1 Introduction, definitions and assumption

Transcription

1 O almost sure covergece rates for the kerel estimator of a covariace operator uder egative associatio H. Jabbari 1 ad M. Erfaiya Departmet of Statistics, Ordered ad Spatial Data Ceter of Excellece, Ferdowsi Uiversity of Mashhad, Mashhad, Ira Abstract Let {X, 1} be a strictly statioary sequece of egatively associated radom variables, with commo cotiuous ad bouded distributio fuctio F. We cosider the estimatio of the two-dimesioal distributio fuctio of (X 1, X k+1 ) based o kerel type estimators as well as the estimatio of the covariace fuctio of the limit empirical process iduced by the sequece {X, 1} where k IN 0. The, we derive uiform strog covergece rates for the kerel estimator of two-dimesioal distributio fuctio of (X 1, X k+1 ) which were ot foud already ad do ot eed ay coditios o the covariace structure of the variables. Furthermore assumig a coveiet decrease rate of the covariaces Cov(X 1, X +1 ), 1, we prove uiform strog covergece rate for covariace fuctio of the limit empirical process based o kerel type estimators. Fially, we use a simulatio study to compare the estimators of distributio fuctio of (X 1, X k+1 ). Key Words: Almost sure covergece rate, Bivariate distributio fuctio, Empirical process, Kerel estimatio. 1 Itroductio, defiitios ad assumptio Estimatio of distributio fuctios of radom pairs (two-dimesioal distributio fuctios) has bee always a subject of iterest of may statisticias. The case of idepedet uderlyig radom variables was studied by [3]. The case of oidepedet radom variables had bee studied, too (see for example [1], [2], [6], [7] ad [8]). Oe of the most applicable cocept of egative depedece i multivariate statistical aalysis ad reliability theory is egative associatio. A fiite family of radom variables {X i, 1 i } is said to be egatively associated (NA) if for every pair of disjoit subsets A ad B of {1, 2,..., }, Cov(f 1 (X i, i A), f 2 (X j, j B)) 0, wheever f 1 ad f 2 are coordiatewise icreasig ad such that the covariace exists. A ifiite family of radom variables is NA if every fiite subfamily is NA. We refer to [1], [8], [9], [10], [11], [12], [14], [15], [18], [19], [20], [21], [22], [23], [24] ad [25] for kowig some of the most importat studies have bee performed o differet aspects of NA radom variables. The metioed commets above motivated the iterest o the estimatio of the bivariate distributio fuctio uder egative associatio. A atural (histogram) estimator of F k (r, s) = P (X 1 r, X k+1 s) with k fixed, is defied by F k (r, s) = 1 {1 (,r] (X i )1 (,s] (X k+i )}. (1) k The asymptotic behavior of this estimator was studied by [6], [7] ad [10]. For depedet sequeces, uder certai coditios (see [16], Theorem 17 ad the first remark of p. 137), the 1 Jabbarih@um.ac.ir 1

