Minimax results for estimating integrals of analytic processes

MINIMAX RESULTS FOR ESTIMATING INTEGRALS OF

ANALYTIC PROCESSES

KARIM BENHENNI AND JACQUES ISTAS

Abstract. The problem of predicting integrals of stochastic processes is considered. Linear estimators have been constructed by means of samples at^N discrete times for processes having a xed Holderian regu- larity^s>0 in quadratic mean. It is known that the rate of convergence of the mean squared error is of order ^N;(2s+1).

In the class of analytic processes H^p, ^p 1, we show that among all estimators, the linear ones are optimal. Moreover, using optimal coe cient estimators derived through the inversion of the covariance matrix, the corresponding maximal error has lower and upper bounds with exponential rates. Optimal simple nonparametric estimators with optimal sampling designs are constructed inH² andH¹and have also bounds with exponential rates.

1. Introduction

In addressing problems involving stochastic processes such as estimation of various quantities from the sample path, parameter inference and control theory, one frequently has access to observations at a nite number of points rather than over an entire observation interval. The following questions then arise: what are the optimal estimators of integrals of random processes and what are the rates of convergence of the mean squared error of these estimators as the number of observations gets large?

This problem was widely studied by Sacks and Ylvisaker (1966, 1968, 1970, 1971), Cambanis (1985) for processes having 0 or 1 quadratic mean derivatives and by Eubank, Smith and Smith (1982) for some class of pro- cesses including the

K

th order iterated integrals of Brownian motion. Ben- henni and Cambanis (1992a, 1992b) considered processes having

K

qua- dratic mean derivatives,

K

a nonnegative integer, and showed that the rate of convergence for the mean squared error in the approximation of random integrals by linear estimators based on the Euler MacLaurin formula for regular sampling design of size

N

, is of order

N

^;(2K+2).

Let

L

N(

X

) the Euler MacLaurin estimator (see Benhenni and Camba- nis (1992a)) of the random integral of the process

X

having a covariance function

R

(

ts

) =^E

X

(

t

)

X

(

s

):

I

(

X

) =

Z b

a

(

t

)

X

(

t

)

dt

URL address of the journal: http://www.emath.fr/ps/

Received by the journal April 30, 1996. Revised November 18, 1997. Accepted for publication June 11, 1998.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

where

is a known nonrandom function in^L2(

ab

]).

The estimator

L

N(

X

) is said to be asymptotically optimal since it has the same asymptotic performance as the optimal linear estimator (see Benhenni and Cambanis (1992a)):

Nlim^!1

N

^2K+2E(

I

(

X

)^;

L

N(

X

))²

= lim_N

!1

N

^2K+2inf_L

N

E(

I

(

X

)^;

L

N(

X

))² (1.1)

=

B

²K⁺²

(2

K

+ 2)!

Z b

a

K(

t

)

²(

t

)

h

^2K+2(

t

)

dt

where

B

m is the

m

th Bernoulli number (e.g. Abramowitz and Stegun (1965)),

h

is the density that generates the sampling design ^f

t

¹

< t

²

<

::: < t

N^g,

K(

t

) dened by

K(

t

) = lim_s

"t

@

^2K+2

@t

^K

@s

^K+1

R

(

ts

)^;lim_s

#t

@

^2K+2

@t

^K

@s

^K+1

R

(

ts

)

is assumed nite and positive on

ab

], and the inmum infL^N is taken among all linear estimators.

Istas and Laredo (1997), Stein (1995) extended the results to processes satisfying a Holder condition, that is, there exists a real number

s

such that for every

t

²

ab

],^E(

X

(

t

+

h

)^;

X

(

t

))^2j

h

^j;2stends to a nite nonvanishing constant as

h

^!0 and showed that the rate for the mean squared error is of order

N

^;(2s+1). In particular, the asymptotic result in (1.1) is obtained by taking

s

=

K

+ 1

=

2, see Stein (1995) for details.

What happens when the process

X

(

t

) is analytic? The following questions are still open. What is the exact rate of convergence for the mean squared error? In fact it is faster than any power of the sample size

N

since in this case, the asymptotic constant in (1.1) is equal to 0 where

K(

t

) = 0 for all

K

. Do nonlinear estimators provide a signicant improvement on the performance of linear estimators?

