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ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА
2009 Математика и механика № 4(8)
УДК 519.2
V. Konev, S. Pergamenshchikov
NONPARAMETRIC ESTIMATION IN A SEMIMARTINGALE REGRESSION MODEL. PART 2. ROBUST ASYMPTOTIC EFFICIENCY1
In this paper we prove the asymptotic efficiency of the model selection procedure proposed by the authors in [1]. To this end we introduce the robust risk as the least upper bound of the quadratical risk over a broad class of observation distributions. Asymptotic upper and lower bounds for the robust risk have been derived. The asymptotic efficiency of the procedure is proved. The Pinsker constant is found.
Keywords: Non-parametric regression; Model selection; Sharp oracle inequality;
Robust risk; Asymptotic efficiency, Pinsker constant, Semimartingale noise.
AMS 2000 Subject Classifications: Primary: 62G08; Secondary: 62G05
1. Introduction
In this paper we will investigate the asymptotic efficiency of the model selection procedure proposed in [1] for estimating a 1-periodic function S: R ^ R, S e L2[0,1], in a continuous time regression model
dyt = S(t)dt + d'%t, 0 < t < n, (1)
with a semimartingale noise | = (|t )0<t<n. The quality of an estimate S (any realvalued function measurable with respect to ct{yt, 0 < t < n}) for S is given by the mean integrated squared error, i.e.
Rq(S,S) = Eq,sI|S-S ||2 , (2)
where Eqs is the expectation with respect to the noise distribution Q given a function S;
II S ||2 =J0 S2(x)dx.
The semimartingale noise (|t )0<t<n is assumed to take values in the Skorohod space D[0, n] and has the distribution Q on D[0, n] such that for any function f from L2 [0, n] the stochastic integral
In (f) = \jsd Is (3)
is well defined with
EQln (f) = 0 and EqI2( f) <a*{0n/s2 ds, (4)
where ct* is some positive constant which may, in general, depend on n, i.e. ct* = a”n,
such that
0 < liminf CTn < limsup CTn <«. (5)
1 The paper is supported by the RFFI - Grant 09-01-00172-a.
Now we define a robust risk function which is required to measure the quality of an estimate S provided that a true distribution of the noise (%t )0<t<n is known to belong to some family of distributions Q* which will be specified below. Just as in [2], we define the robust risk as
K(S n S) = sup rq (S n S) • (6)
QeQn
The goal of this paper is to prove that the model selection procedure for estimating S in the model (1) constructed in [1] is asymptotically efficient with respect to this risk. When studying the asymptotic efficiency of this procedure, described in detail in Section 2, we suppose that the unknown function S in the model (1) belongs to the Sobolev ball
Wk = {f eCkper[0,1],]T Hf(j)||2< r}, (7)
j=o
where r > 0, k > 1 are some parameters, Ckper [0,1] is a set of k times continuously differentiable functions f :[0,1] ^ R such that f (i)(0) = f (i)(1) for all 0 < i < k. The functional class can be written as the ellipsoid in l2, i.e.
Wrk = {f eCkper[0,1]:£ a} 02 < r}, (8)
j=1
k
where aj = ^ (2n[ j/2])2i •
i=0
In [1] we established a sharp non-asymptotic oracle inequality for mean integrated squared error (2). The proof of the asymptotic efficiency of the model selection procedure below largely bases on the counterpart of this inequality for the robust risk (6) given in Theorem 1.
It will be observed that the notion "nonparametric robust risk" was initially introduced in [3] for estimating a regression curve at a fixed point. The greatest lower bound for such risks have been derived and a point estimate is found for which this
bound is attained. The latter means that the point estimate turns out to be robust
efficient. In [4] this approach was applied for pointwise estimation in a heteroscedastic regression model.
The optimal convergence rate of the robust quadratic risks has been obtained in [5] for the non-parametric estimation problem in a continuous time regression model with a coloured noise having unknown correlation properties under full and partial observations. The asymptotic efficiency with respect to the robust quadratic risks, has been studied in [2], [6] for the problem of non-parametric estimation in heteroscedastic regression models. In this paper we apply this approach for the model (1).
The rest of the paper is organized as follows. In Section 2 we construct the model selection procedure and formulate (Theorem 2.1) the oracle inequality for the robust risk. Section 3 gives the main results. In Section 4 we consider an example of the model (1) with the Levy type martingale noise. In Section 5 and 6 we obtain the upper and lower bounds for the robust risk. In Section 7 some technical results are established.