2 limit of the uiform empirical process still is a cetered Gaussia process, but the covariace fuctio chages to Γ k (r, s) = ϕ k (r, s) + ϕ k (r, s) + ϕ k (s, r), (2) where ϕ k (r, s) = F k (r, s) F (r)f (s). [6], [7] ad [10] drove a uiform strog covergece rate of 1/2 for two-dimesioal empirical distributio fuctio of (X 1, X k+1 ) ad covariace fuctio of the limit empirical process assumig a coveiet decrease rate of the covariace. [2] ad [8] cosidered the kerel estimator of F k, defied by ˆF k (r, s) = 1, s X k+i ). (3) k where U is a give bivariate distributio fuctio ad {, 1} is a sequece of positive umbers covergig to zero. They foud the optimal badwidth covergece rate of order 1. I this paper usig ˆF k i (3), we defie the kerel estimator of ϕ k (r, s) ad Γ(r, s) as ˆϕ k (r, s) = ˆF k (r, s) ˆF (r) ˆF (s), ˆΓ(r, s) = ˆϕk (r, s) + ( ˆϕ k (r, s) + ˆϕ k (s, r)) (4) ad derive a uiform covergece rate of order h 2 γ for the above estimators, where ˆF (r) = 1 ) ad 0 < γ < 1/2. For this covergece rate, we eed o coditio o the covariace structure of the variables. The above rate is flexible because of icludig the term which ca be optioally chose. This flexibility makes us able to have a rate that teds to zero (as is ecessary for a covergece rate) ad o the other had, ca be a better rate tha what was foud by [10] ad [8]. It is oted that the proofs are similar to those of [10] I all sectios of this paper pose that C is a positive costat ot depedig o. Also, we use the followig geeral assumptio throughout the article: (A). {X, 1} is a NA ad strictly statioary sequece of radom variables havig bouded desity fuctio ad for ay 1 i ad fixed r, s IR. ) E ) Ch 2 h, a.s. (5) Remark 1.1 It ca be easily checked that (5) holds for ay NA sequece of radom variables metioed i (A), because ) = Y2 Y1 u(t 1, t 2 )dt 1 dt 2, a.s. where Y 1 = r X i, Y 2 = s X i+k ad u is the probability desity fuctio associated to U. By lettig z 1 = r t 1 ad z 2 = s t 2 i the above itegral, we have ) = Xi+k Xi h 2 2 u( r z 1, s z 2 )h 2 h dz 1 dz 2 u( r z 1, s z 2 )dz 1 dz 2. a.s.

3 By further replacemets w 1 = r z 1 ad w 2 = s z 2, we obtai ) = O(h 2 h ). a.s. (6) O the other had for the expected value of U, we ca write E ) = )du(x i, x i+k ). By replacig v 1 = r X i ad v 2 = s X i+k, the above itegral is equal to E ) = h 2 U(v 1, v 2 )u(r v 1, s v 2 )h 2 dv 1 dv 2 u(r v 1, s v 2 )dv 1 dv 2 = O(1). (7) The iequality of (7) holds, sice 0 U(v 1, v 2 ) 1, v 1, v 2 IR ad the last equality holds after some more replacemets. Fially, (5) satisfies by cosiderig (6) ad (7) together. I Sectio 2, we will preset some auxiliary results eeded to establish the above metioed covergece rates. The momet iequality used for the proofs is preseted i this sectio. The strog uiform covergece rates are proved i Sectios 3 ad 4. I Sectio 5, we compare the histogram ad kerel estimators graphically usig a simulatio study ad the coclude the results. 2 Auxiliary results I this sectio, we used the followig momet iequality for NA radom variables ad proved a importat iequality that are eeded for provig our covergece rates. Lemma 2.1 ([13] ad [20]) Let (X 1, X 2,..., X ) be a NA radom vector with EX j = 0 ad EX j p < for some p 2 ad all j = 1,...,. The, there exists a costat C = C(p) > 0, such that E X j p C[ EX j p + ( EXj 2 ) p/2 ]. (8) j=1 j=1 j=1 Lemma 2.2 Let k IN 0 be fixed ad ε be a sequece of positive umbers. Suppose (A) is satisfied. The, there exists a costat C such that, for r, s IR ad p > 2, P ( ˆF k (r, s) F k (r, s) > ε ) Ch 2p ε p. (9) ( k) p/2 Proof. For eac IN, 1 i ad fixed r, s IR defie ad also Z k,i = ) F k (r, s), W k,i = Z k,i E(Z k,i ). 3