We give in this paper some answers to these questions. First we dene the class of analytic processes ^Hp, 1

p

¹, from the Hardy space

H

^p of analytic deterministic functions (e.g. Rudin (1966)). Denote by

L

N(

X

) a linear estimator of

I

(

X

) constructed from the observations

X

(

t

i),

i

= 1

::: N

, at

N

sampling points (

t

¹

::: t

N) taken from the interval ]^;1

1.

The maximal mean squared error in the class ^Hp, 1

p

¹, is given by

E

L^N(^Hp) = sup_X^2HpE(

I

(

X

)^;

L

N(

X

))². We study then the minimax risk error

d

(^Hp) = infL^N

E

L^N(^Hp) (cf. Ibragimov and Has'minskii (1981), Ch.1).

We give in Theorem 3.1 a lower bound for

d

(^Hp). We show in Theorem 3.2 that the maximal error corresponding to the estimator derived from the similar deterministic recovery problem in

H

^p achieves this bound and that this bound has an exponential rate of convergence. Moreover we show in Theorem 4.1 that among all estimators, the linear estimators are optimal.

For a given process in ^Hp, with known covariance function, we consider the best parametric linear estimator

L

^bN obtained through the inversion of the covariance matrix. We show in Theorem 5.1 that the corresponding maximal error

E

_L^bN(^Hp) =

d

(^Hp), and thus has an exponential rate of convergence.

Finally we construct in Section 6 optimal nonparametric estimators with optimal sampling designs in ^H1 and in ^H2.

2. Notations and definitions

First recall the denition of analytic function (e.g. Cartan (1978)). A function

f

(

z

) is said to be analytic on an open domain

D

^C if, for every

z

²

D

, ^f(z+h_h^);f(z) tends to a nite limit as

h

^! 0

h

^{2 C}. An analytic function is equal to its Taylor expansion. From now on, let

= ^f

z

^2C

^j

z

^j

<

1^g

:

We recall the Hardy space

H

^p, 1

p

¹, of analytic functions dened on the unit disc of ^C is such that (e.g. Rudin (1966), Ch. 17):

f

²

H

^p

1

p <

¹

if

M

p(

fr

) =

1 2

Z

;^j

f

(

re

ⁱ)^jp

d

1

pis bounded as

r

^!1

and then ^jj

f

^jjH^p= lim_r

!1

M

p(

fr

)

f

²

H

¹ if

M

¹(

fr

) = sup

;^j

f

(

re

ⁱ)^jis bounded as

r

^!1

and then ^jj

f

^jjH¹= lim_r

!1

M

¹(

fr

)

:

For a given second order process

X

(

z

)

z

² with covariance function

R

, we dene

H

(

R

) the Reproducing Hilbert space generated by

R

(

z

¹

z

²), (

z

¹

z

²) ^{2 2} with inner product

< f

(

:

)

R

(

:z

)

>

H⁽R⁾=

f

(

z

),

z

² for

f

²

H

(

R

), (see Parzen (1959)).

Let^L2(

X

) the^L2-closure of the linear span of the random variables^f

X

(

z

),

z

^2g. Then every

f

²

H

(

R

) is of the form

f

(

z

) = ^EX(

X

(

z

)

) for some

^2L2(

X

) and the correspondence

f

^$

is an isomorphism between

H

(

R

) and ^L2(

X

) so that if

f

i^$

i then

< f

¹

f

²

>

H⁽R⁾=^EX(

¹

²).

Definition 2.1.

X

(

z

) is analytic in if it is dierentiable in the squared mean for all

z

². There exists a process

X

⁰(

z

) such that, as

h

^!0,

h

^2C

E

X

(

z

+

h

)^;

X

(

z

)

h

^;

X

⁰(

z

)

2

! 0

:

It follows that

X

(

z

) is weakly dierentiable

E

X

(

z

+

h

)^;

X

(

z

)

h

! E(

X

⁰(

z

)

)

for all

^{2 L2}(

X

), as

h

^! 0,

h

^2C. Then, if

f

²

H

(

R

),

f

(

z

) = ^E(

X

(

z

)

) for some

^2L2(

X

), we have

f

(

z

+

h

)^;

f

(

z

)

h

^{= E}

X

(

z

+

h

)^;

X

(

z

)

h

! E(

X

⁰(

z

)

)

as

h

^!0,

h

^2C. Thus

f

is analytic in .

