint distribtions with their respecties niariate marginal distribtions, or coplas are joint distribtions whose niariate marginal distribtions are niform in the interal [,]. From now on let s consider the biariate case. Then a copla denoted by C is a fnction C :[,] [,] that satisfy some properties (see Nelsen, 999). By Sklar s theorem (959), there exist a copla C sch that F ( F, F ( ), ( x, R, and this is niqe if F and F are continos. An immediate corollary of this theorem shows that F( F ( ), F ( ), (, [,], where F ( ) inf{ x R : F } ς x and F ( inf{ y R : F ( } ς y are the generalized inerse of F and F, respectiely. If F i is strictly increasing, then its generalized inerse is the ordinary inerse, i,. We assme that F i, i,, are sch that F and F (,, [,], admit a niqe soltion denoted by ς x and ς y, respectiely. Then, rewriting () sing copla, we hae (, (, (,], () whose behaior is eqialent of that shown in Figre.

5 Versão formatada segndo a reista Estadistica 5/dez/8 The properties (i) to ( of gien by (), remain alid for gien by (), with necessary adjstments. Considering (), we obsere that ( ( ϕ ( ) ψ ( Y ) Y if ϕ (.) and ψ (.) are strictly increasing. Estimators and their properties To estimate gien by (), Kole et al. (7) sggest the empirical estimator n, which is bilt by plg-in method sing empirical estimators of the joint and marginal distribtions. To get a better estimator for small samples, we sggest a smoothed estimator ˆ for (), which consider the kernel smoothed estimators for the joint and marginal distribtions. Also, we derie an empirical estimator and a kernel smoothed estimator for gien by (). Let { ( i, Yi ), i, K, n } be a random sample from (, Y). To bild or estimator ˆ, we will consider the biariate kernel of simmetrical type (or prodct) and bandwidths H diag ( h, h ). Then, let k : R R be a biariate kernel fnction sch that k ( dd and x y k ( y; h, h ) k,, with h j >, j,, being fnctions of n sch that h j hh h h as n. Then, the estimator of joint density f ( is gien by n f ˆ ( k( x i, y Yi ; h, h ) n x n i i y Yi k, nhh i h h. Consider the primitie x y fnction of k ( gien by K ( k( dd with x y K ( y; h, h ) K, h h. Then, the smoothed estimator of the joint distribtion F ( is gien by x y F ˆ ( fˆ( dd n n x K( x i, y Yi ; h, h ) n i i y Yi K,. n i h h Let k j (.), j,, be real bonded symmetric fnctions, sch that k ( ) d j and k j ( ; h j ) k j h j. Then, the estimators of the marginal densities h f ( x ) and f ( are j n gien respectielly by f ˆ k( x i ; h ) n x n i k i nh i h and n y Y i fˆ ( k. Now, considering the primitie fnctions K nh i j (.) of k j (.), h j,, gien by w K j ( w) k j ( ) d with w K j ( w; h j ) K j, then the smoothed h j estimators of marginal distribtions F ( x) and F ( are gien respectielly 5

6 Versão formatada segndo a reista Estadistica 5/dez/8 x by Fˆ fˆ d ( ) n y Y i F ˆ ( K n i h. n n K( x i ; h ) i i x K i n n h and Therefore, a smoothed estimator ˆ of gien by () is ˆ ˆ ˆ ˆ F( ( F ( x), F ( ), ( x, R, with F ˆ ˆ Fˆ ( ) ˆ, F ( >, () x F ( where ˆF (.,.), F ˆ (.) and F ˆ (.) are as before. Now, we derie two nonparametric estimators for gien by (). Let the empirical estimators of the joint and marginal distribtions of and Y be n n Fn ( Ι( i Yi, F ( ) ( ) n i n x I i x (or F (, ) n i n Fn x + ) and n F n ( I ( Yi (or F (, ) n i n Fn + y ), respectiely. Then according to Fermanian et al. (), sing the plg-in method, we hae that C (, ) ( ( ), n F F F n n n ( ), (, [,], is the empirical copla. Then, an empirical estimator for () is gien by C n (, ) n (, ), (, (,], () where F n ( ) and F n ( are the empirical qantiles. Bt, to obtain better reslts, we propose the se of a kernel smoothed estimator which is bilt by plg-in method. Let Fˆ, ˆF and ˆF as before. Then, according to Fermanian et al. (), the smoothed estimator for the copla is gien by C ˆ ( Fˆ ( Fˆ ( ), ˆ F ( ) where ˆ F ( ) inf{ : ˆ x F } and ˆ F ( ) inf{ : ˆ y F ( }. As a conseqence, the corresponding smoothed estimator for () reslts in Cˆ( ˆ (, (, (,]. (5) In what follows, we will get the properties of estimators (), () and (5). Consider the following reglarity conditions. (C) F has a bonded qth deriatie; (C) F and F are Lipschitz; (C) h, as n ; (C) lim n h q ; n (C5) x j y l k( dxdy, j + l < q ; R R (C6) x j y l k( dxdy <, j + l q ; R R 6

