WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY FROM OBSERVATIONS OF MIXTURES WITH VARYING MIXING PROPORTIONS. B. L. S.

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1 Indian J. Pure App. Math., 41(1): , February 2010 c Indian Nationa Science Academy WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY FROM OBSERVATIONS OF MIXTURES WITH VARYING MIXING PROPORTIONS B. L. S. Prakasa Rao Department of Mathematics and Statistics, University of Hyderabad, Hyderabad , India e-mai: bsprao@gmai.com Abstract A waveet based inear estimator is proposed for the derivatives of a probabiity density function based on a sampe from a finite mixture of components with varying mixing proportions. It extends the inear estimator of a probabiity density function proposed by Pokhy ko (Theor. Probabiity and Math. Statist, 70 (2005) ). Upper bounds on L 2 and L osses are obtained for such estimators. Key words Estimation of derivatives of a density function, waveets, mixtures of components, varying mixing proportions. 1. Introduction The probem of anaysis of mixtures with varying mixing proportions occur in the study of medica, bioogica, socia and other types of data. The obects of observation J 1,..., J N may beong to any one of M popuations. Let I(J ) denote the indicator of the popuation that contains the obect J. For every obect J, we observe a random variabe X based on the obect J. Note that the distribution function of the random variabe X depends on the indicator I(J ). Suppose that P (X x I(J ) = k) = H k (x), 1 N, 1 k M. Suppose the distribution functions H k (x), 1 k M are unknown and the sequence I(J ), 1 N are aso unknown but we know the probabiity w k () that an obect J beongs to the k-th popuation, that is, w k () = P (I(J ) = k), 1 N, 1 k M. Note that w k () 0, 1 k M and M k=1 w k() = 1, 1 N.

2 276 B. L. S. PRAKASA RAO Observe that the probabiity w k () indicates the mixing proportion of the obects of k-th popuation in the mixture from which the obect J is chosen. It is easy to check that M P (X x) = w ()H (x), 1 N. (1.1) =1 The probem of estimation of the distribution function H (x) was studied in Maiboroda [10] using a weighted empirica distribution function. Maiboroda [11] obtained a generaized version of the Komogorov-Smirnov test for testimg the hypothesis for the homogenity of mixtures with varying mixing proportions. Assuming that the distribution functions H (x) are absoutey continuous with density functions h (x), Pokhy ko [14] constructed inear and adaptive waveet estimators for the density function h (x). Methods of nonparametric estimation of a density function and regression function are widey discussed in the iterature (cf. Prakasa Rao [15, 17]). It is known that the estimation of derivatives of a density as we as that of regression function are aso of importance and interest to detect possibe bumps in the case of a density and to detect concavity or convexity properties in the case of regression function. Asymptotic properties of the kerne type estimators for the derivatives of density have been investigated earier (cf. Prakasa Rao [15], p.237). Our aim in this paper is to discuss waveet inear estimators for the derivatives of a probabiity density function when the sampe of observations come from a mixture of severa components with varying mixing proportions. We propose an estimator for the derivative of the density based on waveets and obtain upper bounds on the L 2 and L osses for the proposed estimator. Estimators of density using waveets was studied for independent and identicay distributed random variabes in Antoniadis et a. [1], for some stationary dependent random variabes in Lebanc [9] and for stationary associated sequnces in Prakasa Rao [19]. Chaubey et a. [2, 3] extended these resuts to derivatives of density estimators for associated sequences and for negativey associated processes. The advantages and disadvantages of the use of waveet based probabiity density estimators are discussed in Water and Ghorai [24] in the case of independent and identicay distributed observations. The same comments continue to hod in this case. However it was shown in Prakasa Rao [16, 18] that one can obtain precise imits on the asymptotic mean squared error for a waveet based inear estimator for the density function and its derivatives as we as some other functionas of the density. Tribouey [21] studied estimation of mutivariate densities using waveet methods. Donoho et a. [6] investigated density estimation by waveet threshoding. For a discussion on statistica modeing by waveets, see Vidakovic [23]. 2. Preiminaries on Waveets A waveet system is an infinite coection of transated and scaed versions of functions φ( ) and ψ( ) caed the scaing function and the primary waveet function respectivey. In the foowing discussion, we assume that φ( ) is rea-vaued. The