4 So, we have ˆF k (r, s) E( ˆF k (r, s)) = = 1 Z k,i + F k (r, s) E( k ˆF k (r, s)) 1 W k,i + 1 E(Z k,i ) + F k (r, s) E( k k ˆF k (r, s)). Regardig 1 E(Z k,i ) = E( ˆF k (r, s)) E(F k (r, s)), we will have ˆF k (r, s) E( ˆF k (r, s)) = 1 W k,i. k Sice (A) is hold, it is clear that W k, are decreasig fuctios of the variables X. So accordig to the properties of NA radom variables (see for more iformatio [12]), {W k,, 1} is NA ad strictly statioary. Also, W k, Ch 2 ad E(W k, ) = 0 the, EW k, p <, for each 1 ad p > 2 ad so we ca apply Lemma 2.1 to the sequece {W k,, 1}. Thus for all 1, we obtai Now for fixed r, s IR, we ca write E W k,i p C[ EW k,i p + ( EWk,i) 2 p/2 ] C p/2 h 2p. (10) P ( ˆF k (r, s) F k (r, s) > ε ) P ( ˆF k (r, s) E( ˆF k (r, s)) > ε 2 ) + P (F k (r, s) E( ˆF k (r, s)) > ε ). (11) 2 Sice 0 < F k (r, s), ˆF k (r, s) < 1 for fixed k IN 0 ad r, s IR, we coclude P (F k (r, s) E( ˆF k (r, s)) > ε ) 0 as +. Now regardig this, usig the Markov iequality ad from 2 (10) ad (11) we fid, for all > k, P ( ˆF k (r, s) F k (r, s) > ε ) 2 p ε p ( k) E p Ch 2p W k,i p ε p. (12) ( k) p/2 To prove the ext results, we should defie the followig otatios as itroduced i [10]. Let t be a sequece of positive itegers such that t +. For eac IN ad each i = 1,..., t, put x,i = Q(i/t ), where Q is the quatile fuctio of F. The for IN ad k IN 0, defie ad D,k = ˆF k (r, s) F k (r, s), D,k = max i,j=1,...,t ˆF k (x,i, x,j ) F k (x,i, x,j ). Furthermore, we will eed the followig result as i Theorem 2 of [6] ad Lemma 2.3 of [10]. Lemma 2.3 If the sequece {X, 1} satisfies (A), the, for eac IN ad each k IN 0, D,k D,k + 2 t a.s.. (13) 4

5 Lemma 2.4 Let ε ad t be two sequeces of positive umbers such that t + ad ε t +, p > 2 ad k IN 0 be fixed. Suppose (A) holds. The, for ay large eoug, P ( ˆF k (r, s) F k (r, s) > ε ) Ct 2 ε p h2p ( k) p/2. (14) Proof. Followig the same steps i Lemma 2.4 of [10] ad applyig Lemma 2.2 ad Lemma 2.3 the result is cocluded. 3 Uiform strog covergece rates of ˆF k I this sectio, we summarize the previous results to get uiform strog covergece rates of ˆF k. Lemma 3.1 Let k IN 0 be fixed ad pose (A) holds. The uder the coditios of Lemma 2.4 ad for every 0 < δ < p 1, we have Proof. Put t = 1 ε h Lemma 2.4 for ε = h 2p 2 2p 2 ˆF k (r, s) F k (r, s) = O(h p 2 2δ 2() ) a.s.. (15) ad let 0 < δ < p 2. Sice t 2 ad t ε whe, from p 2 2δ 2() ad large eough, we obtai P ( ˆF k (r, s) F k (r, s) > ε ) C ε h 2 2p ( C (1+δ). (16) k)p/2 The proof is complete usig the Borel-Catelli Lemma, because for all δ > 0, the sequece o the right-had side above beig summable. If p, ε h 2 1/2. Sice h 2 0 whe, the covergece rate of Lemma 3.1 remais reasoable for a large p. I the ext theorem, we summarize the results of this sectio. Theorem 3.1 Uder the assumptios of Lemma 3.1 ad for every 0 < γ < 1/2, we have ˆF k (r, s) F k (r, s) = O(h 2 γ ) a.s.. (17) Proof. Usig Lemma 3.1 ad alog the lies of Theorem 3.1 i [10], we get the desired result. Remark 3.1 Note that Theorem 4 of [8] holds true for ˆF k defied i (3) uder some regularity assumptios. So for all x, y IR, we have ( k)mse[ ˆF (x, y)] = F (x, y) F 2 (x, y) + 2 (F j (x, y, x, y) F 2 (x, y)) j=2 + O( + h 2 ) + a, where for each positive iteger j, F j is the distributio fuctio of (X 1, X k+1, X j, X k+j ) ad a = 1 ( k) (j 1)(F j (x, y, x, y) F 2 (x, y)) 2 (F j (x, y, x, y) F 2 (x, y)). j=2 j= 1 The, a optimal covergece rate of the MSE is achieved by choosig = C 1. 5

