Asymptotically Efficient Estimation of the Derivative of the Invariant Density

Transcription

1 Statistical Inerence or Stochastic Processes 6: 89 17, 3. 3 Kluwer cademic Publishers. Printed in the Netherlands. 89 symptotically Eicient Estimation o the Derivative o the Invariant Density NK S. DLLYN and YUY. KUTOYNTS Laboratoire de Statistique et Processus, Université du Maine, venue Messiaen, 785 Le Mans, Cedex 9, France. s: arnak.dalalyan.etu@univ-lemans.r, kutoyants@univ-lemans.r eceived in inal orm 17 pril 1; ccepted 3 May ) bstract. The problem o estimation o the derivative o the invariant density is considered or a one-dimensional ergodic diusion process. The lower minimax bound on the L -type risk o all estimators is proposed and an asymptotically eicient up to the constant) in the sense o this bound kernel-type estimator is constructed. MS 1999 Subject Classiication: 6M5, 6G7, 6G. Key words: ergodic diusion, invariant density, derivative estimation, asymptotical eiciency, Pinsker s constant. 1. Introduction In this work we consider a diusion process X given by the stochastic dierential equation dx t = SX t ) dt + dw t, X, t, 1) where W t denotes a standard Wiener process deined on a probability space,f, P) and the initial value X is a random variable independent o the Wiener process. We suppose that the ollowing condition is uliled: { x } GS) = exp Sv)dv dx <. ) By this condition the process 1) is recurrent positive, that is, there exists an invariant probability distribution with the density unction S y) = 1 { y } GS) exp Sv)dv and the process {X t,t } has ergodic properties 3, 5].

2 9 NK S. DLLYN ND YUY. KUTOYNTS The trend coeicient S ) is supposed to be unknown, so the invariant density S ) is unknown unction, too. The problem o estimation o this density by observations X T = {X t, t T } as T has been considered by many authors see 1,, 9, 1, 1] and reerences therein). Particularly, in 9] a lower bound on the minimax risk o all estimators is proposed and several estimators localtime, kernel-type, unbiased) attaining this bound in the limit are constructed. These estimators are called asymptotically eicient when the time o observation tends to ininity. It is also shown that they are T consistent and asymptotically normal. There are several open problems closely related with the results described above. First, to ind an estimator o the density which is asymptotically eicient in the second order, say, as it is done in distribution unction estimation problem in i.i.d. case by Golubev and Levit 8]. Second, to construct an asymptotically eicient estimator o the trend coeicient S ). Third, to ind an asymptotically eicient estimator o the derivative o the invariant density. One o the possible approaches in these problems is the so-called minimax approach that we develop in this paper. emark that there exists a close relation between the problems cited above. In act, the trend coeicient can be written as Sx) = S x) S x). It can be easily shown that this unction can be estimated with a rate depending on the smoothness o S ). Say, i the unction S ) is k-times continuously dierentiable, then the best possible rate has to be T k/k+1). Thereore, i one considers the estimator o Sx) constructed via some eicient estimators o S x) and S x), then, roughly speaking, the error o estimation o the density Sx) is asymptotically smaller than the error o S x) estimation. Thus, the main problem is to construct an asymptotically eicient estimator o the derivative S x). The present work is devoted to the problem o this derivative S ) estimation. similar problem was discussed by Lucas 11] in the case where X T is stationary and continuous but not necessarily a Markov process. The author obtained some conditions under which a T -convergent kernel-type estimator exists, but pointed out that the class o processes satisying these conditions is quite narrow. We present below two results concerning the same problem. The irst one is the lower minimax bound on the L -type risk o all estimators Section ). The second provides a kernel-type estimator, which is asymptotically eicient in the sense o this bound Section 3). We inish by concluding remarks where we propose some possible generalizations o these results Section 4). Note that the statement o the problem and the results obtained in this paper lie in the ramework o Pinsker s bound approach 14] see 13] or the discussion and reerences) and the schemes o the proos ollow the work by Golubev and Levit 8] see as well the paper by Schipper 15]). The main diiculty in our case, which does not exist in the case o density estimation by i.i.d. data, is the ollowing. When we take sup o the risk on the class o unctions S ) satisying condition ),