3 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 277 function φ(x) is a soution of the equation with φ(x) = k= and the function ψ(x) is defined by ψ(x) = k= C k φ(2x k), (2.1) φ(x)dx = 1, (2.2) ( 1) k C k+1 φ(2x k). (2.3) The choice of the sequence {C k } determines the waveet system. It is easy to see that Define k= C k = 2. (2.4) φ k (x) = 2 /2 φ(2 x k), <, k < (2.5) and ψ k (x) = 2 /2 ψ(2 x k), <, k <. (2.6) Suppose the coefficients {C k } satisfy the condition k= C k C k+2 = 2 if = 0 (2.7) = 0 if 0. It is known that, under some additiona conditions on φ( ), the coection {ψ,k, <, k < } is an orthonorma basis for L 2 (R), and {φ,k, < k < } is an orthonorma system in L 2 (R), for each < < (cf. Daubechies [4]). Definition 2.1. The scaing function φ is said to be r-reguar for an integer r 1, if for every nonnegative integer r, and for any integer k 1, φ () (x) c k (1 + x ) k, < x < (2.8) for some c k 0 depending ony on k. Here φ () ( ) denotes the -th derivative of φ( ).

4 278 B. L. S. PRAKASA RAO Definition 2.2. A mutiresoution anaysis of L 2 (R) consists of an increasing sequence of cosed subspaces {V } of L 2 (R) such that (i) = V = {0} ; (ii) = V = L 2 (R); (iii) there is a scaing function φ V 0 such that {φ(x k), < k < } is an orthonorma basis for V 0 ; (iv) for a h( ) L 2 (R), < k <, h(x) V 0 h(x k) V 0 ; and (v) h( ) V h(2x) V +1. Maat [12] has shown that, given any mutiresoution anaysis, it is possibe to find a function ψ( ) (caed the primary waveet function) such that, for any fixed, < <, the famiy {ψ,k, < k < } is an orthonorma basis of the orthogona compement W of V in V +1 so that {ψ,k, <, k < } is an orthonorma basis of L 2 (R) (cf. Daubechies [4]). When the scaing function φ( ) is r-reguar, the corresponding mutiresoution anaysis is said to be r-reguar. Let f L 2 (R). The function f can be expanded in the form (cf. Daubechies [5]): f = k= = P s f + a s,k φ s,k + D f =s =s k= b,k ψ,k (2.9) for any integer < s <. Observe that the waveet coefficients are given by and a s,k = f(x)φ s,k (x)dx (2.10) b,k = f(x)ψ,k (x)dx. (2.11) Suppose that the functions φ and ψ beong to C r, the space of functions with r continuous derivatives for some r 1, and have compact support contained in an interva [ δ, δ] for some δ > 0. It foows, from the Coroary in Daubechies [4], that the function ψ( ) is orthogona to poynomias of degree ess than or equa to r. In particuar ψ(x)x dx = 0, = 0, 1,..., r. The above introduction to waveets is based on Antoniades et a. [1]. For a detaied discussion, see Daubechies [5]. For a brief survey on waveets, see Strang [20].

5 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY Introduction to Soboev Spaces and Besov Spaces Let f be a function defined on the rea ine which is integrabe on every bounded interva. It is said to be weaky differentiabe if there exists a function g defined on the rea ine which is integrabe on every bounded interva such that y x g(u)du = f(y) f(x). The function g is defined amost everywhere and is caed the weak derivative of f (cf. Harde et a. [8]). It is known that, if f is weaky differentiabe with weak derivative g, then f(u)φ (u)du = g(u)φ(u)du for any φ D(R) where D(R) denotes the space of infinitey differentiabe functions, on the rea ine R, with compact support. Definition 3.1. Let 1 p and m 0 be an integer. A function f L p (R) beongs to the Soboev space Wp m (R), if it is m-times weaky differentiabe and f (m) L p (R). In particuar Wp 0 (R) = L p (R). The space Wp m (R) is equipped with the norm f W m p = f p + f (m) p where f p denotes the norm for L p (R). Let W m p (R) = W m p (R) if 1 p < and W m (R) = {f : f W m (R) : f (m) uniformy continuous }. Note that W 0 p (R) = L p (R), 1 p <. Let f L p (R) for some 1 p. Let ( h f)(x) = f(x h) f(x) and define 2 h f = h h f. For t 0, define and ωp(f, 1 t) = sup h f p h t ωp(f, 2 t) = sup 2 h f p. h t Let 1 q. Suppose there exists a function ɛ(t) on [0, ) such that ɛ q < where ɛ q = ( 0 = ess sup t t 1 ɛ(t) q dt) 1/q, if 1 q < ɛ(t), if q =. (3.1) Definition 3.2. Let 1 p, q and s = n + α where n 0 is an integer and 0 < α 1. The Besov space B s p,q is the space of a functions f such that f W n p (R) and ω 2 p(f (n), t) = ɛ(t)t α where ɛ q <.