6 If k = 0 ad s = r the estimator ˆF k (r, s) becomes to the oe-dimetioal kerel distributio fuctio ˆF (r). The results of Theorem 3.1 hold true for ˆF. So, we ca write ˆF (r) F (r) = O(h 2 γ ) a.s.. (18) r IR Remark 3.2 From the results of Theorem 3.1, we uderstad that the covergece rate h 2 γ for every 0 < γ < 1/2 ad is very faster tha those obtaied later by [10] (i.e. γ ). So, the kerel estimator of two-dimesioal ad oe-dimetioal distributio fuctio F k ad F is better tha empirical oe, respectively. Now, we ca obtai the covergece rate of the kerel estimator of ϕ k. Theorem 3.2 Uder the assumptios of Theorem 3.1 ad for every 0 < γ < 1/2, we have ˆϕ k (r, s) ϕ k (r, s) = O(h 2 γ ) a.s.. (19) Proof. The proof is similar to that of Theorem 3.2 i [10] ad the we omit it. 4 Uiform strog covergece rates of ˆΓ As [10], we will itroduce uiform strog covergece rates for the kerel estimators of the sum ϕ k (r, s) ad the covariace fuctio Γ(r, s). Regardig that the covariace structure of a sequece of NA radom variables highly determies its approximate idepedece (see [16]), it is commo to have assumptios o the covariace structure of the radom variables. For this, we use the same defiitio of [10] as v() = j=+1 Cov(X 1, X j ) 1/3. (20) I the followig lemma, we prove the uiform strog covergece rate for the sum ˆϕ k (r, s) which is sufficiet to obtai the desired result for the kerel estimator of Γ. Lemma 4.1 Let (A) holds, θ > 0 ad pose that a = p 2 2δ p 2 +3p 0 < δ < p 2. If 2 for some p > 2 ad for each v(a ) Ch 4θ(p 1) (p 2)(p+3) a θ (21) for all 1, we have ˆϕ k (r, s) 2p 2 (p 2)(p 2 2δ) ϕ k (r, s) = O( 2p() ) a.s.. (22) Proof. The idea is essetially the same as the proof of Lemma 4.1 of [10]. So, we repeat their proof usig our required otatios ad defiitios. Take ε = h 2p 2 (p 2)(p 2 2δ) 2p() for each 0 < δ < p 2 2 ad t = a ε h. Now, we ca write P ( ( ˆF k (r, s) F k (r, s)) > ε ) P ( ˆF k (r, s) F k (r, s) > ε ). (23) a Sice 0 < δ < p 2 2p() t +, ε a t + ad a 2, (p 2)(p 2 2δ) > 0 ad 0 < p 2 2δ p 2 +3p 0 as +. 6 < 1, it is easy to see ε 0, a +,

