View of Bernstein-type approximations of smooth functions

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BERNSTEIN-TYPE APPROXIMATION OF SMOOTH FUNCTIONS Andrea Pallini

1.INTRODUCTION

The Bernstein polynomials are generally used for the approximation of con-tinuous functions, uniformly, on a closed interval of interest. The Bernstein poly-nomials were introduced to provide a simple proof of the Weierstrass approxima-tion theorem. See Lorentz (1986), chapter 1, and Pinkus (2000). The Bernstein polynomials are worth of being applied to practical contexts, if they are able to agree with a reasonable number of derivatives of the function to approximate. In any case, the Bernstein polynomials can be regarded as a relevant unifying con-cept in approximation theory. See also Korovkin (1960), chapter 1, Davis (1963), chapter 6, Feller (1971), chapter 7, Rivlin (1981), chapter 1, Cheney (1982), chap-ters 1 to 4, and Timan (1994), chapter 1. Here, following the Bernstein polynomi-als, we propose new Bernstein-type approximations based on the binomial distri-bution and the multivariate binomial distridistri-bution. The Bernstein-type approxima-tions can be viewed as the generalizaapproxima-tions of the Bernstein polynomials obtained by considering a convenient approximation coefficient in linear kernels. The Bernstein-type approximations are shown to be uniformly convergent in the sense of Weierstrass. The Bernstein-type approximations are shown to yield a de-gree of approximation that is better than the dede-gree of approximation of the Bernstein polynomials.

In section 2, we overview the main theoretical features of the Bernstein poly-nomials, focussing on the binomial distribution. We propose the Bernstein-type approximations. In section 3, we study the multivariate Bernstein polynomials that are defined by the multivariate binomial distribution. We propose the multi-variate Bernstein-type approximations. In section 4, we study the degrees of ap-proximation by the Bernstein polynomials and the Bernstein-type approxima-tions. In section 5, we study the Bernstein-type estimates for smooth functions of population means. In section 6, we discuss the results of a simulation study on examples of smooth functions of means. Finally, in section 7, we conclude the contribution with some remarks.

We refer to Serfling (1980), Barndorff-Nielsen and Cox (1989), and Sen and Singer (1993), chapter 3, for asymptotics and results in classical theory of statisti-cal inference. We also refer to Aigner (1997), for results on combinatorial theory.

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2. BINOMIAL DISTRIBUTION

2.1. Bernstein polynomials

Let 5 be the space of polynomials ( )_m P x of degree at most m , for all real numbers x . Let g be a bounded, real-valued function defined on the closed in-terval [0,1] . The Bernstein polynomial B_m( ; )g x of order m for the function g is defined as 1 0 ( ; ) ( ) (1 ) m m v v m v m B g x g m v x x v § · ¨ ¸ © ¹

¦

, (1)

where m is a positive integer, and x [0,1]. See Lorentz (1986), chapter 1. Point

x in (1) works as a probability, in the binomial distribution, where x [0,1]. It is seen that B_m( ; )g x 5_m, for every x [0,1].

The Bernstein polynomial B_m( ; )g x , where x [0,1], is a well known linear positive operator. The general approximation theory for the Bernstein polynomial

( ; )

m

B g x as a linear positive operator, where x [0,1], is provided by Korovkin (1960), chapters 1 to 4. See also Appendix (8.1).

The Bernstein polynomial Bm( ; )g x , where x [0,1], was introduced to prove the Weierstrass approximation theorem. See Pinkus (2000). In particular, if ( )g x

is continuous on x [0,1], then we have that lim _m( ; ) ( )

mofB g x g x , (2)

uniformly at any point x [0,1]. The basic proofs of the uniform convergence (2) are detailed in Rivlin (1981), chapter 1, and Lorentz (1986), chapter 1. See also Korovkin (1960), chapters 1 to 4, Davis (1963), chapter 6, Feller (1971), chapter 7, and Cheney (1982), chapters 1 to 4.