6 280 B. L. S. PRAKASA RAO For properties of Besov spaces, see Meyer [13] and Triebe [22] (cf. Lebanc [9], Harde et a. [8]). Suppose that the function f beongs to the Besov cass F s,p,q (L) = {f B s p,q, f B s p,q L} for some 0 < s < r + 1, p 1 and q 1, where f B s p,q = P 0 f p + [ 0( D f p 2 s ) q ] 1/q. Given a doube indexed sequence {γ,k } define the norm γ,. p = ( k γ p,k )1/p. (3.2) In view of the representation (2.9), it be can shown that the function f B s p,q if and ony if a s,. p <, and ( s[ b s,. p 2 (s+(1/2) (1/p)) ] q ) 1/q <. (3.3) Let φ( ) be a scaing function as defined earier. Define θ φ (x) = k= Suppose the foowing conditions hod: (C1) The ess sup x θ φ (x) < where φ(x k). ess sup g(x) = inf{y : λ([x : g(x) > y]) = 0} x and λ is the Lebesgue measure on the rea ine. (C2) There exists a bounded nondecreasing function Φ( ) such that φ(u) Φ( u ) amost every where and for some integer r 0. 0 u r Φ( u )du <. Lemma 3.1. Suppose that the scae function φ( ) is such that the coection {φ(x k), < k < } is an orthonorma system in L 2 (R) and the spaces V, < < are nested. Further suppose that the function φ satisfies the condition (C2) and it is r + 1 times weaky differentiabe. If φ (r+1) satisfies the condition (C1), then the norm. B s p,q is equivaent to the norm. Bp,q s in the space of the waveet coefficients for a s, p, q such that 0 < s < r + 1 and 1 p, q where f Bp,q s = a 0 p + ( (2 (s+(1/2) (1/p)) b p ) q ) 1/q. =0

7 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 281 (Here a 0 p denotes [ k= a 0,k p ] 1/p and b p denotes [ k= b,k p ] 1/p ). For a proof of Lemma 3.1, see Theorem 9.6 in Harde et a. [8], p Estimation of the d-th Derivative of a Probabiity Density Function Let {Y i, 1 i n} be independent and identicay distributed random variabes with probabiity density function f which is d-times differentiabe. Suppose that f (d) is bounded and has compact support. Suppose that f (d) L 2 (R). Let us first consider the case d = 0. The probem now is the estimation of the probabiity density function f. A waveet based density estimator of the density function f can be motivated in the foowing way from the expansion given in (2.9) (cf. Prakasa Rao [19]). We can estimate f(x) by f(x) where where f(x) = k N s α s,k φ s,k (x), (4.1) α s,k = 1 n n φ s,k (Y i ). (4.2) i=1 Here N s is the set of integers k such that supp(f) supp(φ s,k ) is nonempty. Since the functions f and φ have compact supports, the cardinaity of the set N s is finite and it is of the order O(2 s ). Let us now consider the probem of estimation of the derivative f (d) of f. As in Prakasa Rao [16], we assume that the scaing function φ( ) generates a r-reguar mutiresoution anaysis for some r (d + 1) and that there exists C m 0 and β m 0 such that f (m) (x) C m (1 + x ) β m, 0 m r. (4.3) This assumption impies that that the derivative φ (d) is bounded for every d 0 (cf. Prakasa Rao [16]). Furthermore the proection of f (d) on V s is f s (d) (x) = a s,k φ s,k (x), (4.4) k N s where a s,k = ( 1) d f(x)φ (d) s,k (x)dx. The equation given above can be ustified by using integration by parts since the function φ( ) is r-reguar (cf. Prakasa Rao [16]). This expression motivates the foowing estimator for f (d) (x) : f (d) s (x) = k N s â s,k φ s,k (x), (4.5)