We consider in this paper the class of processes ^Hp, 1

p

¹, dened as follows

1

p <

¹,

X

is analytic in and

rlim^!1

1 2

Z

;

R

^p=2(

re

ⁱ

re

ⁱ)

d

1

p

1

p

=¹,

X

is analytic in and

rlim^!1 sup

^2;]j

R

(

re

ⁱ

re

ⁱ)^j 1

:

Clearly,

R

(

re

ⁱ

re

ⁱ) =^Ej

X

(

re

ⁱ)^j2 0, then, for any

>

0,

R

(

re

ⁱ

re

ⁱ) is dened as the positive real number

such that

¹⁼=

R

(

re

ⁱ

re

ⁱ).

We have the following relation between the two spaces ^Hp and

H

(

R

),

p

1.

Lemma 2.2. Let

X

^{2 Hp} be a stochastic process with covariance function

R

, and let

H

(

R

) be the associated Reproducing Hilbert space. Then

H

(

R

)

H

^p and ^jj

:

^jjH^pjj

:

^jjH⁽R⁾.

Proof. Let

f

²

H

(

R

) there exists

^{2 L2}(

X

(

z

)

z

² ) such that

f

(

z

) =

E(

X

(

z

)

) and ^jj

f

^jj2_H⁽_R⁾=^Ej

^j2. From Cauchy-Schwarz inequality, we have

j

f

(

z

)^{j2 Ej}

X

(

z

)^{j2 Ej}

^j2

=

R

(

zz

)^jj

f

^jj2_H⁽_R⁾

so that

21

Z

;^j

f

(

re

ⁱ)^jp

d

^jj

f

^jjp_H⁽_R⁾ ₂^{1 Z}

;

R

^p=2(

re

ⁱ

re

ⁱ)

d:

It follows that

f

^2Hp and^jj

f

^jjHpjj

f

^jj_H⁽_R⁾.

Likewise we can show that

H

(

R

)

H

¹ and ^jj

f

^{jjH1 jj}

f

^jj_H⁽_R⁾.

We wish to estimate the random integral of a real valued process dened on ]^;1

1

I

(

X

) =

Z

1

;1

(

t

)

X

(

t

)

dt

(2.1) where

is a weight nonrandom function by linear estimators

L

N(

X

) ^,

L

(

X

(

t

¹)

::: X

(

t

N)) that are based on observations of

X

at the sampling points (^;1

< t

¹

< ::: < t

N

<

1).

In what follows we assume that the weight function

is analytic with sup^j

^j 1, then the weighted process

X

^{2 Hp} if

X

^{2 Hp}. Indeed, we have for 1

p <

¹

21

Z

;

R

X(

re

ⁱ

re

ⁱ)]^p2

d

_{= 12} ^Z

;^j

^jp(

re

ⁱ)

R

X(

re

ⁱ

re

ⁱ)]^2p

d

and for

p

=¹

^2;sup^]j

R

X(

re

ⁱ

re

ⁱ)^j sup

^2;]j

²(

re

ⁱ)^j sup

^2;]j

R

X(

re

ⁱ

re

ⁱ)^j

:

By Cauchy-Schwarz inequality, we have

E Z

1

;1

(

t

)

X

(

t

)

dt

² 2

Z

1

;1

²(

t

)

R

(

tt

)

dt:

Since

H

^p

L

¹ (e.g. Duren (1970)), we deduce that ^R;11

(

t

)

X

(

t

)

dt

is well- dened.

The maximal mean squared error corresponding to such estimators is dened by

E

L^N(^Hp) = sup

X^2HpEX(

I

(

X

)^;

L

N(

X

))²

and the minimax linear risk error is given by

d

(^Hp) = inf_L

N

E

L^N(^Hp)

:

(2.2)

We denote by ^V0 the class of processes in^Hp that vanish at the sampling points (

t

¹

::: t

N):

V

0 = ^f

X

^2Hp

X

(

t

i) = 0

i

= 1

::: N

^g

:

(2.3)

3. Rates of convergence for linear estimators Introduce the following rates

R

^p_N = sup

jjf^jjHp1

Z

1

;1

B

N(

u

)

f

(

u

)

du

² for 1

p

¹

(3.1) where

B

N(

u

) =^QN_i^=11;u;_ut^tii is the nite Blaschke product (cf. Duren (1970) p.20).