2. Oracle inequality for the robust risk
The model selection procedure is constructed on the basis of a weighted least squares estimate having the form
W 1 n
Sy = ZY( j)0 jJj with 0 j,n = - j0n 4i(t) dyt, (9)
j=1 n
where (4 j) j>1 is the standard trigonometric basis in i2[0,1] defined as
4 = 1, 4 j (x) = V2 Trj (2n[ j/2]x), j > 2, (10)
where the function Trj (x) = cos(x) for even j and Trj (x) = sin(x) for odd j ; [x]
denotes the integer part of x. The sample functionals 0 jn are estimates of the
corresponding Fourier coefficients
0j = (S,4j) = 10 s(t)4j(t)dt. (11)
Further we introduce the cost function as
W W
Jn (Y) = S Y2 (j)0 2,n - 2 S Y(j) 0 j,n + p Pn(Y) • j=1 j=1
Here
- -2 a.
with an = S0, l = [J"]+1;
P (y) is the penalty term defined as
Jj,n ,n j, °n ^ 'ji,<
n j=i
\ = a n 1 Y 1 P n(Y) = n ‘
As to the parameter p, we assume that this parameter is a function of n , i.e. p = pn such that 0 <p< 1/3 and
lim n5 pn = 0 for all 8 > 0.
n——W
We define the model selection procedure as
s. = sy (12)
where y is the minimizer of the cost function Jn (y) in some given class r of weight sequences y = (y(j))j>1 e [0,1]W , i.e.
Y = argminYer Jn (y) • (13)
Now we specify the family of distributions Q*n in the robust risk (6). Let Pn denote the class of all distributions Q of the semimartingale (|t) satisfying the condition (4).
It is obvious that the distribution Q0 of the process |t = Va^wt, where (wt) is a standard Brownian motion, enters the class Pn, i.e. Q e Pn . In addition, we need to
impose some technical conditions on the distribution Q of the process (|t )0<t<n . Let denote
^(Q) = limmax eq I ,
1< j< n
where
I j,n = ~T=In (4 j ) , Vn
( In (4 j ) is given in (3)) and introduce two Pn ^ R+ functionals
and
A,n(Q) = sup
xgH ,# ( x)<n
Z Xj {Eq j-°(Q))
j=i
(14)
L2,n (Q) = suP Eq
| x|<1, # ( x)< n
l^j,n V j=1 y
where H = [-1,1]W , 1 x |2 = S°°=J x;2 , #(x) = ZW=1 1{|xj |>0} and
1 j,n = j - EQj •
Now we consider the family of all distributions Q from Pn with the growth restriction on Z,,n (Q) + ¿2,n (Q), i.e.
Pn*={Q e Pn : A,n(Q) + L2,n(Q) <ln},
where ln is a slowly increasing positive function, i.e. ln —+w as n — +w and for any
5 > 0
lim \ = 0 •
n—w n5
It will be observed that any distribution Q from P* satisfies conditions Cl) and C2) on the noise distribution from [1] with c* < ln and c2 n < ln. We remind that these conditions are
C1) <n = A,n(Q) <w;
C2) c2,n = L2,n (Q) <w
In the sequel we assume that the distribution of the noise (1t) in (1) is known up to its belonging to some distribution family satisfying the following condition.
C2) Let Q*n be a family of the distributions Q from P^ such that Q0 e Q*n .
An important example for such family is given in Section 4.
Now we specify the set r in the model selection procedure (12) and state the oracle inequality for the robust risk (6) which is a counterpart of that obtained in [1] for the mean integrated squared error (2). Consider the numerical grid
An = {1,..., k*} x ft,..., tm }, (15)
where ti = is and m = [1/s2]; parameters k* > 1 and 0 < s< 1 are functions of n , i.e. k* = k* (n) and s = s(n), such that for any 8 > 0
lim k * (n) =+ro, lim k * (n)ln n = 0,
n^ro n^ro
lim s(n) = 0 and lim n8s(n) = +ro.
(16)
For example, one can take
s(n) =-----1------ and k * (n) = ^ ln(n +1)
ln(n +1)
for n > 1 .
Define the set r as
r = {Ya,ae An }, (17)
where Ya is the weight sequence corresponding to an element a = (P, t) e An, given by the formula
Ya (j) = 1{1< j < Jc,} +(1 - (j/raa )P)1{ j < j <®a} (18)
where j0 = j0(a) = [raa/(1 + lnn)], raa = (Tp tn)1'(2p+:) and
t = (P + 1)(2P +1)
P n2pp ‘
Along the lines of the proof of Theorem 1 in [1] one can establish the following
result.