2.2. Bernstein-type approximations

The Bernstein polynomial B_m( ; )g x is given by (1), where x [0,1]. The Bern-stein-type approximation B( )_ms ( ; )g x of order m for the function ( )g x is defined as

( ) 1 0 ( ; ) ( ( ) ) (1 ) m m v s s v m v m B g x g m m v x x x x v § · ¨ ¸ © ¹

¦

, (3)

where s ! 1/ 2 is a convenient approximation coefficient, m is a positive inte-ger, and x [0,1]. It is seen that B( )_ms ( ; )g x 5_m , where s ! 1/ 2, for every

[0,1]

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The Bernstein polynomial B_m( ; )g x can be obtained as B_m( 0 )( ; )g x , by setting 0

s in the definition (3) of B( )_ms ( ; )g x , for every x [0,1].

The Bernstein-type approximation B_m( )s ( ; )g x , where s ! 1/ 2, and x [0,1], is a linear positive operator, with the properties outlined in Appendix (8.1).

If ( )g x is continuous on x [0,1], then we have that B( )_ms ( ; )g x o g x( ), where s ! 1/ 2 is fixed, as m o f , uniformly at any point x [0,1]. In Ap-pendix (8.2), under the condition s ! 1/ 2, we provide a proof of this uniform convergence.

The uniform convergence (2) of the Bernstein polynomial B_m( ; )g x , for every [0,1]

x  , can also be proved by setting s 0 in B( )_ms ( ; )g x , given by (3), for every x [0,1], in the proof in Appendix (8.2).

3.MULTIVARIATE BINOMIAL DISTRIBUTION

3.1. Multivariate Bernstein polynomials

Let g be a bounded, real-valued function defined on the closed k-dimensional cube [0,1]k. We let x (x₁,!,x_k)T , where x [0,1] k. The multivariate Bern-stein polynomial B_m( ; x)g for the function g is defined as

1 1 1 1 1 1 1 1 1 m 1 1 1 0 0 ₁ ( ; x) (1 ) (1 ) k k k k k m m k v m v v m v k k k v v k k m v m m B g g x x x x v v m v § · ¨ ¸ § · § · ¨ ¸ ¨ ¸ ¨ ¸ © ¹ © ¹ ¨ ¸ © ¹

¦ ¦

" # " " , (4) where m (m₁,!,m_k)T are positive integers, and x [0,1] k. See Lorentz (1986), chapter 2. Points x₁,!,x_k in (4) work as probabilities, in a multivariate binomial distribution characterized by the product of k mutually independent binomial distributions, where x [0,1] k. It is seen that the multivariate Bernstein polyno-mial B_m( ; x)g 5 , where _m

1 k

i i

m

¦

m is the total degree in B_m( ; x)g , for every x [0,1] k.

The multivariate Bernstein polynomial B_m( ; x)g , is a linear positive operator, where x [0,1] k. See Appendix (8.1).

The definition of the multivariate Bernstein polynomial B_m( ; x)g , where x [0,1] k, is suggested in Lorentz (1986), chapter 2, without proving its uniform

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convergence. It can be shown that the multivariate Bernstein polynomial

m( ; x)

B g converges to (x)g uniformly, at any k -dimensional point of continuity x [0,1] k, as m o f_i , where i ! .1, ,k

3.2. Multivariate Bernstein-type approximations

The multivariate Bernstein polynomial B_m( ; x)g is given by (4), where x [0,1] k. The multivariate Bernstein-type approximation B_m( )s ( ; x)g for the func-tion (x)g is defined as 1 1 1 1 1 1 1 1 ( ) m 0 0 ₁ ( ) ( ; x) ( ) k k s m m s v v _s k k k k k m m v x x B g g m m v x x § · ¨ ¸ ¨ ¸ ¨ ¸ © ¹

¦ ¦

" # 1 1 1 1 1 1 1 (1 ) k(1 ) k k k v m v v m v k k k m m x x x x v v § · § · _¨ _¸ _¨ _¸ © ¹"© ¹ " , (5)

where s ! 1/ 2 is a convenient approximation coefficient, m (m₁,!,m_k)T are positive integers, and x [0,1] k. It is seen that the multivariate Bernstein-type approximation B( )_ms ( ; x)g 5 , where _m

1 k

i i

m

¦

m is the total degree in B_m( )s ( ; x)g , where s ! 1/ 2, for every x [0,1] k.

The multivariate Bernstein polynomial B_m( ; x)g can be obtained as B_m( 0 )( ; x)g , by setting s 0 in the definition (5) of B_m( )s ( ; x)g , for every x [0,1] k.