The lower bound for the minimax error is given by the following result.

Theorem 3.1. For a xed sampling

T

N = (

t

¹

::: t

N) of size

N

, we have

d

(^Hp)

R

^p_N

:

Proof. Let

X

^2V0 and

L

N a linear estimator. Then

EX(

I

(

X

))² = ^EX(

I

(

X

)^;

L

N(0

:::

0))²

_Xsup

2H p

EX(

I

(

X

)^;

L

N(

X

(

t

¹)

::: X

(

t

N))²

:

It follows that

Xsup^2V0EX(

I

(

X

))² _Xsup

2H p

EX(

I

(

X

)^;

L

N(

X

(

t

¹)

::: X

(

t

N))²

:

Moreover, by Rudin (1966), Th.17.9, we have

Xsup^2V0EX(

I

(

X

))² = sup

2H p

E

Z

1

;1

B

N(

u

)

(

u

)

du

²

:

In particular, let the process

(

t

) =

f

(

t

)

v

where

f

²

H

^p,^jj

f

^jjH^p 1 and

v

is a random variable with mean 0 and variance 1. Then we obtain

sup^2HpE

Z

1

;1

B

N(

u

)

(

u

)

du

2 sup

jjf^jjHp1E

Z

1

;1

f

(

u

)

vB

N(

u

)

du

2

= sup

jjf^jjHp1

Z

1

;1

B

N(

u

)

f

(

u

)

du

2

:

The upper bound for the minimax error is given by the following result.

Theorem 3.2. For any xed sampling design

T

N = (

t

¹

::: t

N) of size

N

, we have

d

(^Hp)

R

^p_N

and the best linear estimator in the class of stochastic process ^Hp is given by the best linear estimator in the class of deterministic functions

H

^p. Proof. For any linear estimator

L

N and

X

^{2 Hp} we have, Parzen (1959), Wahba (1990)

EX(

I

(

X

)^;

L

N(

X

))²

= sup

(

Z

h

^;

L

N(

h

)

2

h

²

H

(

R

)

^jj

h

^jjH⁽R⁾1

)

sup

(

Z

h

^;

L

N(

h

)

2

h

²

H

^p

^jj

h

^jjH^p 1

)

:

The last inequality follows from Lemma 2.2. Then

inf_L

N

Xsup^2HpEX(

I

(

X

)^;

L

N(

X

))²

inf_L

N

sup

(

Z

h

^;

L

N(

h

)

2

h

²

H

^p

^jj

h

^jjH^p1

)

:

(3.2) The linear estimator that minimizes the right-hand side of (3.2) is there- fore the best linear estimator for deterministic functions in

H

^p. From Mic- cheli and Rivlin (1984), we have

d

(^Hp) sup

(

Z

h

²

h

(

t

i) = 0

i

= 1

::: N

^jj

h

^jjH^p1

)

and the result follows from Rudin (1966), Th.17.9.

Remark 3.3. It follows from Theorems 3.1 and 3.2 that

d

(^Hp) =

R

^p_N = sup_X^2V0EX(

I

(

X

))². The optimal sampling design is therefore given by (

t

^?1

::: t

_?N) =Argmin⁽t¹:::t^N)

R

^p_N. Newman (1979), Loeb and Werner (1974) and Miccheli and Rivlin (1984) gave upper and lower bounds for the rate of convergence of the minimax error associated with the optimal sampling design for approximating integrals of deterministic functions in

H

^p, 1

p

¹. Dene the linear optimal minimax risk associated with the optimal sampling design

d

^?(^Hp) = inf

(t¹:::t^N)

d

(^Hp) (3.3)

= inf_B

N

sup

jjf^jjHp1

Z

1

;1

B

N(

u

)

f

(

u

)

du

2

:

Then we obtain for the random case for 1

p <

¹

49 exp

(

;12

s

N

(

p

^;1)

p

)

d

^?(^Hp) 121exp

(

;

s

N

(

p

^;1)

p

)

exp

(

;10

r

N

2

)

d

^?(^H1) exp

(

;2

r

N

2

)

:

As pointed out by Newman (1979) for approximating integrals of determin- istic

H

¹-functions, no quadrature formula \works" for

p

= 1 in the random case.