7 Usig ε a i place of ε i Lemma 2.4, we obtai for all large eough, P ( ( ˆF k (r, s) F k (r, s)) > ε ) p 2 2δ 2p+4. Isertig this o the right- 0. So, we have by Borel-Catelli 2p 2 By elemetary calculatios, we may write ε = had side of (24) leads to summable upper boud as a Lemma Now, as [10], we ca write = p+3 a Ct 2 a p ε p h2p ( k) p/2 Ct 2 a p+1 ε p ( a ) ε Ca p+3 ( a ) h2p p/2 p/2 h2p 2. (24) ( ˆF 2p 2 (p 2)(p 2 2δ) k (r, s) F k (r, s)) = O( 2p() ) a.s.. (25) ˆϕ k (r, s) ϕ k (r, s) ( ˆF k (r, s) F k (r, s)) + 2a ˆF (r) F (r) r IR + ϕ k (r, s). (26) k=a +1 For the first term o the right-had side of (26), we use (25). Sice p+3 > 1 by usig Lemma 3.1 for the secod term, we have a r IR 2p 2 ˆF (r) F (r) = O(a h p 2 2δ 2p+4 ) 2p 2 (p 2)(p 2 2δ) = O( 2p() ) a.s.. (27) For the third term o the right-had side of (26), we use Corollary of Theorem 1 i [17] ad relatio (21) i [15] as those applied i [10]. So by (21) for θ = (p 2)(p+3) > 0 ad a 2p+4 = p 2 2δ p 2 +3p, we obtai ϕ k (r, s) C Cov 1/3 (X 1, X k+1 ) k=a +1 Hece the proof is completed. k=a +1 4(p 1)(p 2)(p+3) 2()(p 2)(p+3) = Cv(a ) Ch a (p 2)(p+3) 2() 2p 2 (p 2)(p 2 2δ) = C 2p(). (28) We ow summarize the above result i the followig theorem. Theorem 4.1 Uder the assumptios of Lemma 4.1 ad coditio (21) for all 1, θ > 0 ad 0 < γ < 1/2, we have a ˆϕ k (r, s) ϕ k (r, s) = O(h 2 γ ) a.s.. (29) Proof. As i proof of Theorem 4.1 of [10], we apply the lies of proof of Theorem 3.1 ad use Lemma 4.1 istead of Lemma 3.1. So, for δ > 0 ad p > 2 we have (p 2)(p 2 2δ) > γ ad the 2p() the proof is cocluded. 7

8 Now, applyig the lies of proof of Theorem 4.2 i [10] ad usig Theorems 3.1 ad 4.1, we ca state the followig theorem which summarizes the results for ˆΓ. Theorem 4.2 Suppose (A) holds. 0 < γ < 1/2, we have Uder coditio (21) for all 1, θ > 0, p > 2 ad ˆΓ(r, s) Γ(r, s) = O(h 2 γ ) a.s.. (30) Remark 4.1 As stated i Remark 3.2, our covergece rate h 2 γ for every 0 < γ < 1/2 ad i Theorem 4.2 is very faster tha those obtaied later by [10] (i.e. γ ). So, the kerel estimator of Γ is better tha empirical oe. 5 Simulatio study I this sectio, we ited to compare the behavior of our estimator with those of [10] via a simulatio study. As oted i [4], [5] ad [12] a umber of well kow multivariate distributios such as multivariate ormal distributio witegative correlatios possess the NA property. So for geeratig the NA radom variables, pose that X 1,..., X have multivariate ormal joit distributio with zero mea vector ad the followig covariace matrix 1 ρ ρ 2 ρ 1 Σ = 1 ρ 1 ρ ρ 2 1 ρ ρ 1 ρ 2 ρ 3 1 where ρ > 0. For = 20, 100, we geerate oe sample from -dimesioal multivariate ormal distributio with zero mea vector ad covariace matrix Σ assumig ρ = 0.1, The for k = 0, 1, 2, we compute the histogram estimator F k i (1) ad the kerel estimator ˆF k i (3) usig = 1 ad = log 1 () ad U(.,.) as bivariate ormal distributio with zero mea vector ad covariace matrix [ ] 1 1 ρ. (31) 1 ρ 2 ρ 1 Results for k = 0, 1, 2 ad differet values of, ρ ad are preseted i Figures 1-3, respectively. Also for simplicity of comparig, we compute the followig mea square distaces (MSDs) betwee F k (r, s) ad ˆF k (r, s) (or F k (r, s)) for all r, s: MSD 1 = 1 N MSD 2 = 1 N ( ˆF k (r, s) F k (r, s)) 2 r,s ( F k (r, s) F k (r, s)) 2 (32) r,s where N is the product of all umbers r ad s. The results are also reported i Figures 1-3. Figure 1 shows that for k = 0 (oe-dimesioal distributio fuctio): a) Whe is small ( = 20) ad large ( = 100), kerel estimator (gree) of F (r) is better tha histogram estimator (black) for all values of ρ ad badwidth rates. b) Whe becomes large, the kerel estimator has a good fit. c) Whe is small, the badwidth rates = log 1 () is better tha = 1. d) Whe is large, the badwidth rates = 1 ad = log 1 () have the same behaviors. e) Sice the kerel estimator is smooth, the best estimator of F (r) is the kerel estimator. 8