3 ESTIMTION OF THE DEIVTIVE 91 we touch the case GS) =, which corresponds to the null recurrent process, and in this case we have no more the law o large numbers. Thereore to prove the asymptotic eiciency o the estimators we need to control the divergence o this integral. It is done using three conditions 3) 5), given below. To ormulate the main results we need some notation. Let C T be the space equipped with the topology o the uniorm convergence) o all continuous unctions deined on,t] and let B T be the σ -algebra o its Borelian subsets. The process X T = {X t, t T } induces a probability measure on the space C T, B T ). This measure and the mathematical expectation with respect to this measure will be denoted by P S and E S, respectively. In the cases where the unction S ) depends on a parameter, we will write P and E instead o P S and E S. The set o all trend coeicients S ), which provide the weak) existence, uniqueness and ergodicity o X t see 3]) will be denoted by. We suppose that the initial value X has the density S ). Under this condition {X t,t} is a stationary process. We consider the nonparametric problem o S ) derivative unction estimation rom continuous path observations and give the exact asymptotics o the quadratic minimax risk. That is, we ind a constant C) such that lim in sup T k/k+1) E S ϑ T x) S x)) dx = C), T ϑ T S where the inimum is taken over the class o all possible estimators. To deine the set we need some additional notation. Let χ denote the indicatoroanevent and F S ) be the probability distribution o the random variable ξ with density unction S ),thatis, F S x) = Pξ x). Throughout this paper the constants do not depending on T and S) will be denoted by C. The parameter space is deined in the ollowing way. Let us ix a real number >and an integer k and deine the ellipsoid k,) = {S ) k+1) S ] } x) dx. We can deine now the ininite-dimensional parameter space as the set o all S k,), such that or a constant D does not depending on S) the ollowing conditions are uliled: 1. There exists apositive constant B does not depending on S) such that sup B S x) ] dx<d, 3) B>B sup B B>B x >B x <B S x) ] ES ξ χ {y>x} F S y) S y) dy] dx<d. 4)

4 9 NK S. DLLYN ND YUY. KUTOYNTS. The ollowing estimate holds: S x) ] ] χ{ξ>x} F S ξ) ES dx<d. 5) S ξ) O course, the set depends on the constants k,, D and B. In the sequel this set will be denoted by D.. Lower Bound In this section we will establish a lower bound on the minimax risk in sup T ϑ T, S ) = in sup E S ϑ T x) S x)) dx, 6) ϑ T S ϑ T S where the in is taken over all estimators o the derivative. THEOEM 1. The ollowing inequality is true lim lim in in sup T k/k+1) T ϑ T, S ) Pk, ), 7) D T ϑ T S D where the Pinsker constant Pk, ) is deined by ) πk + 1)k + 1) k/k+1) Pk, ) = k + 1) 1/k+1). 8) 4k Proo. To prove this inequality we minorate the minimax risk 6) by the minimax risk over a properly chosen parametric amily. The dimension o this amily depends on T and tends to ininity when T. Thus, let us introduce the ollowing parameterizations S x) = i e i x)u x ), x, ], 9) i L where = ln1 + T)and Ux) is k + 1 times dierentiable increasing unction vanishing or x and equal to 1 or x 1. The unctions {e i } i Z are the elements o the trigonometric basis o L, ], thatis, ) πix sin i i>, e i x) = 1 1 i i =, ) πix cos i i<. The positive number L = L T will be chosen later. In the outside o the interval, ] we put S x) = S x) = sgnx)k + ) x ) k+1.