Thus we see that the rate of convergence for the optimal minimax error for estimating integrals of analytic processes is of order exponential and can even be faster for

p >

1. This generalizes the case when the processes to integrate have a

s

-Holder smoothness in quadratic mean studied by Sacks and Ylvisaker (1966, 1968, 1970), Benhenni and Cambanis (1992a), Istas and Laredo (1997), Stein (1995) where the rate of convergence of the mean squared error is of order

N

^2s+1. It is a di cult task to nd the optimal sampling design of size

N

that minimizes the minimax error for any 1

p <

¹. The case

p

= ¹ is covered by Bojanov (1974) and is treated in Section 6.2. It remains the problem of nding optimal linear estimators.

They are not available when integrals of deterministic functions in

H

^p

1

p <

¹

p

⁶= 2, are approximated.

4. Rates of convergences for non-linear estimators We wish now to estimate

I

(

X

) by non-linear estimators, and we want to know if these estimators could improve actually the performance of linear estimators. Let

N(

X

) ^,

(

X

(

t

¹)

::: X

(

t

N)) be a (general) estimator.

The \non-linear minimax" risk is dened by

e

(^Hp) = inf

N

Xsup^2HpEX(

I

(

X

)^;

N(

X

))²

:

The linear estimators are optimal among all the estimators.

Theorem 4.1. For any xed sampling design

T

N = (

t

¹

::: t

N) of size

N

, we have

e

(^Hp) =

d

(^Hp) =

R

^p_N

:

Proof. For all

X

^2V0,

EX(

I

(

X

))² = ^EX(

I

(

X

)^;

N(0

:::

0)+

N(0

:::

0))²

= ^EX(

I

(

X

)^;

N(0

:::

0))^2;

²_N(0

:::

0)

EX(

I

(

X

)^;

N(0

:::

0))²

_Xsup

2H p

EX(

I

(

X

)^;

N(

X

))²

then

Xsup^2V0EX(

I

(

X

))² _Xsup

2H p

EX(

I

(

X

)^;

N(

X

))²

Xsup^2V0EX(

I

(

X

))² inf

N

Xsup^2HpEX(

I

(

X

)^;

N(

X

))²

:

The proof then follows from the proofs of Theorems 3.1 and 3.2 and the straightforward bound

e

(^Hp)

d

(^Hp).

5. Parametric optimal linear estimator in ^{Hp 1 p 1} Let

f

²

H

(

R

) of the form

f

(

t

) = ^R;11

(

s

)

R

(

st

)

ds

, and for a sampling design

T

N = ^f

t

¹

::: t

N^g of size

N

we dene the covariance matrix, as- sumed non-singular,

R

T^N =^f

R

(

t

i

t

j)^g1ijN. Then it is known, Sacks and Ylvisaker (1966, 1968, 1970), Benhenni and Cambanis (1992a), that the esti- mator dened by

L

^bN(

X

) =

f

_T^0N

R

^;1_T^N

X

T^N minimizes the mean squared error

EX(

I

(

X

)^;

L

N(

X

))²with respect to the class of all linear estimators, where

f

_T⁰N = (

f

(

t

¹)

::: f

(

t

N)) and

X

_T⁰N = (

X

(

t

¹)

::: X

(

t

N)). The linear esti- mators

L

^bN(

X

), dened in the class ^Hp

p

1, are optimal in the sense of minimax error. This is given by the following result.

Theorem 5.1. For a xed sampling design

T

N = (

t

¹

::: t

N) of size

N

, we

have _Xsup

2H p

EX(

I

(

X

)^;

L

^bN(

X

))² =

R

^p_N

:

Proof. For any linear estimator

L

N and any process

X

^2Hp, we have

EX(

I

(

X

)^;

L

^bN(

X

))^{2 E}X(

I

(

X

)^;

L

N(

X

))²

Xsup^2HpEX(

I

(

X

)^;

L

^bN(

X

))² inf_L

N

Xsup^2HpEX(

I

(

X

)^;

L

N(

X

))²

=

R

^p_N

by Theorems 3.1 and 3.2.

For any process

X

^2V0, we have

EX(

I

(

X

))² = ^EX(

I

(

X

)^;

L

^bN(

X

))²

Xsup^2V0EX(

I

(

X

))² _Xsup

2H p

EX(

I

(

X

)^;

L

^bN(

X

))²

R

^p_{N X}sup

2H p

EX(

I

(

X

)^;

L

^bN(

X

))²

by Theorem 3.1.