Theorem 1. Assume that the unknown function S is continuously differentiable and the distribution family Q* in the robust risk (6) satisfies the condition C*). Then the estimator (12), for any n > 1, satisfies the oracle inequality
K (S - S) <1 + 3p-2p2min r; (S y, S) +1 Dn (p), (19)
1 - 3p Yer ' n
where the term Dn (p) is defined in [10] such that
lim = 0 (20)
n^ro n8
for each 8 > 0 .
Remark 1. The inequality (19) will be used to derive the upper bound for the robust risk (6). It will be noted that the second summand in (19) when multiplied by the optimal rate n2k/(2k+:) tends to zero as n for each k > 1. Therefore, taking into account that p^ 0 as n , the principal term in the upper bound is given by the minimal risk over the family of estimates (SY)Yer . As is shown in [7], the efficient
estimate enters this family. However one can not use this estimate because it depends on the unknown parameters k > 1 and r > 0 of the Sobolev ball. It is this fact that shows an adaptive role of the oracle inequality (19) which gives the asymptotic upper bound in the case when this information is not available.
3. Main results
In this Section we will show, proceeding from (19), that the Pinsker constant for the robust risk (6) is given by the equation
( ^ 2k / (2k +1)
~ ' . (21)
Rkn = ((2k + 1)r)17(2k+1}
I (k + 1)n
It is well known that the optimal (minimax) rate for the Sobolev ball Wrk is
n2k/ (2k+1) (see, for example, [8, 9]). We will see that asymptotically the robust risk of
the model selection (12) normalized by this rate is bounded from above by R»,n.
Moreover, this bound can not be diminished if one considers the class of all admissible estimates for S.
Theorem 1. Assume that, in model (1), the distribution of (|t ) satisfies the condition C» ). Then the robust risk (6) of the model selection estimator S » defined in (12), (17), has the following asymptotic upper bound
limsup n2k'(2k+1:i-^ sup R» (S», S) < 1. (22)
n^œ Rk,n S^wk
Now we obtain a lower bound for the robust risk (6). Let nn be the set of all
estimators Sn measurable with respect to the sigma-algebra ct{yt, 0 < t < n} generated
by the process (1).
Theorem 2. Under the conditions of Theorem 1
liminf n2k'(2k+»-L inf sup R*n(Sn,S) > 1. (23)
n^o> Rk,n S"GUn SGWrk
Theorem 1 and Theorem 2 imply the following result Corollary 3. Under the conditions of Theorem 1
lim n2k/(2k+i:i^ „inf sup Rl (§S) = 1. (24)
Rk,n S»en» SEW?
Remark 1. The equation (24) means that the sequence Rl n defined by (21) is the Pinsker constant (see, for example, [8, 9]) for the model (1).
4. Example
Let the process (^) be defined as
^ = QWt + Q2zt , (25)
where (wt )t >0 is a standard Brownian motion, (zt )t >0 is a compound Poisson process
defined as
N
z, = EY,- •
i=1
where (Nt )t>0 is a standard homogeneous Poisson process with unknown intensity
1
X > 0 and (Yj ) j>1 is an i.i.d. sequence of random variables with
EYj = 0, EY2 = 1 and EY4 < œ.
Substituting (25) in (3) yields
e in ( f )=(ft2 + ¿x)imi2.
In order to meet the condition (4) the coefficients gx, g2 and the intensity X > 0 must satisfy the inequality
q2 + £22 X<ct». (26)
Note that the coefficients ^ , g2 and the intensity X in (4) as well as ct» may
depend on n , i.e. g{ = gt (n) and X = X(n).
As is stated in [1], Theorem 2, the conditions C1) and C2) hold for the process
(25) with ct = ct(Q) = ft2 + defined in (14), c» (n) = 0 and
c»(n) < 4ct(ct + £2E Y14).
Let now Q» be the family of distributions of the processes (25) with the coefficients satisfying the conditions (26) and
where the sequence ln is taken from the definition of the set P*. Note that the distribution Q0 belongs to Q*. One can obtain this distribution putting in (25) gl = Va* and g2 = 0. It will be noted that Q*n c P* if
4a*(ct* +^finE Y/) < ln .