5 ESTIMTION OF THE DEIVTIVE 93 It is evident that the unction S ) deined in this way is k times dierentiable and the kth derivative is continuous. Consequently, the density ) is k + 1 times continuously dierentiable. Since or any ixed, the unction S ) is locally Lipshitz and sup x xs x) x <, there exists a unique solution o 1) see 3]) corresponding to the case S = S. In the sequel we consider the parametric space Ɣ T o all sequences { i } i L such that i G σ i or all i Z such that i L.Here σ i = σ i,t = α ) k 1, i = 1) T i + with ) Tk + 1)k + 1) 1/k+1) α = α T = 11) 4kπ k and σ =. The integer L is chosen to be equal α]. LEMM 1. For any m =, 1,..., k, we have the ollowing estimates x m) sup S v) dv) C k+ T 1/4k+). 1) sup Ɣ T x,] Proo. For m =, using the inequality i G σ i and the deinitions 1) and 11), we obtain x S v) dv sup S x) sup i e i x) i x x i = i = = 4G L α ) k 1/ 1 4Gαk/)+1 Ck+ T i i=1 + T T. 1/4k+) emark that this chain o inequalities proves the bound 1) or m = 1 as well. Suppose that this bound holds or 1,,...,m 1 and prove it or m. Since all the derivatives o the unction U ) up to the order m are bounded, we have S m) x) 1 πi ) m C k+ i + T 1/4k+) i = C i m σ m i + i = C m 1 T α i=1 Cα k m 1 T + T Ck+ T 1/4k+) i k 1 α ) k/ C k+ + i T 1/4k+) Ck+ Ck+ 1/4k+) T 1/4k+).

6 94 NK S. DLLYN ND YUY. KUTOYNTS COOLLY 1. The ollowing relation holds x) = 1 + o T 1) { exp { x ) k+ } i x >, 1 i x. The proo is easy and immediately ollows rom Lemma 1. Let {ξ i } i Z be i.i.d. random variables with common probability density px) such that p ξ i <G, Eξ i ) =, Eξi ) = 1, I = x)] dx = 1 + ε, px) where ε wheng. We introduce a prior distribution on Ɣ T putting with i = σ i ε) ξ i σ i ε) = 1 + ε) T α1 ε) i k 1 or each i dierent rom. The coeicient will be deterministic and equal to. The Fisher inormation I i o this prior distribution with respect to i or i = ) is then equal to 1 + ε)/σ i ε). Since the minimax risk can be evaluated below by the Bayesian one with a correction term which is shown to be small in order), we seek now a lower bound o the Bayesian risk corresponding to a prior distribution, deinedby T ) = in ϑ T E ) + ϑ T x) x)) dx, where E is the mathematical expectation with respect to the probability measure P dx T ) d). Let ψ i, and ψ i,t be the Fourier coeicients on, ] o ) and ϑ T ), respectively, that is, ψ i, = x)e ix) dx, ψ i,t = ϑ T x)e i x) dx. Since {e i )} is an orthonormal sequence and the unction ) belongs to the Hilbert space generated by this sequence, using the Parseval s identity one has T ) in ϑ T E ϑ T x) x)) dx in ϑ T E ) ψ i,t ψ i,. i <L

7 ESTIMTION OF THE DEIVTIVE 95 By two-dimensional van Trees inequality see 4]) T ) 1 + o T 1) ) ]) E ψi, / i + ψ i, / i T E I i ) + I i ) ), 13) + I i + I i <i L where I i ) is deined by the ormula ] S I i ) = E ξ). i The elementary calculations show that x) = x) S x) + i x) E i Using Lemma 1, one can easily prove that x ] sup x) E x which gives us ξ S i v) dv x ξ ] S v) dv. i = o T 1), 14) x) = x)e i x)u x ) + o T 1). 15) i So the partial derivative o ψ,i with respect to i can be evaluated like ollows ψ i, i = = = e i x) i x) dx e i x)u x ) x) dx + 1 o T 1) e i x) x) dx + 1 o T 1). This equality and the elementary identity ei x) + e i x) = 1/ imply ψ i, i + ψ i, i = 1 + o T 1)). 16) It is not diicult to show that I i ) + I i ) = e i x) + e i x)) x) dx + O T 1) = 1 + o T 1). Now the inequality 13) can be rewritten as T ) o T 1) ) <i L σ i ε) Tσ i ε) ε). 17)