Then we deduce from Remark 3.3 the rate of convergence for the maximal error when optimal estimators

L

^cN(

X

) associated with optimal sampling design are used in the class ^Hp,

49 exp

(

;12

s

N

(

p

^;1)

p

)

inf_T

N

Xsup^2HpEX(

I

(

X

)^;

L

^bN(

X

))²

121exp

(

;

s

N

(

p

^;1)

p

)

1

p <

¹

exp

(

;10

r

N

2

)

inf_T

N

Xsup^2H1EX(

I

(

X

)^;

L

^bN(

X

))² exp

(

;2

r

N

2

)

:

These estimators also do not \work" for

p

= 1. Therefore the rate of convergence is of order exponential or faster in the class of analytic process in ^Hp,

p >

1. These optimal estimators are highly parametric because they depend on the covariance function assumed known. Besides they are unstable since they require the inversion of the covariance matrix.

6. Construction of optimal estimators

There are no simple estimators available for any 1

p <

¹

p

⁶= 2.

However, there are estimators for the cases

p

= 2 and

p

=¹. 6.1. Nonparametric optimal estimators in^H2

Let

f

(

z

) =^P_p ⁰

f

p

z

^p be an analytic function. The space

H

²is a Hilbert space with norm^jj

f

^jj2_H²=^P_p ⁰

f

_p²(Rudin (1966), ch.17). For any sampling design

T

N =^f;1

< t

¹

< ::: < t

N

<

1^g, dene the subspace

V

⁰ = ^f

f

²

H

²

f

(

t

i) = 0

i

= 1

::: N

^g

and let

W

be the orthogonal supplement of

V

⁰ in

H

²:

V

^0L

W

=

H

²

V

^{0 ?}

W

. An inspection of the kernel of the linear application from

H

² onto^RN,

f

^! (

f

(

t

¹)

::: f

(

t

N)) makes it clear that dim(

W

) =

N

. Consider the following

N

functions of

H

²

g

i(

z

) = ^X

p ⁰

t

^p_i

z

^p

z

^2;1

1]

i

= 1

::: N:

For any function

f

²

V

⁰,

< fg

i

>

_H²= ^P_p ⁰

t

^p_i

f

p =

f

(

t

i) = 0, and therefore

V

^{0 ?}span^f

g

i

i

= 1

::: N

^g. Moreover, ^f

g

i

i

= 1

::: N

^gis an independent family so that span^f

g

i

i

= 1

::: N

^g=

W

. The orthogonal projection

P

of

H

² onto

W

can be written as

P

(

f

)(

z

) = ^XN

ij⁼¹

ij

f

(

t

j)

g

i(

z

)

where the coe cients

ij are determined using

P

(

g

k) =

g

k and

g

k(

t

j) =

1

1;t^kt^j. They are given by the following system

N

X

j⁼¹

ij 1

1^;

t

k

t

j =

ik for

ik

= 1

::: N:

We then construct the estimator through

L

N(

f

) =

Z

1

;1

P

(

f

) and we ob- tain

L

N(

f

) = ^XN

ij⁼¹

ij

f

(

t

j)

Log

^1+1;t_tⁱi

t

i

:

(6.1)

Following Theorems 3.1 and 3.2, Miccheli and Rivlin (1977), Th.3, the optimality of (6.1) is obtained, and is stated by the following Theorem.

Theorem 6.1. For any sampling design

T

N = (

t

¹

::: t

N) of size

N

, we

have sup

X^2H2EX(

I

(

X

)^;

L

N(

X

))²=

d

(^H2) =

R

²_N

:

The estimators

L

N(

X

) are nonparametric as they do not require knowl- edge of the covariance function.

It follows from Remark 3.3 with

p

= 2 that these estimators have also an exponential rate of convergence.

Example 6.2. Let

X

be a centered process with covariance function

R

(

z

¹

z

²) = exp^f;(

z

^1;

z

²)^2g,^j

z

^1j

^j

z

^2j

<

1. Then the covariance function

R

can be expanded as

R

(

z

¹

z

²) = ^X

nm ⁰

r

nm

z

¹ⁿ

z

^2m

with ^X

nm ⁰

j

r

nm^j

<

¹

:

It follows that the process can be expanded as

X

(

z

) = ^P_n ⁰

x

n

z

ⁿ, where

f

x

n

n

0^g are centered random variables satisfying ^E

x

n

x

m =

r

nm, and thus belongs to^H2. Let

T

N = (_Nⁱ

i

=^;

N

+1

:::

0

::: N

^;1) be a regular sampling. The normalized mean squared error^EX(

I

(

X

)^;

L

N(

X

))²

=

^E

I

²(

X

) was computed for

T

Nwith several values of

N

(cf. Figure 1 in the Appendix).