The multivariate Bernstein-type approximation B_m( )s ( ; x)g , where s ! 1/ 2, and x [0,1] k, is a linear positive operator, with the properties outlined in Ap-pendix (8.1).

The multivariate Bernstein-type approximation B_m( )s ( ; x)g converges to (x)g

uniformly, where s ! 1/ 2 is fixed, at any k -dimensional point of continuity x [0,1] k, as m o f_i , where i ! . In Appendix (8.2), under the condition 1, ,k

1/ 2

s !  , we provide a proof of this uniform convergence.

The uniform convergence of the multivariate Bernstein polynomial B_m( ; x)g , for every x [0,1] k, can be proved by setting s 0 in B( )_ms ( ; x)g , given by (5), for every x [0,1] k, in the proof in Appendix (8.2).

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4.DEGREE OF APPROXIMATION

4.1. Bernstein polynomials

Let ( )Z G be the modulus of continuity of the real-valued function g , for every G ! . The modulus of continuity ( )0 Z G of the function ( )g x , where

[0,1]

x  , is defined as the maximum of g x( ₀) g x( ) , for x₀ x , where G

0, [0,1]

x x  . If the function g is continuous, then ( )Z G o , as 0 G o .0

Setting G m1/ 2, for every x [0,1], it can be shown that the Bernstein polynomial B_m( ; )g x , given by (1), has degree of approximation

2 1/ 2 1 ( ; ) ( ) 1 ( ) 4 m B g x g x d§_¨ m ·_¸Z m © ¹ . (6) We let 1/ 2 2 1 x k i i x § · ¨ ¸ ©

¦

¹ , where x [0,1] k

. The modulus of continuity ( )Z G of the real-valued function (x)g , x [0,1] k, is defined as the maximum of

0

(x ) (x)

g g , for x -x₀ , where G x , x [0,1]₀ k. If the function g is continu-ous, then ( )Z G o , as 0 G o .0

Setting G m1/ 2, for every x [0,1] k, it can be shown that the multivariate Bernstein polynomial B_m( g;x) , given by (4), has degree of approximation

1 1 1/2 1 1 ( ; x) (x) 1 ( ) 4 k i i B g g d_¨§ m m ·_¸Z m ©

¦

¹ m , (7) where 1 k i i m

¦

m .

We can observe that the coefficient 1m2/4 in the upper bound (6) corre-sponds to the coefficient 1 1

1 1 /4 k i i m m

¦

in the upper bound (7), by consider-ing the dimension k in the upper bound (7). 1

4.2. Bernstein-type approximations

In Appendix (8.3), for every x [0,1], it is shown that the Bernstein-type ap-proximation B_m( )s ( ; )g x , given by (3), where s ! 1/ 2, has degree of approxima-tion

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( ) 1 2 1 1/ 2 ( ) 1 ( ; x) (x) 1 4 s s m B g g d§_¨ m m ·_¸Z m © ¹ . (8)

The upper bound (6) can be obtained from the upper bound (8), by setting 0

s .

In Appendix (8.3), for every x [0,1] k, it is also shown that the multivariate Bernstein-type approximation B_m( )s ( ; x)g , given by (5), where s ! 1/ 2, has de-gree of approximation ( ) 1 2 1 1/ 2 m 1 1 ( ; x) (x) 1 ( ) 4 k s s i i B g g d_¨§ m m ·_¸Z m ©

¦

¹ , (9) where 1 k i i m

¦

m .

The upper bound (7) can be obtained from the upper bound (9), by setting 0

s . We can observe that the coefficient in the upper bound (8) corresponds to the dimension k for the coefficient in the upper bound (9). 1

4.3. A comparison

Given a convenient value for the approximation coefficient s , the Bernstein-type approximations B_m( )s ( ; )g x and B( )_ms ( ; x)g , given by (3) and (5), where

1/ 2

s !  , can typically outperform the corresponding Bernstein polynomials ( ; )

m

B g x and B_m( ; x)g , given by (1) and (4), for any function g to approximate, for every x [0,1] and x [0,1] k, respectively.

The value for s , where s ! 1/ 2, can only modify the coefficients in the de-grees of approximation (8) and (9), without affecting their modulus of continuity

1/ 2

(m )

Z , for any fixed m and any fixed m (m₁,!,m_k)T, respectively.