8 96 NK S. DLLYN ND YUY. KUTOYNTS Simple calculations show that the last sum is equal to 4kα1 ε) Tk + 1) 1 + o T 1)). eplacing α by expression 11) we get the inequality T ) T k/k+1) Pk,)1 + o T 1))1 ε) with ) πk + 1)k + 1) k/k+1) Pk,)= k + 1) 1/k+1). 18) 4k emark 1. It is important to emphasize that the values 1) and 11) are the solutions o the maximization problem related with the unctional y) = y i Ty i> i + over the set { Ek, ) = y = y i ) ) i> πi k y i }. i> This choice is based on relation 17) and Lemma. Now we prove some lemmas which will permit us to complete the proo o the lower bound. LEMM. There exists a constant D>such that the unctions { S ) } Ɣ T satisy conditions 3) 5), i T is suiciently large. Proo. We show irstly that there exists a constant D such that u) ] ] χ{ξ>u} F ξ) E du<d 19) ξ) sup Ɣ T or all T>T. emark that or z>y>the ollowing inequality is true z ) k+ y ) k+ z y)y ) k+1, it ollows that 1 F y) y) y This leads us to the inequality 1 F ξ) u)e ξ) exp { z y)y ) k+1} dz = ) χ {ξ>u}] du 1 y ) k+1. u) E χ {ξ>u} ξ ) k+ ] du u) u ) k du<1 )

9 ESTIMTION OF THE DEIVTIVE 97 or T large enough. In the same way, the inequality y ) k+ x ) k+1 y ) or y x, implies that Hence u) u F y) y) dy u u) dy y) 1 u ). k+1 u e u )k+1 y u) dy = C F ξ) u)e ξ) ) χ {<ξ<u}] du C u) u) 1 du u) du = C ) < 1 1) i T is large enough. Then, we have and 1 F y) y) dy 1 1 y) dy = o T 1) ) u)e F ξ) ξ) ) χ { 1<ξ<}] du 8 Proceeding like in the proo o ), one can show that Thus, 1 F y) dy y) sup F y) y 1 y) 1 = ) k+ u) du <C. ) ) F ξ) u)e χ {ξ< }] du 1 3) ξ) or T large enough. Combining the inequalities ) 3), we obtain ) χ{ξ>u} F ξ) ] u)e du C. 4) ξ) In the same way it can be shown that ) χ{ξ>u} F ξ) ] u)e du C. 5) ξ)

10 98 NK S. DLLYN ND YUY. KUTOYNTS For u, ], using Lemma 1 we have u) = o T 1) and ) χ{ξ>u} F ξ) ] E ξ) ) 1 F ξ) E χ {ξ>+1}] + ξ) ) 1 + E χ { 1 ξ +1}] + ξ) + E F ξ) ξ) ) χ {ξ< 1}] C. Hence ) χ{ξ>u} F ξ) ] u)e du C ξ) and the inequality 19) is proved. We pass now to the proo o the ollowing estimate u) ] E ξ χ {y>u} F y) y) dy] du<d. 6) To prove it, one has to consider the cases ξ<, ξ, ], ξ, u] and ξ > u. In the irst, third and ourth cases we obtain 6) proceeding like in the proo o 19). For the second one, we have u) ] E ξ C 4 u) ] ] χ {y>u} F y) dy χ { ξ } du y) ] 1 y) dy du u) ] du = D. 7) So we proved 6) which implies the estimate 4). It remains to check the condition 3). We suppose that >andb>. I >B/, then B x) ] dx x) ] dx = C D B.