8 282 B. L. S. PRAKASA RAO where â s,k = ( 1)d n n i=1 φ (d) s,k (Y i). Note that the estimator defined above reduces to the density estimator given in (4.1) (d) for d = 0. We now rewrite the expression for the estimator f s (x) in a sighty different form. Note that f s (d) (x) = â s,k φ s,k (x) k N s where and = n [ ( 1)d n k N s i=1 n = ( 1)d n = ( 1)d n = ( 1)d n = ( 1)d n = ( 1)d n φ (d) i=1 k N s n i=1 φ (d) s,k (Y i)]φ s,k (x) s,k (Y i)φ s,k (x) 2 (s/2)+ds φ (d) (2 s Y i k)2 s/2 φ(2 s x k) k N s n [ φ (d) (2 s Y i k)φ(2 s x k)]2 s+ds k N s n K (d) (2 s Y i, 2 s x)2 s+ds i=1 i=1 n i=1 K (d) s (Y i, x), (4.6) K s (x, y) = 2 s K(2 s x, 2 s y) K(x, y) = k N s φ(x k)φ(y k). Here K (d) s (x, y) denotes the d-th partia derivative of K s (x, y) with respect to x. 5. Estimation of the d-th Derivative of a Component Probabiity Density Function from a Mixture with Varying Mixing Proportions Let {X i, 1 i N} be random variabes as described Section 1. The probem is to estimate the d-th derivative of the component probabiity density function h (x) corresponding to the mixing proportion w ( ) based on the observations X 1,..., X N. Note that the probabiity density function of X is given by p (x) = M w ()h (x) =1

9 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 283 for 1 N. For x, y R N, define the inner product < x, y > N by the reation < x, y > N = 1 N N x k y k, whenever x = (x 1,..., x N ) and y = (y 1,..., y N ). Let w k = (w k (1),..., w k (N)). Suppose that the vectors w k, 1 k M are ineary independent in R N. Then it foows that the matrix Γ N = ((< w k, w > N )) is nonsinguar and det(γ N ) > 0. Let a = (a (1),..., a (N)) be a vector such that (i) < a, w k >= δ k, 1 k, N; and k=1 (ii) < a, a >= 1 N N =1 a () 2 is minimum. Here δ k is the Kronecker deta function. By using Lagrange mutipiers, it can be checked that a () = 1 det(γ N ) N ( 1) +k γk N w k(), (5.1) k=1 where γ N k denotes the determinant of the minor (, k) of the matrix Γ N. We now construct the waveet inear estimator, for the d-th derivative of the density h (x) of the -th component, at resoution eve s. It is defined by ] s (x) = ( 1)d N N =1 a ()K (d) s (X, x). (5.2) We now study the properties of the estimator ] s (x). For d = 0, it can be shown that this estimator is essentiay the same as the density estimator studied in Pokhy ko [14]. 6. Properties of the Estimator ] s (x) Lemma 6.1. Let P r Vs g g s denote the proection of a function g L 2 (R) on the space V s. The estimator ] s (x) is an unbiased estimator of ] s (x). Since X is a random variabe with the density function p (x) = M =1 w ()h (x), it foows that E( ] s (x)) = ( 1)d N N =1 a ()E[K (d) s (X, x)]. (6.1) Note that E[( 1) d K (d) s (X, x)] = ( 1) d K (d) (u, x)p (u)du s

10 284 B. L. S. PRAKASA RAO = = = = M ( 1) d K (d) (u, x)[ w ()h (u)]du M w ()[ =1 M =1 M =1 s w ()[ w ()P r Vs h (d) (x) =1 =1 M = P r Vs [ w ()h (d) ](x) ( 1) d K (d) (u, x)h (u)du] s K s (u, x)h (d) (u)du] = P r Vs p (d) (x) = [p (d) ] s (x). (6.2) The second equaity foows by integration by parts and the ast three equaities use the fact that proection and derivative are inear operators. Therefore E( ] s (x)) = 1 N = 1 N = M N =1 =1 a ()[p (d) ] s (x) N M a ()( w k () k ] s(x)) k k=1 k=1 ] s(x) < a, w k > N = ] s (x). (6.3) Lemma 6.2. Suppose the scaing function φ( ) has the property that the function K(x, y) is d-times differentiabe with respect to x and there exists F L 2 (R) such that K(x, y) F (x y). Suppose that Z is a random variabe with density function h L 2 (R) which is d-times differentiabe and h (d) L 2 (R). Then E ( 1) d K s (d) (Z, ) ] s ( ) ds+2s F 2 (u)du. (6.4) Proof : Observe that E[( 1) d K s (d) (Z, x)] = ] s (x)