7 Versão formatada segndo a reista Estadistica 5/dez/8 (C7) R ( x + y ) dk( <. R Theorem. Let (, Y ) be a continos random ector. Under assmptions (C) to (C7), then ˆ ˆ ˆ P ( F, F ( ) ( F, F ( ), for each fixed ( x, R, n with F ˆ ˆ, F ( > and F, F ( >. Proof. Assming the conditions (C) and (C) to (C6) alid, then the Lemma 8 of Fermanian et al. () applies, which reslts in one of the conditions of Lemma 7 of these athors. Now, also consider alid the conditions (C), (C) and (C7), then the referred Lemma 7 is alid, ie, the smoothed process { n( Fˆ ( F( ), x, y R} conerges weakly to a tight Brownian bridge in D ( R ), as n, where D ( R ) is the space of fnctions in R that are right continos and hae left limits. Therefore, by Theorem... of Lehmann (999), ˆ P F( F( as n. Set F ˆ ˆ F( ), F ˆ ˆ ( F(,, F ( x) F( ) and F ( F(,, then by Theorem 5.. of Fller, we hae that ˆ P F F and P Fˆ ( F(, as n. Now, consider θ ˆ ( ˆ θ, ˆ, ˆ ) ( ˆ (, ), ˆ ( ), ˆ θ θ F x y F x F ( ), θ ( θ, θ, θ) ( F(, F, F ( ) and g (w) : R R a continos fnction defined by g ( w ) w /( ww ) with w ( w, w, w ) and w, w >. We saw that ˆ P θi θi, i,,, as n. Then, by Lemma 5.. of Fller (976), θ ˆ P θ, and by Theorem 5.. of Fller, P g( θˆ) g( θ ), ie, ˆ ˆ ˆ P F( /( F F ( ) F( /( F F ( ). Therefore, as n, we hae that ˆ ˆ ˆ P ( F, F ( ) ( F, F ( ), for each fixed ( x, R. Theorem. Let (, Y) be a continos random ector. Sppose that F has continos marginal distribtion fnctions and that the copla fnction C ( has continos partial deriaties. Then P n ( (, for each fixed (, (,]. n Proof. Assming that F has continos marginal distribtion fnctions and that the copla fnction C ( has continos partial deriaties, then by Theorem of Fermanian et al. (), the empirical copla process { Z n ( n( Cn ( ), < } conerges weakly to the Gassian process { G C (, < } in l ((,] ), which is the space of almost shre bonded fnctions in (,] with the spremm norm. Then, by Theorem.. P of Lehmann (999), if n then Cn (, for each fixed (, (,]. Becase a fnction of this estimator also conerge in probability, the reslt follows. Theorem. Let (, Y) be a continos random ector. Sppose that F has continos marginal distribtion fnctions and that the copla fnction C ( has continos partial deriaties. Then 7