11 ESTIMTION OF THE DEIVTIVE 99 For B/, we have B x) ] dx C x ) k+ e x )k+ dx C = D B B x ) k+3 e x )k+ dx B ) k+ y e y dy D B ) e y dy D e B ) <De B < D B. 8) This inequality completes the proo o Lemma. LEMM 3. The ollowing relation holds { } 1 πi ) k C T = Ɣ T i <1 ε) i Z { } S D 9) i T is large enough. Proo. Since C T is a subset o Ɣ T, the conditions 3) 5) are satisied or any C T. So, only the condition sup C T k+1) x) ] dx needs to be checked. Note that k+1) x) = S k) x) + P S k 1) x),..., S x) )] x), where Pz 1,...,z k ) is a polynomial. By Lemma 1 S m) x) = o T 1), m =, 1,...,k 1. So, on the one hand, P S k 1) x),..., S x) )] = ot 1) or x, ]. On the other hand, using the orthonormality o the trigonometric basis e i and the deinition o C T, we obtain 4 S k) x) x) ] 1 + o T 1) dx = = 1 + o T 1) i Z < 1 ε) 1 + o T 1) ) S k) x) ] dx πi ) k i

12 1 NK S. DLLYN ND YUY. KUTOYNTS or any C T. Consequently, k+1) x) ] dx = 4 S k) x) x) ] dx1 + ot 1)) 1 ε)1 + o T 1)). 3) It can be easily checked that k+1) x) ] dx C. 31) x > Combining 3) and 31) we obtain k+1) x) ] dx 1 ε) + ot 1). This completes the proo o Lemma 3. LEMM 4. The probability o the event S consequently S D ) = ot 1 ). Proo. The relation 9) implies S D ) C T ). D is exponentially small and emark now that 1 πi ) ] k E i i Z = 1 πi ) kσi ε) i Z = 1 ε) k o T 1) ) 1 + ε). Hence, or T suiciently large 1 πi ) ] k E i 1 ε). i Z The Hoeding s inequality gives us the ollowing upper bound { } 1 πi ) k C T ) = i 1 ε) i Z { } 1 πi ) k i Ei ) ε1 ε) i Z exp { ε 1 ε) }, Q

13 ESTIMTION OF THE DEIVTIVE 11 where Q = So we have G4 4 i Z πi ) 4kσ i C T i <α CT α 4k+1 = CT 1/k+1). i 4k α i S D ) C T ) exp{ CT 1/k+1) }. ) k 1 C i k α k T i <α TheLemma4isproved. Now everything is ready to inish the proo o Theorem 1. Note irst that we can consider only those estimators ϑ T or which T ϑ T, ) = E ϑ T x) x)) dx<1 with ) = S ) and S is the unction S corresponding to the value =. The set o all estimators satisying this inequality will be denoted by W T. For the other estimators the result is evident. We have the ollowing obvious inequalities: in ϑ T W T sup T ϑ T, S ) S D in ϑ T W T sup T ϑ T, S ) S D Ɣ T in T ϑ T,)d) ϑ T W T Ɣ T sup T ϑ T,)d) ϑ T W T Ɣ T \ D T ) sup T ϑ T,)d). ϑ T W T Ɣ T \ D We have already ound a lower bound or the irst term. The second term can be bounded as ollows sup ϑ T W T Ɣ T \ D T ϑ T,)d) 8 sup Ɣ T + )S D ). It ollows rom Lemma 1 that the L norm o is bounded uniormly on Ɣ T. Consequently, using the Lemma 4 we obtain in ϑ T W T Thereore sup T ϑ T, S ) T ) ot 1 ). S D lim lim in in sup T k/k+1) T ϑ T, S ) Pk,)1 ε). D T ϑ T S D

