WAVELET-BASED ESTIMATORS OF THE INTEGRATED SQUARED DENSITY DERIVATIVES FOR MIXING SEQUENCES

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1 Pa. J. Statist. 009 Vol. 5(3), WAVELET-BASED ESTIMATORS OF THE INTEGRATED SQUARED DENSITY DERIVATIVES FOR MIXING SEQUENCES N. Hosseinioun, H. Doosti an H.A. Nirouan 3 Departent of Statistics, School of Matheatical Sciences, Ferowsi University of Mashha, Iran Eail: na_ho8@stu-ath.u.ac.ir; oosti@ath.u.ac.ir; 3 niruan@ath.u.ac.ir ABSTRACT The proble of estiation of the square erivative of a probability ensity f is consiere using wavelet orthogonal bases. We obtain the precise asyptotic expression for the ean integrate error of the wavelet estiators when the process is strongly ixing. We show that the propose estiator attains the sae rate as when the observations are inepenent. Certain wee epenence conitions are ipose to the { X i } efine in {, N, P}. an KEYWORDS Nonparaetric estiation of a ensity; Wavelet; Mixing process. The otivation for estiation f. INTRODUCTION I ( f ) = f ( x) x, where f is a probability ensity is the -th erivative is well nown. Kernel-type estiation for the functional I ( f ) has been investigate by Hall an Marron (987), Rao (997) an Bicel an Ritov (988) aong others. In Praasa Roa (996), we have stuie nonparaetric estiation of the erivative of a ensity by wavelets an a precise asyptotic expression for the ean integrate square error, following techniques of Masry (99). Praasa Roa (999) also obtaine the precise asyptotic expression integrate square error of the wavelet estiators. We now exten the result to the case of strongly ixing process. We show that the propose estiator attains the sae rate as when the observations are inepenent. Certain wee epenence conitions are ipose to the { X i } efine in {, N, P}. Let N enote the -algebra generate by events { X A,..., X A }. We consier the following classical ixing conitions:. Uniforly strong ixing (u.s..), also calle ixing : p( AB) p( A) p( B) sup sup = ( s ) 0 as s. A N B N p( A) s, 009 Paistan Journal of Statistics 3

2 3 Wavelet-base estiators of the integrate square ensity. -ixing: sup corr( X, Y ) = ( s) 0 as s X L N Y L N s sup,. A very well nown easure of epenence in probabilistic literature is escribe by the ixing conitions. Aong various ixing conitions use in the literature, -ixing is reasonably wea, an has any practical applications. Many stochastic processes an tie series are nown to be ixing. Uner certain wea assuptions autoregressive an ore generally bilinear tie series oels are strongly ixing with exponential ixing coefficients. The proble of ensity estiation fro epenent saples is often consiere. For instance quaratic losses were consiere by Ango Nze an Douhan (993). Bosq (995), an Douhan an Loen (990). Linear wavelet estiators were also use in context: Douhan (998) an Douhan an Loen (990). Leblance (99,996) also establishe that the L -loss ( p < ) of the linear wavelet ensity estiators for a stochastic process converges at the rate of f belongs to the Besov space B, erivative of a ensity. p s p q N s (s) ( s = s / p / p), when the ensity. Doosti et.al (006) extene the above result for. DISCUSSION OF THEOREM'S ASSUMPTIONS Consier the following conitions: C : The istribution of ( Xi, X ) has a oint ensity fi, such that for all i an, v / v i ( fi, ( x, y) xy) = fi, (.,.) Fv < for soe v > M : The process is -ixing an t= ( t ) R <. M : The process is -ixing an / = ( t) <. t / Since the inequality ( t) ( t) hols (see Douhan (99)), M iplies M. Also note that if X an Y are rano variables, then the following covariance inequalities hol.(see Douhan (99), section..) cov( X, Y ) ( i) X. Y, (.) i / p i p q cov( X, Y ) ( i) X. Y, for any p, q an / p / q =.