8 Versão formatada segndo a reista Estadistica 5/dez/8 { Wn ( n( n ( ( ), < } conerges weakly to the centered Gassian process { G (, < } with ariance { C ( C ( + { } ( ) + { } ( ) { C ( } ( ) { } ( + { }{ }{ }} { / }, in l ((,] ), for each fixed (, (,]. Note:, and represent the deriatie with respect to C, and, respectiely. Proof. The fnction (, / can be represented by the mapping C a ( C, C, C ) a C /( C C ) where C ) and C,. The first map is linear and continos and then Hadamard differenciable (see the proof of Lemma.9.8 in an der Vaart and Wellner, 996), and the second map is Hadamard differenciable on the domain of fnctions that are away from zero (see section.9.. in an der Vaart and Wellner). Then, by the chain rle (Lemma.9. in an der Vaart and Wellner), the map φ : D((,] ) l ((,] ) where φ ( C)( C /{ C ( C ( )} C ( / (or ( C a C / CC ) is Hadamard differenciable at C tangentially to C ((,] ). Assming that F has continos marginal distribtion fnctions and that the copla fnction C ( has continos partial deriaties, then the Theorem of Fermanian et al. is alid (see the proof of theorem aboe), where G C takes its ales in C ([,] ). Also, these athors (page 85) show that the limited Gassian process can be written as GC ( BC ( BC ( ) BC (,, where B C is a Brownian bridge in [,] with coariance fnction E[ B ( B (, )] C ( ^, ^ ) C (, ) C C, for each,,,. Then, E[ G C ( ] and Var[ GC ( ] C ( C ( + + { } ( ) + { } ( { } ( ) { } ( + { }{ }{ }. By the delta method (Theorem.9. of an der Vaart and Wellner) the empirical process { W n (, < } of conerges weakly to the centered Gassian process G with ariance gien aboe. Theorem. Let (, Y ) be a continos random ector. Under assmptions (C) to (C7), and spposing that C has continos partial deriaties, then ˆ P ( (, for each fixed (, (,]. n Proof. Assming the conditions (C) to (C7) satisfied, with C haing continos partial deriaties, then the Theorem of Fermanian et al. () is alid, ie, the smoothed copla process { Z ˆ( n( Cˆ( ), } conerges weakly to a Gassian process { G C (, } in l ([,] ). Becase ˆ P, as n, then by Theorem 5.. of Fller we hae that ˆ P ( (, for each fixed (, (,]. Theorem 5. Let (, Y) be a continos random ector. Under assmptions (C) to (C7), and spposing that C has continos partial deriaties, then { W ˆ ( n( ˆ ( ( ), < } 8

9 Versão formatada segndo a reista Estadistica 5/dez/8 conerges weakly to the centered Gassian process { G (, < } with ariance { C ( C ( + { } ( ) + { ( } { } ( ) { } ( + { }{ }{ }} { / } in l ([,] ), for each fixed (, (,]. Proof. Assming (C) to (C7) alid, with C haing continos partial deriaties, then the Theorem of Fermanian et al. is alid (see proof of Theorem aboe). Also, in the proof of Theorem, we deried E[ G C ( ] and Var[ GC ( ]. Becase is differentiable at C, then by the delta method, { Wˆ ( n ( ˆ (, ( ), < } conerges weakly to a centered Gassian process { G (, < } with ariance like aboe, in l ((,] ). Simlations In this section we wish to inestigate the sample properties of or kernel estimators. Initially we consider ˆ gien by () and then gien by (5). For each estimator, we consider, samples (with different sizes) obsered from a biariate normal random ector with correlation.7. We calclated the bias, the mean sqared error and the p-ale of normality test of Jarqe Bera in some points of the biariate grid (5 5 points). In these simlations, we se prodct of two Gassian kernel, optimal bandwidth of Hansen () and 99% of the central data. Let s firstly consider the local measre written in terms of distribtion fnctions. Considering a random ector (, Y ) with normal distribtion with mean (.5;6.), ec ( Σ) (.;.9;.9;.99) and correlation.7, we calclate and we estimate ˆ sing, samples of size, obsered from this random ector. See their graphs in Figre. The reslting ales of sal statistics of the estimator are shown in Table to Table. Figre (a) Theoretical fnction of dependence of Sibya gien throgh distribtion fnctions, and their (b) graphs of leel cres, considering a random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). Also, we present the (c) estimated fnction of dependence of Sibya and their (d) graph of leel cres, sing, random samples of size,. F( F(x) F( F(x) ˆ F( F(x) F( F(x) (a) (b) (c) (d) This simlation was repeated for samples of sizes 5 and,, whose reslts are also shown in these tables. Comparing the three simlations, we conclde that the bias, the mean sqared error and the rejection of normality decreases with increasing the size of the sample. 9