9 f) I all graphs, MSD of kerel estimator is less tha histogram estimator. g) I all cases, the histogram estimator has a over estimate. Figure 2 shows that for k = 1 (two-dimesioal distributio fuctio with lag oe): a) Whe is small ( = 20), we have over estimate for weak depedece (ρ = 0.1) ad = log 1 (). Also, this wrog fit holds true whe is small ( = 20), ρ = 0.1 ad = 1 for some values of r ad s (that is r, s [ 2, 4], approximately). b) MSD of kerel estimator is less tha histogram estimator for all cases. c) Whe is large ( = 100), the differece betwee kerel ad histogram estimators is very small. d) Whe is small ( = 20) or large ( = 100), the badwidth rate = 1 has a better role tha = log 1 () for estimatig F 1 (r, s) i weak (ρ = 0.1) depedece case ad i strog (ρ = 0.36) depedece case, the badwidth rate = log 1 () is almost better tha = 1 for estimatig F 1 (r, s). Figure 3 shows that for k = 2 (two-dimesioal distributio fuctio with lag two): a) Whe is small ( = 20) ad ρ = 0.1, we have over estimate for large values of r ad s (that is r, s [0, 4], approximately). b) MSD of kerel estimator is less tha histogram estimator for all cases. c) Whe is large ( = 100), the differece betwee kerel ad histogram estimators is very small. d) Whe is small ( = 20) or large ( = 100), the badwidth rate = log 1 () has a better role tha = 1 for estimatig F 2 (r, s), approximately. Refereces [1] Alam, K. ad Saxea, K.M.L. (1981). Positive depedece i multivariate distributio. Comm. Stat-Theor. Meth., A 10, [2] Azevedo, C. ad Oliveira, P.E. (2000). Kerel-type estimatio of bivariate distributio fuctio for associated radom variables. New Treds i Probability ad Statistics, [3] Dosker, M.D. (1951). A ivariace priciple for certai probability limit theorems. Mem. Amer. Math. Soc., 6, [4] Dubhashi, D., Priebe, V. ad Raja, D. (1996). Negative depedece through the FKG iequality. Basic Research i Computer Sciece, RS-96-27, [5] Ebrahimi, N. ad Ghosh, M. (1981). Multivariate egative depedece. Comm. Stat-Theor. Meth., 4, [6] Heriques, C. ad Oliveira, P.E. (2003). Estimatio of a two dimesioal distributio fuctio uder associatio. J. Statist. Plaig If., 113, [7] Heriques, C. ad Oliveira, P.E. (2005). Strog covergece rates for the estimatio of a covariace operator for associated samples. Preprit, Pre-Publicacoes do Departameto de Matematica da Uiversidade de Coimbra [8] Jabbari, H. (2009). Almost sure covergece of kerel bivariate distributio fuctio estimator uder egative associatio. Joural of Statistical Research of Ira, 6, [9] Jabbari, H. (2013). O almost sure covergece for weighted sums of pairwise egatively quadrat depedet radom variables. Stat. Papers, 54(3),

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