11 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 285 and where Since E ( 1) d K s (d) (Z,.) ] s ( ) 2 2 = E( = it is easy to see that which impies that Therefore Hence ( 1) d K (d) s (Z, x) E[( 1) d K (d) (Z, x)] 2 dx) E[Y 2 (x)]dx, (6.5) Y (x) = ( 1) d K (d) s (Z, x) E[( 1) d K (d) (Z, x)]. K s (x, y) = 2 s K(2 s x, 2 s y), K (d) s (x, y) = 2 ds+s K(2 s x, 2 s y) K s (d) (x, y) 2 ds+s F (2 s x 2 s y). E[Y 2 (x)] E([( 1) d K s (d) (Z, x)]] 2 ) 2 2ds+2s F 2 (2 s x 2 s y)h(y)dy. (6.6) E ( 1) d K s (d) (Z,.) ] s ( ) ds+2s [ 2 2ds+2s [ s s F 2 (2 s x 2 s y)h(y)dy]dx F 2 (2 s x 2 s y)dx]h(y)dy = 2 2ds+s F 2 (v)dv. (6.7) Lemma 6.3. Suppose the scaar function φ satisfies the conditions stated in Lemma 6.2. In addition, suppose that the functions h (d) k L 2 (R), 1 k M. Define the estimator ] s (x) of ] s (x) as given in (5.2) for 1 M. Then E ( ] s (x) ] s (x)) 2 dx 1 N < a, a ) N 2 2ds+s F 2 (v)dv (6.8) for 1 M and for a s 0.

12 286 B. L. S. PRAKASA RAO Proof : Let Y (x) = ( 1) d K s (d) (X, x) [p (d) ] s (x), 1 N. Then the random variabes Y (x), 1 N are independent with mean zero for every x. Appying Lemmas 6.1 and 6.2, we get that E ( ] s (x) ] s (x)) 2 dx = E[ 1 N 2 N =1 a 2 ()Y 2 (x)dx] 1 N < a, a > N 2 2ds+s F 2 (v)dv. We now obtain bounds on the mean integrated squared error ]] 2 2). (6.9) Theorem 6.1. Suppose the conditions stated in Lemma 6.2 and Lemma 6.3 hod for some r 0.. Suppose that the functions h (d) k F s,p,q (L), 1 k M for some L > 0. Further suppose that s (0, r + 1) and 1 q 2. Then there exists a constant C = C(q, s, L) > 0 such that for a = 1,..., M, h (d) 2 2 2d+ 2) C[(< a, a > N N + 2 2s ]. (6.10) Proof. For any function g L 2 (R), et P r V g denote the proection of g onto the subspace V. Since and P r V h (d) E( V, it foows that h (d) 2 2) = (x)) = (x) = P r V h (d) (x), E( ) 2 2) + E( ) h (d) 2 2. We now obtain a bound on the second term on the right side of the above equation by appying the Lemmas 3.1 and 6.1 to 6.3. Note that the second term on the right side of the above equation is given by E( ) h (d) 2 2 = h (d) = b i,k 2 i= k= s 2 2is b i 2 2 i= 2 2s ( h (d) B s 2,2 )2 (6.11)

13 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 287 since B2,q s Bs 2,2 (cf. Harde et a. [8], p.124). Here b i,k denote the waveet coefficients of the function h (d) in the space orthogona to V. Furthermore, for a L > 0 and a q such that 1 q 2, there exists a constant C 1 = C 1 (s, q, L) such that for a h (d) F s,2,q (L), h (d) B s 2,2 C 1. Hence E h (d) 2 2 C 1 2 2s. (6.12) We now obtain an upper bound on the first term Appying Lemma 6.3, we get that E 2 2). E ] 2 2) 1 N < a, a ) N 2 2d+ F 2 (v)dv (6.13) = C 2 < a, a ) N 2 2d+ N. Combining the reations (6.12) and (6.13), we get that there exists a constant C > 0 such that h (d) 2 2 2d+ 2) C[< a, a > N N + 2 2s ]. (6.14) Remark 6.1. If the integer is chosen so that 2 N 1/(2s+2d+1), then the bound on the right side wi be minimum and is of the order N 2s/(2d+2s+1) [< a, a > N +1]. This resuts extend Theorem 1 in Pokhy ko [14] on the L 2 -oss in the probem of density estimation to the estimation of the d-th derivative of a density function constructed from observations of a mixture with varying mixing proportions. The next resut deas with L -oss of the estimator as an estimator of h (d). We first state a emma due to Pokhy ko [14]. Lemma 6.4. Suppose that a function g C 1 (R) with g <, g 2 < and g <, where g denotes the derivative of the function g. Then g (4 g g 2 2) 1/3. Theorem 6.2. Suppose the conditions stated in Lemma 3.1 and Lemma 6.3 hod for some r 0. Suppose that the functions h (d) k W r+1, 1 k M for some r d and there exists γ > 0 such that h (d) k W r+1 γ. Further suppose that