5. Upper bound
1. Known smoothness
First we suppose that the parameters k > 1, r > 0 and a* in (4) are known. Let the family of admissible weighted least squares estimates (Sy)yer for the unknown function
S e Wk be given (17), (18). Consider the pair
a0 = (k, *0)
where t0 = [rn/e]e, rn = r/an and e satisfies the conditions in (16). Denote the corresponding weight sequence in r as
Y0 = Ya0 • (28)
Note that for sufficiently large n the parameter a0 belongs to the set (17). In this section we obtain the upper bound for the empiric squared error of the estimator (6). Theorem 1. The estimator SYo satisfies the following asymptotic upper bound
limsup n2k/(2k+1:i-^ sup R* (Sy0 , S) < 1. (29)
n^w Rk,n SEwr
Proof. First by substituting the model (1) in the definition of § . in (9) we obtain
,,n
§ j,n = § j + n ^ j,n >
vn
where the random variables \are defined in (14). Therefore, by the definition of the estimators S Y in (9) we get
II §y0 - S II2 = £ (1 -Y0(,))2 §2 - 2Mn + £ y2(j) j 1=1 j=1
with Mn =-p£ (1 -Y0(1))Y0(1)§1 j .
Vn , =1
It should be observed that
EQ,S Mn = 0
for any Q e Q*. Further the condition (4) implies also the inequality Eq %,n < a*n for each distribution Q e Q*n. Thus,
R* (sJo, S) <£ (1 -y0(j))2 §2 + an £ Y 2( 1), (30)
j=10 n 1=1
where i0 = j0 (a0). Denote
Un = n2k/(2k+1) sup(1 - y0( j))2 /aj ,
j>10
where aj is the sequence as defined in (8). Using this sequence we estimate the first summand in the right hand of (30) as
72k/(2k+1) £ (1 _y0(j))2 02 <Un X a} 02
02 •
j=10 j >
From here and (8) we obtain that for each S e
Y^ (S) = n2k/(2k+1) £ (1 _ y0(j))2 02j <unr.
j=10
Further we note that
1 • .— 2k/ (2k+1) 1
limsup (rn ) Un < 2k ( \2k/(2k+1) »
n^œ n (,Tk )
where the coefficient Tk is given (18). Therefore, for any n > 0 and sufficiently large
n > 1
sup Y,n(S)<(1 + n)(CT»n)2k/(2k+1) Y», (31)
where Y, =
r1 (2k+1)
1 n2 k (Tk )2 k/ (2k+1)
To examine the second summand in the right hand of (29) we set
1
j=i
Since by the condition (5)
2(тк )y(2k+1) k2
(к + 1)(2k +1)
Note that by the definition (22)
Therefore, for any n > 0 and sufficiently large n > 1
Hence Theorem 1. □
2. Unknown smoothness Combining Theorem 1 and Theorem 1 yields Theorem 1. □
6. Lower bound
First we obtain the lower bound for the risk (2) in the case of "white noise" model (1), when ^ = 4CTwt. As before let Q0 denote the distribution of (§t )0<t<n in _D[0, n].
Theorem 1. The risk (2) corresponding to the the distribution Q0 in the model (1) has the following lower bound
where Rq(v) = Rq0 (•,•)•
Proof. The proof of this result proceeds along the lines of Theorem 4.2 from [2]. Let
| x |> 1. For each 0 < n< 1 we introduce a smoother indicator of the interval [-1 + n, 1 - n] by the formula
It will be noted that In e C" (R), 0 < In< 1 and for any m > 1 and positive constant
liminf n2k/(2k+1) inf —L sup R0(sn,S)>1,
(32)
V be a function from Cш (R) such that V(x) > 0 , f V(x)dx = 1 and V(x) = 0 for
J — 1
c > 0
lim sup f f (x)I’m (x) dx - f 11 f (x) dx = 0
(33)
where | f |* = sup-1£x£1 | f (x) |. Further, we need the trigonometric basis in L2[-1,1], that is
e, (x) = 1/V2, e} (x) = Tr} (n[ j/2]x), j > 2. (34)
Now we will construct of a family of approximation functions for a given regression function S following [2]. For fixed 0 < s< 1 one chooses the bandwidth function as
h = hn = (u*)2i+1 Nnn 2k+1 (35)
with
_* 2k
* °nkn j A T 1 4
us =------------^------------------ and Nn = ln n
(1 -s) r 2 +'(k + 1)(2 k + 1)
and considers the partition of the interval [0,1] with the points xm = 2hm , 1 < m < M , where
M = [1/(2 h)] -1.