14 1 NK S. DLLYN ND YUY. KUTOYNTS This completes the proo o inequality 7), since ε can be taken as small as we want. 3. Eiciency o a Kernel-type Estimator Now we have a lower bound or the minimax risk, to prove its optimality we have to ind an estimator achieving it. So, we investigate in this section the behavior o the ollowing estimator T ϑ K,T x) = K T x X t )χ { Xt <B T T } dx t, where K T ) is a kernel-type unction and B T is a positive number. Let us denote with and K T x) = α,t K xα,t ) 3) K x) = 1 π 1 1 u k ) cosux) du ) πtk + 1)k + 1) 1/k+1) α,t =. 4k We show in this section that, or B T = T, the estimator ϑ K,T is asymptotically eicient, that is, this estimator achieves the lower bound obtained in the previous section. This means that the ormula 3) gives the optimal kernel in the problem o the irst derivative o the invariant density estimation. Particularly, i k =, then the optimal kernel has the ollowing orm K sin x x cos x) x) =. πx 3 For x = this unction is equal to /3. THEOEM. We have lim lim sup in sup T k/k+1) T ϑ T, S ) Pk,). D T ϑ T S D Proo. It is evident that in ϑ T sup T ϑ T, S ) in S D K sup T ϑ K,T, S ). S D Hence, it is suicient to evaluate the risk T K, S ) = E S ϑ K,T x) S x)) dx,

15 ESTIMTION OF THE DEIVTIVE 13 where S ) D is the unknown trend coeicient. To evaluate this risk we will use the Fourier transormations. Let us denote ϕ S λ) = e iλx S x) dx, ϕ K,T λ) = e iλx ϑ K,T x) dx, T ϕ K λ) = e iλx K T x) dx, ϕ T λ) = 1 T By Parseval s identity T K, S ) = 1 π E S ϕ K,T λ) ϕ S λ) dλ. e iλx t χ { Xt <B T } dx t. s the estimator ϑ K,T is a convolution, its Fourier transorm is product o two Fourier transorms. Indeed ϕ K,T λ) = T e iλx K T x X t )χ { Xt <B T T } dx t dx = T e iλx K T x X t ) dx χ { Xt <B T T } dx t T = e iλx t ϕ K λ)χ { Xt <B T T } dx t = ϕ K λ) ϕ T λ). So the quadratic risk can be rewritten as T K, S ) = 1 π E S ϕ K λ)ϕ T λ) ϕ S λ) dλ = 1 E ϕk S λ)ϕ T λ) ϕ S λ) dλ π = ϕ K λ) Var S ϕ T λ)] dλ + π + 1 ϕk λ)e S ϕ T λ) ϕ S λ) dλ. π For the mathematical expectation o ϕ T λ),wehave E S ϕ T λ) = 1 T ] T E S e iλx t SX t )χ { Xt <B T } dt = E S e iλξ ] Sξ)χ { ξ <BT } = 1 ϕ Sλ) 1 e iλu S u) du. u >B T The ollowing lemma describes the behavior o the bias and variance terms and tells us how to choose B T.

16 14 NK S. DLLYN ND YUY. KUTOYNTS LEMM 5. I the unction K T ) is such that ϕ K λ) 1 or each λ, then there exists a constant C = C D such that, or any S D, ϕ K λ) ϕk Var S ϕ T λ)] dλ + C + CB ] T, T T T ϕk λ)e S ϕ T λ)] ϕ S λ) ϕk dλ 1)ϕ S + C ], where denotes the L -norm over. Proo. We prove here only the irst inequality, the second one can be proved in the same way. Note that Itô ormula gives us the ollowing representation with ϕ T λ) E S ϕ T λ)] = H S λ, X T ) H S λ, X ) + T + e iλx t g S λ, X t ) ] dw t BT g S λ, y) = e iλu S u) χ {u<y} F S y) du B T S y) and H S λ, x) = x g Sλ, y) dy. Consequently, Var S ϕ T λ)] = T HS E S λ, X T ) H S λ, X ) + T + e iλx t χ { Xt <B T } g S λ, X t ) ) dw t. Using the triangle inequality we get ϕ K λ) Var S ϕ T λ)] dλ ) 3 33) with 1 = 1 T ϕ T K λ) E S e iλx t χ { Xt <B T } dw t dλ 1 ϕ K λ) dλ, T = 1 ϕ T K λ) E HS S λ, ξ) dλ, 3 = 1 T ϕ T K λ) E S g S λ, X t ) dw t dλ. B T