Very large values of s do not bring any advantage, with typical examples of application of the Bernstein-type approximations B_m( )s ( ; )g x and B( )_ms ( ; x)g , de-fined by (3) and (5), where s ! 1/ 2, x [0,1] and x [0,1] k . Convergence to unity of the coefficients that determine the degrees of approximation (8) and (9) is rather fast, as s increases.

In Figure 1 we compare the Bernstein polynomial B_m( ; )g x , given by (1), with the Bernstein-type approximation B_m( )s ( ; )g x , given by (3), where m 2, the function g is defined as g x( ) x4, g x( ) x3, and g x( ) x2, for values of x in the interval [0.01, 0.99] , and s 0.5, 0.6, 0.8,1.1,1.5, 2. The value s 2 gives the best performance; the value s 0.5 gives the worst performance. The per-formance of the Bernstein-type approximation (3) can be even more substantial for values of m larger than the chosen value m 2.

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(a) 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 (b) 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 (c) 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4 0 100 300 500 0.0 0.1 0.2 0.3 0.4

Figure 1 – The difference between B_m( ; )g x g x( ) (dotted line), and the differences ( )s _{( ; )} _{( )}

m

B g x g x , (solid lines), for the Bernstein polynomial Bm( ; )g x , given by (1), and the Bern-stein-type approximations B( )_ms ( ; )g x , given by (3), for 500 equidistant values of x that range in the interval [0.01, 0.99] , where s 0.5, 0.6, 0.8,1.1,1.5, 2, and m 2; g x( ) x4, panel (a),

3 ( ) g x x , panel (b), g x( ) x2, panel (c). (a) 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 (b) 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 (c) 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15 0 100 300 500 0.0 0.05 0.10 0.15

Figure 2 – The difference between B_m( ; )g x g x( ), (dotted line), and the differences ( )s _{( ; )} _{( )}

m

B g x g x , (solid lines), for the Bernstein polynomial Bm( ; )g x , given by (1), and the Bernstein-type approximations B( )ms ( ; )g x , given by (3), where s 0.5, 0.6, 0.8,1.1,1.5, 2, and

2 ( )

g x x  , for 500 equidistant values of x that range in the interval [0.01,0.99] ; x m 2, panel (a), m 3, panel (b), m 5, panel (c).

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In Figure 2, we show the performance of the Bernstein-type approximation

( )

( ; )

s m

B g x given by (3), where the function g is defined as g x( ) x2 , for x

values of x in the interval [0.01, 0.99] , and s 0.5, 0.6, 0.8,1.1,1.5, 2. The Bern-stein-type approximation B_m( )s ( ; )g x , given by (3), clearly performs better than the Bernstein polynomial B_m( ; )g x , given by (1), as m increases, from m 2 to

5

m , for values of x in the interval [0.01, 0.99] , and s 0.5, 0.6, 0.8,1.1,1.5, 2.

5. ESTIMATION OF SMOOTH FUNCTIONS OF MEANS

5.1. Bernstein-type estimators

The Bernstein-type approximations B( )_ms ( ; )g x and B_m( )s ( ; x)g , given by (3) and (5), respectively, where x [0,1] and x [0,1] k, can be used for estimating smooth functions of population means in the statistical inference from a random sample of n independent and identically distributed (i.i.d.) random observations.

Let X be a univariate random variable with values x [0,1], with distribution function F , and finite mean P E X[ ]. We want to estimate a population char-acteristic T of the form T g( )P , where g is a smooth function g: [0,1]oR1. The natural estimator of T is Tˆ g x( ), where x is the sample mean, calculated on a random sample of n i.i.d. observations X_j, j ! , of X. That is, 1, ,n

1 1 n

j j

x n

¦

X . An alternative estimator of T g( )P can be obtained as the Bernstein-type estimator B( )_ms ( ; )g x defined as

( ) 1 0 ( ; ) ( ( ) ) (1 ) m v m v s s m v m B g x g m m v x x x x v § · ¨ ¸ © ¹

¦

, (10)

where s ! 1/ 2. The Bernstein-type estimator (10) follows from the definition (3) of B( )_ms ( ; )g x , by substituting the argument x [0,1] with the sample mean

x , where x ranges in [0,1].