For each interval [xm - h, xm + h] we specify the smoothed indicator as In (vm (x)), where vm (x) = (x - xm)/h . The approximation function for S(t) is given by
M N
Sz,„ (x) = ££ Zm, jDm, j (x), (36)
m=1 j=1
where z = (zm,j\<m<M,!<j<N is an array of real numbers;
Dm, j (x) = ej (vm (x))In (vm (x))
are orthogonal functions on [0,1].
Note that the set Wk is a subset of the ball
Br = {f e L2[0,1]:|| f ||2< r}.
Now for a given estimate S'n we construct its projection in L2[0,1] into Br
Fn := P% (§n) •
In view of the convexity of the set Br one has
IIS n - S||2 >||F n - S||2
for each S e Wk c Br.
From here one gets the following inequalities for the the risk (2)
sup R0 (S n, S) > sup R0 (f n, S) > sup R0 (f n, S),
SeWk SeWk {zeRd:Sz,neWrk}
where d = MN .
In order to continue this chain of estimates we need to introduce a special prior distribution on Rd . Let k = (Km, j )1<m<M,1£ j<N be a random array with the elements
Km,j = tm,j <j , (37)
where K*m,j are i.i.d. gaussian N(0,1) random variables and the coefficients
V* *
c
t ^nyj
m j ‘
We choose the sequence (y*)1£j<N in the same way as in [2] ( see (8.11)) , i.e.
y* = Nknj k -1.
We denote the distribution of k by |aK . We will consider it as a prior distribution of the random parametric regression SK,n which is obtained from (36) by replacing z with k. Besides we introduce
I = jz e Rd : max max —m,j- < ln nL (38)
^ 1< m<M 1< j< N tm, j J
By making use of the distribution |aK, one obtains
suPR0(SnS) ~\{zeRd :Sz „ewk }nS„ EQ0 A,n 11 Fn - Sz,n ||2 ^k (dz) •
Further we introduce the Bayes risk as
R(Fn) = JRd R0(F n Sz,n )^K (dz) and noting that || pn ||2 < r we come to the inequality
sup Ro(Sn,S)>R(Fn)-®n, (39)
SeWk
where TOn = E(1{Sk niWk } +lsn )(r + 11 SK,n ||2) •
By Proposition A. 1 from Appendix A. 1 one has, for any p > 0 ,
lim np wn = 0.
Now we consider the first term in the right-hand side of (39). To obtain a lower bound for this term we use the L2 [0,1] -orthonormal function family (Gm, j )1£ m<M ,1£ j < N
which is defined as
-Jh
We denote by g mj. and gm,j (z) the Fourier coefficients for functions pn and Sz respectively, i.e.
g m, j = J0 Fn(x)Gm, j (x)dx and gm,j (z) = J0 Sz ,n (x) Gm,j (x)dx •
Now it is easy to see that
M N
11 F n - Sz,n ||2 > ZZ ( gm j - gm, j ( z))2 •
Gm„, (X) = — ej (vm (x) )1(|vm (x)|<1) •
’ m, j ôm> j' m=1 j=1
Let us introduce the functionals Kj (•) : L1[-1,1] ^ R as
Kj ( f ) = i-! e2 (v) f (v) dv.
In view of (36) we obtain that
5 gm, j ( z ) = |0 Dm, j ( x) Gm, j ( x) dx = ^ Kj (1 n ) •
dz . ’j J0
m, j
Now Proposition A.2 implies
M N M N K2( I )
R ( F n ) > Z Z ÎRd ES„n (g - S.,j (z))2 *K №) > h Z Z K . ,2..nj. , •
m=1 j=1 m=1 j=1 Kj (1n ) nh + ^ j CT
Therefore, taking into account the definition of the coefficients (tm j ) in (37) we get
m, j >
* N
with
( ) K j2 (In ) y
T,(n, y) =
Kj (In2)y +1
jn
Moreover, the limit equality (33) implies directly
limsupsup
n^° j>1 y>0
(y + 1)tj (n, y)
-1
y
Therefore, we can write that for any v > 0
*
y,
= 0.
R ( F n)
2nh(1 + v) j= y. +1 It is easy to check directly that
CT* N y* _L_
lim-----n— Z —-— = (1 -e)2k+1,
n^œ 2nhRk ,n j=1 y.+1
where the coefficient R. n is defined in (21). Therefore, (39) implies for any 0 <s< 1
. . 2k 1 1
lim inf inf n2k+1 —— sup Rj(Sn, S) > (1 -e)2k+1.