14 288 B. L. S. PRAKASA RAO the scaing function φ C r with compact support. Then there exists a constant C = C(r, γ) > 0 such that for a = 1,..., M, Proof : Since it foows that ]] ) C[(< a, a > N ) E( 1/2 2d+ (x)) = (x) = P r V h (d) (x), N 1/3 + 2 (r+1) ]. (6.15) h (d) ) = E E( ) ) + E( ) h (d) + h (d). (6.16) Since φ C r (R) for some r 1 with a compact support, the condition (C3) hods for a r 1. Furthermore φ W (R). r Hence there exists a constant C = C(γ) > 0 depending on γ such that h (d) C(γ)2 (r+1), (6.17) whenever h (d) W r+1 (R) with h (d) γ. This foows from Theorem 8.1(ii), Lemma 8.6 and Theorem 8.2 in Harde et a. [8]. Since φ C r (R), r d with a compact support, the function K(x, y) is a uniformy bounded and continuousy d-times differentiabe function with respect to x and there exists a constant C > 0 such that K (d) (x, y) C2 d+. Therefore E(( 1) d K (d) (X i, x)) C2 d+. Then Let Y d, (x) = Y d, (x) = 1 N N i=1 (x) E( a (i)(( 1) d K (d) 2 N N C2d+ a () =1 (x)). (X i, x) [p (d) (x)) 2C2 d+ [< a, a > N ] 1/2. (6.18) i The same argument shows that Y d 1, C2 d N =1 a () <.

15 WAVELET LINEAR ESTIMATION FOR DERIVATIVES OF A DENSITY 289 Appying Lemma 6.4, we get that Y d, [C2 d+ (< a, a > N ) 1/2 Y d, 2 2] 1/3 for a X i, 1 i N. Observe that K (d) (x, y) C2 d+ for some constant C > 0 and K (d) (x, y) = 0 if x y > L = 2 diam(supp φ). Let F (x) = C2 d+ I [0,L] ( x ) where I(A) denotes the indicator function of the set A. Then F ( ) L 2 (R). Appying arguments simiar to those in Lemma 6.3, we get that E E ] [C2 d+ (< a, a > N >) 1/2 2 2d+2 < a, a > N > F 2 (x)dx] 1/3 C 2d+ N 1/3 (< a, a > N ) 1/2 (6.19) for some constant C > 0 independent of and d. Combining the inequaities (6.17) and (6.19), we get that there exists a constant C > 0 such that E[ h (d) ] C[2 (r+1) + 2d+ N 1/3 (< a, a > N ) 1/2 ]. (6.20) Remark 6.2. If the integer is chosen so that 2 N 1/[3(r+d+2)], then the bound on the right side wi be minimum and is of the order N 2(r+1)/[3(r+d+2)] {[< a, a > N ] 1/2 + 1}. This resut extends Theorem 2 in Pokhy ko [14] on the L -oss in the probem of density estimation to the estimation of the d-th derivative of a density function constructed from observations of a mixture with varying mixing proportions. Remark 6.3. Let p max(p, 2). Foowing the techniques in Lebanc [9] and Prakasa Rao [19], one can get bounds on the L p -oss h (d) 2 p ), whenever h k F s,p,q (L), 1 k M by noting that and E[ h (d) 2 p ] 2[ E( E 2 p ) + E( ) h (d) 2 p ] ) h (d) 2 p C 12 2s for some positive constant C 1 whenever s 1 p and s = s + 1 p 1 p (cf. Lebanc [9], p.83). If 1 p 2, then a bound on the L p -oss h (d) p p )

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