3 Hosseinioun, Doosti an Niruan INTRODUCTION TO WAVELET A wavelet syste is an infinite collection of translate an scale versions of functions an calle the scaling function an the priary wavelet function respectively. The function ( x) is a solution of the equation with ( x) = C ( x ) = ( x) x = an the function ( x) is efine by ( x) = ( ) C ( x ). Note that the choice of the sequence see that C eterines the wavelet syste. It is easy to Define an C =. = /, ( x ) = ( x ), <, < (3.) /, ( x ) = ( x ). <, <. Suppose that the coefficients C satisfy the conition C C K l = if l = 0 = 0. if l 0. It is nown that, uner soe aitional conition on {, <, < } is an orthonoral basis for, orthonoral syste in, the collection L ( R) an {,, < < } is an L ( R) for each < < (cf. Doubachies (99)). Definition 3.. A scaling function non-negative integer l ( r) c is sai to be r-regular for an integer r r an for any integer, if for every ( l) ( x) c ( x ), < x < for soe c 0 epening only on where ( l ) (.) enotes the l-th erivative of.

4 3 Wavelet-base estiators of the integrate square ensity Definition 3.. A ultiresolution analysis of subspaces V of V i) = = {0}; L ( R) such that L ( R) contains of increasing sequences of close ii) = V = L ( R); iii) there is a scaling function V0 ( x ), < < such that is an orthonoral basis for V 0 ; an for all iv) For all < <, h( x) V0 h( x ) V0 v) h( x) V h( x) V. h L ( R), Let H enote the space of all functions g(.) in erivatives are absolutely continuous an efine the nor L ( R) whose first ( S ) H ( ) / g = [ g ( t ) t ]. Lea 3.. (Mallat (989)) Let a ultiresolution analysis be r-regular. Then for every 0 < s < r, any function g L ( R) belongs to H iff sl et e t= <, where l = e g g an gl is the orthogonal proection of g on V t. l Rears. The above introuction is base on Antoniais (99). For a etaile introuction to wavelet, see Chui (99) or Daubechies (99). For a brief survey, see Strang (989).. ESTIMATION BY THE METHODS OF WAVELETS Suppose X,... X n is a ixing, ientically istribute rano variables with ensity f, f interpret (0) f is -ties ifferentiable an f as f. The proble of interest is the estiation of enotes the -th erivative of f. We

5 Hosseinioun, Doosti an Niruan 35 I ( f ) = f ( x) x. Assue that f L ( R) an there exist D 0, 0 such that ( ) where >. f ( x) D x for x, 0, Consier a ulitiresolution as iscusse in Section 3. Let be the corresponing scaling function. Suppose that the ultiresolution is r-regular for soe r efinition, ( r) C, an its erivative ( ) for every integer, there exists a constant A > 0 such that. Then by up to orer r are rapily ecreasing i.e., Let Then an ( ) A ( x), 0 r. ( x ) l/ l l, = ( x ), <, t <. ( ) l / l ( ) l l, = ( x ), 0 r ( l / ) l ( ) A l, ( x). 0 r. (.) ( x ) If, then it is clear that ( ) ( ) li l, x f ( x) = 0, 0, for any fixe l an. Let fl is the orthogonal proection of f on V l. Note that where f ( x) = a ( x), l l, l = al = f ( u) l, ( u) u l, = ( ) f ( u) ( u) u. (.) by (3.) for. Clearly the equation (.) hols for = 0. Hence for all 0 a = ( ) E ( X ) l Further ore l,.

6 36 Wavelet-base estiators of the integrate square ensity l l l = e f f = f a 0 as l, by the properties of ultiresolution ecoposition. Hence Note that p / p g = g x, p. p Let I ( f ) = f. K f ( x) = a ( x), K, l, l l, = K where K = K n is a sequence of positive integers epening on l = l n tening to infinity as n an l l as n. Note that f,, ( x) is a truncate proection of f = n on V t. Given a saple X,... X n, let A = ( x ) ( x ), n n l l i l n( n ) i= = an we estiate I ( f ) by K Iˆ ( f ) = A. (.3) Note that l = K K l an E( A ) = a l l E( Iˆ ( f )) = K a. = K l Suppose that as l n 5. MAIN RESULTS n {( ) 0 s}{ l n /( 0 )}} = log n. Define I ˆ ( f ) as an estiator of I ( f ) where I ˆ ( f ) is given by the equation (.3), then we have the following two results: Theore 5.. If { X n } satisfies the conition C, then n( n ) ( ) ˆ E I ( ) ( ) ( ) l ( ) f I f x x as n n.