T^.œ Sn Rk,n Sew'k
Taking here limit as e ^ 0 implies Theorem 1. □
7. Appendix
A.1. Properties of the parametric family (36)
In this subsection we consider the sequence of the random functions S defined in (36) corresponding to the random array k = (Km,j )1<m<M,1£j<N given in (37). Proposition A.1. For any p > 0
hm n lim E||SK,n ||2 [l{S t + Lc ] = 0•
n^œ n^œ ^n )
This proposition follows directly from Proposition 6.4 in [6].
A.2. Lower bound for parametric “white noise” models
In this subsection we prove some version of the van Trees inequality from [10] for the following model
dyt = S(t, z)dt + vct dwt, 0 < t < n, (A.1)
where z = (z1v.., zd)' is vector of unknown parameters, w = (wt)0<t<T is a Winier process. We assume that the function S (t, z) is a linear function with respect to the parameter z , i.e.
S(t, z) = ltzjSj (t). (A.2)
j=1
Moreover, we assume that the functions (Sj )1£j<d are continuous.
Let ® be a prior density in Rd having the following form:
d
®cz) = ®c zlv.,zd)=n^j(zj) *
j=1
where 9j is some continuously differentiable density in R. Moreover, let g (z) be a continuously differentiable Rd ^ R function such that for each 1 < j < d
lim g(z)9j(zj) = 0 and [ , | gj(z) |®(z)dz <<», (A.3)
JR
where
dg (z)
gj (z) = ■
dzj
Let now Xn = C[0, T] and B(Xn) be ct - field generated by cylindric sets in Xn . For any B(Xn) ® B(Rd) - measurable integrable function §=§(x,9) we denote
JR JX
where ^z is distribution of the process (A.1) in Xn . Let now v = ^0 be the distribution
of the process (ct*wt )0£i£n in X. It is clear (see, for example [11]) that |az << v for any
z e Rd. Therefore, we can use the measure v as a dominated measure, i.e. for the observations (A.1) in Xn we use the following likelihood function
rs x d|j.z ifnS(t,z) rn S2(t,z) )
fiy,z) = — = exp|io -J=rdyt -J0 ~2^dt\- <A'4)
Proposition A.2. For any square integrable function g measurable with respect to ct{yt, 0 < t < n} and for any 1 < j < d the following inequality holds
CT* B2
,r<g n-g(z))2 a „ , j . ■ (A*5)
I0" Sj (t) dt + CT*Ij
where
bj =\Rdgj(z) °(z) dz and ¡j=Jr ^ dz •
Proof. First of all note that the density (A.3) is bounded with respect to 9j e R for
any 1 < j < d , i.e. for any y = (yt )o<t<n e X
limsup f (y, z) <»•
\zj l^w
Therefore, putting
d
Y j = Y j (y, z) = — ln( f (y, z)®( z)) j j 89 j and taking into account condition (A.3) by integration by parts one gets
E ((gT - g(z))yj) = JRwxRd (gT(y) - g(z))(f (y>z)0(z))dz dv(y) =
= JRW xRd g j (z) f (y z)0(z) dzdv(y) = Bj •
Now by the Bounyakovskii-Cauchy-Schwarz inequality we obtain the following lower bound for the quiadratic risk
B2
E(gT - g(z))2 > j
EY 2
Note that from (A.4) it is easy to deduce that under the distribution |az
d ¡n Sj(t) tn S(t,z)Sj(t) f
^lnf(y,z) =Jo-rTdyt-Jo ----------------------^------dt = Jo
uzj VCT CT VCT
This implies directly
d
Ez —ln f (y> z ) = 0
dzj
and
Ez
Therefore,
( d ^ d ln f (y>z)
Kdzj
S2 <t)d • EY 2 =J7i0'S';(') dt+¡j •
Hence Proposition A.2. □
8. Acknowledgments
This research has been executed in the framework of the State Contract 02.740.11.5026.
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СВЕДЕНИЯ ОБ АВТОРАХ:
Konev Victor, Department of Applied Mathematics and Cybernetics, Tomsk State University,
Lenin str. 36, 634050 Tomsk, Russia, e-mail: [email protected]
Pergamenshchikov Serguei, Laboratoire de Math'ematiques Raphael Salem, Avenue de
l’Universit'e, BP. 12, Universit'e de Rouen, F76801, Saint Etienne du Rouvray, Cedex France
and Department of Mathematics and Mechanics,Tomsk State University, Lenin str. 36, 634041
Tomsk, Russia, e-mail: [email protected]
&атья принята в печать 16.11.2009 г.