10 Versão formatada segndo a reista Estadistica 5/dez/8 Table Bias for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and,, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5, -,8 -,8 -,9 -,,6,,,5 -,8 -,778 -,59 -,6,,,,5 -,6 -,6 -,97 -,5 -,,,,5 -,9 -, -,5 -, -,7,,,75,6, -, -,8 -,5,,,95,,,,,,,,99,,,,,,,,, -, -,8 -,69 -,,5,,,5 -,8 -, -,89 -,,,,,5 -,86 -,88 -,6 -, -,,,,5 -,7 -, -, -, -,,,,75,5,, -, -,,,,95,,,,,,,,99,,,,,,,,, -,85 -,56 -, -,,,,,5 -,586 -,9 -,55 -,7,,,,5 -,8 -,56 -,5 -, -,,,,5 -, -,5 -, -,9 -,,,,75,, -, -, -,,,,95,,,,,,,,99,,,,,,, Table Mean sqared error for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5, 7,77,68,9,8,,,,5,8,5,87,7,,,,5,8,95,6,6,,,,5,8,7,6,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,,,,97 6,6,69,,,,,5 6,8,8,6,,,,,5,75,,,,,,,5,5,,,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,,, 57,7,5,,,,,,5,587,6,,,,,,5,9,,7,,,,,5,,,,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,

11 Versão formatada segndo a reista Estadistica 5/dez/8 Table P-ale of normality test for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and,, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5,,,,,,,,,5,,,,,65,,,5,,,7,,,,,5,,6,,,,,,75,9,65,579,5,,,,95,8,,5,6,,,,99,,,,,,,,,,,,,,,,,5,,,857,978,,,,5,,,98,66,,8,,5,,8,6,5,,9,,75,,9,,5,,,,95,,,,,7,,,99,,,,,,,,,,,,7,,5,7,69,5,,,687,7,,686,6,5,,,6,68,968,788,6,5,,5,,876,89,88,9,75,,,65,896,9,98,7,95,67,69,656,7,99,87,,99,9,9,86,,6,88, Next, we consider the measre of local dependence written in terms of copla. From the random ector (, Y ) ~ N ( µ, Σ) with correlation.7 considered aboe, we obsered, samples of size,. Then, we obtain the graphs of the theoretical and estimated fnctions (Figre ), and the sal statistics which are presented in Table to Table 6. In these tables we see also the reslts of simlations with samples of sizes 5 and,, showing that as n increases the bias, mean sqared error and the rejection of normality generally decreases, bt the last ery slowly. Figre (a) Theoretical fnction of dependence of Sibya gien throgh copla and their (b) graph of leel cres, considering a random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). Also, we present the (c) estimated fnction of dependence of Sibya and their (d) graph of leel cres, considering, random samples of size, ˆ (a) (b) (c) (d) Comparing the behaior of the two estimators, we see that the estimator sing coplas has lower bias, mean sqared error and less rejection of normality.

12 Versão formatada segndo a reista Estadistica 5/dez/8 Table Bias for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and,, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5, -,56 -, -,8 -, -, -, -,,5 -,98 -,56 -,6 -, -, -, -,,5 -,79 -,6 -,7 -, -,8 -,,,5 -, -, -, -, -, -,,,75 -, -, -,9 -, -,8 -,,,95 -, -, -, -, -, -, -,,99 -, -,,,,,,,, -, -,599 -,5 -,7 -, -, -,,5 -,58 -,67 -,66 -, -, -, -,,5 -,6 -,6 -,8 -, -,5,,,5 -, -,5 -,9 -,5 -,7 -,,,75 -, -, -,5 -,7 -,6 -,,,95 -, -,, -, -, -,,,99 -, -,,,,,,,, -, -, -,5 -,7 -, -, -,,5 -,89 -, -,9 -,9 -, -, -,,5 -,6 -,8 -,7 -, -,,,,5 -,6 -,8 -, -, -,,,,75 -, -, -, -,5 -, -,,,95 -, -,,, -, -,,,99 -, -,,,,,, Table 5 Mean sqared error for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5, 8,596,795,87,,,,,5 9,75,78,5,,,,,5,8,57,5,,,,,5,,,,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,,, 5,8 6,85,6,,,,,5 5,56,99,9,,,,,5,8,8,8,,,,,5,,,,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,,, 5,8,,5,,,,,5,6,55,,,,,,5,,5,,,,,,5,,,,,,,,75,,,,,,,,95,,,,,,,,99,,,,,,,