We see that the mean squared error decreases quickly even with a very small sample size

N

.

6.2. Nonparametric optimal estimators in ^H1

For

p

= ¹, using Bojanov's formula (Bojanov (1974)), we can dene simple linear optimal estimators that use only observations at the optimal sampling points generated by

T

_?N = (

t

^?1

::: t

_?N) = Argmin⁽t¹:::t^N)R;11

B

_N²(

u

)

du

, and thus are nonpara- metric. They are of the form

L

N(

X

) = ^XN

i⁼¹

c

iN (

X

)(

t

_?i)

where the coe cients

c

iN are dened by

c

iN =

Z

1

;1

Q

iN(

x

)

dx

where

Q

iN(

x

) =

!

_i²(

x

)

!

_i²(

t

_?i)

(

1^;

W

_i⁴(

x

)^;2

!

_i⁰(

t

_?i)

!

i(

t

_?i)

W

_i⁰(

t

_?i)

W

i(

x

)(1^;

W

_i²(

x

))

)

and

W

i(

x

) =

x

^;

t

_?i 1^;

xt

_?i

!

i(

x

) = ^YN

k⁼¹k⁶⁼i

W

k(

x

)

The optimality of these estimators is given by the following result.

Theorem 6.3. For the optimal sampling

T

_?N of size

N

, we have

Xsup^2H1EX(

I

(

X

)^;

L

N(

X

))² =

Z

1

;1

B

_N²(

u

)

du

2

:

The optimal sampling design

T

_?N is obtained numerically by minimizing the constant^R;11

B

_N²(

u

)

du

. For instance when

N

= 4,

T

^4?=^f;

t

^?1

^;

t

^?2

t

^?1

t

^?2g where

t

^?1 0

:

892

t

^?2 0

:

401 in which case ^R;11

B

²⁴(

u

)

du

0

:

0518. When

N

= 12,

T

^12? = ^f;

t

^?1

^;

t

^?2

::: t

^?1

t

^?2g where

t

^?1 0

:

986,

t

^?2 0

:

963,

t

^?3 0

:

904,

t

^?4 0

:

777,

t

^?5 0

:

544,

t

^?6 0

:

198 in which case ^R;11

B

¹²² (

u

)

du

8

:

02 10^;5. Round-o problems occur in computing the coe cients of Bo- janov's estimator for large sample sizes and hence it is unstable.

Proof. We have from Lemma 2.2 and Bojanov (1974), for

X

^2H1,

EX(

I

(

X

)^;

L

N(

X

))²

= sup

(

Z

1

;1

h

^;

L

N(

h

)

2

h

²

H

(

R

)

^jj

h

^jjH⁽R⁾1

)

sup

(

Z

1

;1

h

^;

L

N(

h

)

2

h

²

H

¹

^jj

h

^jjH¹ 1

)

=

Z

1

;1

B

_N²(

u

)

du

2

:

The equality follows from Theorem 3.1.

Likewise, from Remark 3.3, we deduce upper and lower bounds for the rate of convergence of the maximal error when these estimators are used

exp

(

;10

r

N

2

)

_Xsup

2H 1

EX(

I

(

X

)^;

L

N(

X

))²exp

(

;2

r

N

2

)

:

Therefore for

p

=¹, using simple nonparametric optimal estimators, the rate of convergence for the maximal error is of the same exponential order as the optimal coe cient estimators

L

^bN(

X

).

Acknowledgements We thank Gabriel Lang for his help in Section 6.1.

Appendix ^A. Figure

Figure 1. 1: Normalized error for simple estimator

L

N(

X

) in ^H2. 2: Normalized error for optimal parametric estimator

L

bN(

X

) in ^H2

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Karim Benhenni: Laboratoire de Statistique et Analyse des Donnees, BSHM, Universite Pierre Mendes France, 1251 Av. Centrale, BP 47 38040 Grenoble Cedex 09, France. E-mail: [email protected].

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