17 ESTIMTION OF THE DEIVTIVE 15 To evaluate the term we use the act that ϕ K λ) is less than 1. Thus, according Parseval s identity and condition 4) we have 4 T E BT ξ S e iλu S u) χ {y>u} F S y) dy du B T S y) dλ = 8π BT T S u) ] ξ ] χ {y>u} F S y) ES dy du 8πDB T. B T S y) T epeating exactly the same arguments one can check that 3 8πD T. Lemma 5 is proved. We choose the kernel-type unction K T ) in the ollowing way ϕ K λ) = ϕ α λ) = 1 λ ) k α, + where α = α T is a positive number. s we will see below, the integrals o the righthand sides in Lemma 5 converge both to zero with the rate T k/k+1). Thereore the choice B T = T gives us the ollowing upper estimate or quadratic risk T K, S ) L T ϕ α,ϕ S ) 1 + o T 1) ), where o T 1) tends to zero uniormly on S and L T ϕ α,ϕ S ) = 1 4 ϕα λ) + T ϕ α λ) 1 ϕ S λ) ) dλ. πt Since the unction S ) is in the ellipsoid k,), its Fourier transorm should belong to the ollowing set { } 1 = ϕ λ k ϕλ) dλ. π eplacing in L T the unction ϕ α by its explicit expression, we obtain L T ϕ α,ϕ S ) = α 1 λ k ) πt α dλ + 1 α λ k ϕ πα k S λ) dλ. α Since ϕ S belongs to, the second term o the right-hand side is less than /α k and the irst term can be calculated explicitly: α ) 1 λ 4αk dλ = k α k + 1)k + 1). α α

18 16 NK S. DLLYN ND YUY. KUTOYNTS It leads us to the ollowing inequality { in sup L T ϕ α,ϕ S ) in α> S α> 8k α πtk + 1)k + 1) + α k } = in α> Gα). The unction Gα) is continuously dierentiable and strictly convex, consequently it attains the minimum at the point α satisying the ollowing equation 8k πtk + 1)k + 1) = k α k+1 which leads to πt k + 1)k + 1) α = 4k, ) 1/k+1) and in Gα) = Gα k + 1) ) = α> α k = Pk,)T k/k+1). This completes the proo o Theorem. 4. emarks 1. One can use exactly the same arguments to ind the Pinsker s constant in the problem o l) S ) estimation when S k + l 1,). The optimal rate o convergence ϕ T and the Pinsker s constant are then ϕ T = T β, ) πk + l 1)k + l 1) β P l k, ) = 1 + β) 1 1+β 4k with β = k/k + l 1).. The condition 5) can be replaced by B ] sup B τ S x) E χ{ξ>x} F S ξ) S dx<, S B S ξ) where τ is positive and less than /k + 1). This condition is a little weaker than 5), but it does not enlarge signiicantly the class o diusion processes. eerences 1. Bosq, D. and Davydov, Y.: Local time and density estimation in continuous time, Math. Meth. Statist. 81) 1999), 45.. Castellana, J. V. and Leadbetter, M..: On smoothed density estimation or stationary processes, Stoch. Proc. ppl ), Durett,.: Stochastic Calculus: Practical Introduction, CC Press, Boca aton, FL, 1996.

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