13 Versão formatada segndo a reista Estadistica 5/dez/8 Table 6 P-ale of normality test of Jarqe Bera for some points of the biariate grid for the estimated fnction of dependence of Sibya obtained throgh, random samples of size 5,, and,, obsered from the random ector (,Y) with normal distribtion with mean (.5;6.) and ec(σ)(.;.9;.9;.99) (correlation.7). n Grid,,5,5,5,75,95,99 5,,,,,,,,,5,9,59,,,,,,5,,,86,,,,,5,,,,565,5,,,75,,,,9,9,,,95,,,,,,58,,99,,,,,,,,,,,5,,,,,,5,55,88,,,,,,5,,,7,6,,,,5,,,6,587,,,,75,,,,59,859,,,95,,,,,,6,99,99,,,,,,7,,,,77,67,,,,,,5,67,689,9,,,,,5,,,,5,,,,5,,,,,58,,,75,,,5,,,,,95,,,,,5,5,,99,,,,,,7,9 Applications to real data This section illstrates the implementation of the procedre described in Section. Considering the largest companies in Brazil in 6, according to the criterion of the brazilian magazine Exame, we select all priate and state companies (except banks and insrers) which reslt in,8 companies (see sericos/melhoresemaiores/). Then, we analyzed some ariables concerning the rates of annal performance. Firstly, we se the estimator () and then the estimator (5). In this estimation we sed biariate grid with points, 9% of the central data and bandwidht also according to Azzaline (98). Sales ($ million) and net profit ($ million) for the 88 companies that contains this information, present Spearman s correlation eqal to 55, indicating global positie association. In Figre we see the scatter plot of these two ariables in (a), the graph of ˆ gien by () in (b) and (c) which show mild positie dependence, the scatter plot of normalized ranks and the kernel smoothed copla density in (d), (e) and (f) clearly show positie dependence, and the kernel smooted copla in (g) and (h) indicate positie dependence. In Figre 5 we see the graphs of ˆ gien by (5), which reslt in a similar behaior of the graph (b) and (c) in Figre. Spearman s correlation between net profit ($ million) and sales margin (%) for the 86 companies considered, reslts in 8. In Figre 6, we see the scatter plot of the original ariables in (a) and the remaining graphs indicate strong positie dependence. For these data, ˆ gien by (5) are shown in Figre 7.

14 Versão formatada segndo a reista Estadistica 5/dez/8 Figre Considering sales (, in $ million) and net profit (Y, in $ million) of the companies in 6, we hae (a) scatter plot, (b) estimated fnction of dependence of Sibya gien throgh distribtion fnctions and their (c) graph of leel cres. For the corresponding normalized ranks, we hae the (d) scatter plot, the (e) kernel smoothed copla density and their (f) leel cres, the (g), kernel smoothed copla and their (h) leel cres. Y ˆ. F( F(x) F( F(x) (a) (b) (c) V.. U ĉ.. (d) (e) (f) Ĉ (g) (h) Figre 5 Considering sales (, in $ million) and net profit (Y, in $ million) of the companies in 6, we hae the estimated fnction of dependence of Sibya gien throgh copla in (a) and their graph of leel cres in (b). ˆ (a) (b)

15 Versão formatada segndo a reista Estadistica 5/dez/8 Figre 6 Considering net profit (, $ million) and sales margin (Y, %) of the companies in 6, we hae (a) scatter plot, (b) estimated fnction of dependence of Sibya gien throgh distribtion fnctions and their (c) graph of leel cres. For the corresponding normalized ranks, we hae the (d) scatter plot, the (e) kernel smoothed copla density and their (f) leel cres, the (g) kernel smooted copla and their (h) leel cres. Y ˆ 6 F( F(x) F(.5.5 F(x) (a) (b) (c) V.. U ĉ (d) (e) (f). Ĉ (g) (h) Figre 7 Considering net profit (, $ million) and sales margin (Y, %) of the companies in 6, we hae the estimated fnction of dependence of Sibya gien throgh copla in (a) and their graph of leel cres in (b). 6 ˆ.5.5 (a) (b) 5