7 Hosseinioun, Doosti an Niruan 37 Theore 5.. If { X n } satisfies the conition M, then Let n( n ) ( ) ˆ E I ( ) ( ) ( ) l ( ) f I f x x as n n. 6. PROOFS J = E Iˆ ( f ) I ( f ) = Var Iˆ ( f ) EIˆ ( f ) I ( f ) n ( ) = ( ˆ Var I ( f ) al ( ) n f x x ˆ = Var I ( f ) f, ln, f. Following along the lines of Roa (999), we get sl = n f, ln, f o Proof of Theore 5.. Observe that ˆ ( ) = l =, n ln ln. (6.) Var I f Var A cov A A, (6.) where cov( X, Y ) is interprete as var( X ). It is straightforwar to chec that EA l = n A ln ln i l i n ln ln ( ) E x n n x x x, (6.3) where the last suation runs over all i,, i,. Using (.) in (6.) leas to EA A ln ln / / ( ) ( ) ( i ) ( ) ( ) ( ) ( ) l x i f x i x i l x i f x i x i i n n n n ( n ) ( ) ( ) ( ) ( ) l i i n ln ( ) E x E n n x. (6.) i < Note that it suffices to boun the right-han sie of (6.3). By (.) an Masry (99), one ay easily get

8 38 Wavelet-base estiators of the integrate square ensity / / l ( i ) ( i ) ( i ) ( i ) ( i ) ( i ) n x f x x l n x f x x / / l ( ) ( ) n ln u ln ln v l i i n x f u ln x f v ln ln ( ) ( ) ( ) ( ) l ( ) n ln u = ( u) f u l n l ( ) n ln ln = ( u) u O( ). (6.5) By siilar arguent as in Rao (999), we get l ( ) ( ) n l i i n ln E ( x ) E ( x ) ( u ) u l ( ) n al naln O ln ( ). (6.6) Substituting (6.5) an (6.6) in (6.), one ay easily obtain ln ln ln EAl ( ) ( ) ( ) n Al n u u O n ( n ) n l ( ) ( ) n ( u) u a ( ) ln ( ) lnaln O n n ln ( ). Since ( ) < an al = () naln o, (Roa (999)), ln ( ) 3 EA A = O n ( u) u o() O(). (6.7) n ( n ) ln ( ) ln ln So we ay easily conclue n( n ) ˆ VarI ( ) ( f ) = O n l ( u ) u o () O n ln ( ). (6.8) Applying (6.8) in (6.), yiels the esire result. Proof of Theore 5.. Applying Holer inequality for v an v with / v / v =, one ay obtain

9 Hosseinioun, Doosti an Niruan 39 l ( i ) ( ) ( i, ) i n x ln x f x x x x So it is easy to obtain /v /v l ( ) ( ) n ln v v v l ( ) ( ) n ln F u u v v /v /v v v ln ln A A v v v F u v ( u) ( v). l ( i ) ( ) ( i, ) i n x ln x f x x x x /v /v l n l n v u v v v v F A u v u v /v /v l = n l n v u v Fv A u v v v u v F u v ( ) /v ( ) /v v v ln ln v u v v A u v v v / v / v ln ln v u v Fv A u v v v = F v ln v A l l v ( v) v n n = O( ) = o(). (6.9) Using (6.6), (6.9) an (6.) in (6.), conclue the result. ACKNOWLEDGEMENT The aurhors are grateful to the referees an Eitor for useful coents. REFERENCES. Ango Nze, P. an Douhan, P. (993). Functional estiation for tie series: a general approach. Pr epublication e I Universit e Paris-Sun No Antoniais, A. (99). Soothing noisy ata with coiflets. Statistica Sinica,,

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