16 Versão formatada segndo a reista Estadistica 5/dez/8 Figre 8 Considering net profit (, $ million) and general indebtedness (Y, %) of the companies in 6, we hae (a) scatter plot, (b) estimated fnction of dependence of Sibya gien throgh distribtion fnctions and their (c) graph of leel cres. For the their normalized ranks, we hae the (d) scatter plot, the (e) kernel smoothed copla density and their (f) leel cres, the (g) kernel smoothed copla and their (h) leel cres. Y 5 6. ˆ. F( F(x) F( F(x). (a) (b) (c) V.. U ĉ (d) (e) (f). Ĉ (g) (h) Figre 9 Considering net profit (, $ million) and general indebtedness (Y, %) of the companies in 6, we hae the estimated fnction of dependence of Sibya gien throgh copla in (a) and their graph of leel cres in (b)... ˆ (a) (b) 6

17 Versão formatada segndo a reista Estadistica 5/dez/8 Figre Considering growth in sales (, %) and net profit (Y, $ million) of the companies in 6, we hae (a) scatter plot, (b) estimated fnction of dependence of Sibya gien throgh distribtion fnctions and their (c) graph of leel cres. For the corresponding normalized ranks, we hae the (d) scatter plot, the (e) kernel smoothed copla density and their (f) leel cres, the (g) kernel smoothed copla and their (h) leel cres. Y ˆ.5. F( F(x) F( F(x) (a) (b) (c) V.. U.5 ĉ... (d) (e) (f)..... Ĉ (g) (h) Figre Considering growth in sales (, %) and net profit (Y, $ million) of the companies in 6, we hae the estimated fnction of dependence of Sibya gien throgh copla in (a) and their graph of leel cres in (b)..5 ˆ (a) (b) 7

18 Versão formatada segndo a reista Estadistica 5/dez/8 The net profit ($ million) and general indebtedness (%) for 88 companies, reslt in -. for the Spearman s correlation. In Figre 8 and Figre 9 we see the graphs of local dependence which indicate negatie dependence. The correlation of Spearman between growth in sales (%) and net profit ($ million) for 88 companies reslt in 7, and the local graphs indicate independence between these ariables. Acknowledgements. The athors acknowledge the spport of FAPESP grant 8/ References Azzalini, A. (98). A note on the estimation of a distribtion fnction and qantiles by a kernel method. Biometrika, 68,, 6-8. Bairamo, I., Kotz, S. and Kozbowski, T. J. (). A new measre of linear local dependence. Statistics, 7(), -58. Bjere, S. and Doksm, K. (99). Correlation cres: Measres of association as fnctions of coariate ales. Annals of Statistics, (), Fermanian, J. D., Radloic, D., Wegkamp, M. (). Weak conergence of empirical copla processes. Bernolli (5), Fller, W. A. (976). Introdction to Statistical Time Series. New York: John Wiley & Sons. Hansen, B. E. (). Bandwidth selection for nonparametric distribtion estimation. Working paper. Holland, P. W. and Wang, Y. J. (987). Dependence Fnction for Continos Biariate Densities. Commnications in Statistics Theory and Methods, 6(), Kole. N. V., Gonçales, M. Dimitro, B. (7). Probabilistic Properties of Sibya s Dependence Fnction. Preprint. Lehmann, E. L. (999). Elements of Large-Sample Theory. New York: Springer. Nelsen, R. B. (999). An Introdction to Coplas. New York: Springer-Verlag. Sklar, A. (959). Fonctions de répartition à n dimensions e lers marges. Pblications de l Institt de Statistiqe de l Uniersité de Paris 8, 9-. Sibya, M. (96). Biariate Extreme Statistics. Annals of the Institte of Statistical Mathematics,, 95-. an der Vaart, A. W., Wellner, J. A. (996). Weak Conergence and Empirical Processes. New York: Springer-Verlag. Note: () and Y are PQD - positiely qadrant dependent (NQD - negatiely qadrant dependent), if for all fixed ( x, R, F( ( ) F F (. 8