Let X be a k -variate random variable with values x [0,1] k, where

1

X (X ,!,X_k)T , with distribution function F , and finite k -variate mean [ X ]

E

P , P (P₁,!,P_k)T . We want to estimate T g( )P , where

1

: [0,1]k

g oR . The natural estimator of T is Tˆ g( x ), where x (x₁,!,x_k)T is the k -variate sample mean on a random sample of n i.i.d. observations X ,_i

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1, , i ! , of X , n 1 1 n i ij j x n

¦

X , i ! . An alternative estimator of 1, ,k ( ) g

T P can be obtained as the multivariate Bernstein-type estimator B( )_ms ( ; x )g

defined as 1 1 1 1 1 1 1 1 ( ) m 0 0 ₁ ( ) ( ; x ) ( ) k k s m m s v v _s k k k k k m m v x x B g g m m v x x § · ¨ ¸ ¨ ¸ ¨ ¸ © ¹

¦ ¦

" # 1 1 1 1 1 1 1 (1 ) k(1 ) k k k v m v v m v k k k m m x x x x v v § · § · _¨ _¸ _¨ _¸ © ¹"© ¹ " , (11)

where s ! 1/ 2. The multivariate Bernstein-type estimator (11) follows the defi-nition (5) of B( )_ms ( ; x)g , by substituting the argument x [0,1] k with the sample

x , where x ranges in [0,1]k.

5.2. Orders of error of Bernstein-type estimators

In Appendix (8.4), it is shown that the Bernstein-type estimator B_m( )s ( ; )g x , given by (10), where s ! 1/ 2, can be an accurate substitute for the natural esti-mator ( )g x . In particular, if we consider the sample size n as fixed, then we have that ( ) 2 1 ( ; ) ( ) ( ) s s m B g x g x O m , (12) where s ! 1/ 2, as m o f .

In Appendix (8.4), it is also shown that the Bernstein-type estimator B_m( )s ( ; x )g , given by (11), where s ! 1/ 2, and m (m₁,!,m_k)T, can be an accurate substi-tute for the natural estimator ( x )g . In particular, if we consider the sample size n as fixed, then we have that

( ) 2 1 m 1 ( ; x ) ( x ) ( ) k s s i i B g g

¦

O m , (13) where s ! 1/ 2, as m o f_i , i ! .1, ,k

We know that x P O n_p( 1/2), as n o f . We also know that

1/ 2

( ) ( ) _p( )

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In Appendix (8.4), it is shown that the Bernstein-type estimator B_m( )s ( ; )g x , given by (10), where s ! 1/ 2, is a consistent estimator of ( )g P , as m o f , and n o f . In particular, it is shown that

( ) 2 1 1/2 ( ; ) ( ) ( ) ( ) s s m p B g x g P O m O n , (14) where s ! 1/ 2, as m o f , and n o f .

We know that x P O (_p n1/ 2), where x_i P_i O n_p( 1/ 2), for every 1, ,

i ! , as n o f . We also know that k g( x ) g( )P O n_p( 1/ 2), as n o f . In Appendix (8.4), it is also shown that the multivariate Bernstein-type estima-tor B_m( )s ( ; x )g , given by (11), where s ! 1/ 2, and m (m₁,!,m_k)T, is a consis-tent estimator of ( )g P , as m o f_i , where i ! , and n o f . In particular, 1, ,k

it is shown that ( ) 2 1 1/ 2 m 1 ( ; x ) ( ) ( ) ( ) k s s i p i B g g P

¦

O m O n , (15) where s ! 1/ 2, as m o f_i , i ! , and n o f .1, ,k

5.3. Asymptotic normality of Bernstein-type estimators

The Bernstein-type estimator B_m( )s ( ; )g x is defined by (10), where s ! 1/ 2, and m is a positive integer. We denote by V2 the asymptotic variance of

1/ 2 ( ) n g x , as n o f . That is, 2 2 2 { '( )}g E X[( ) ] V P P ,

where g x'( ) (dx)1dg x( ), x [0,1]. We suppose that m 2s 1 dn1/ 2. Then, the distribution of the Bernstein-type estimator B( )_ms ( ; )g x is asymptotically normal,

1/ 2 ( ) 2

{ _ms ( ; ) ( )} d (0, )

n B g x g P oN V , (16)

where s ! 1/ 2, as m o f , and n o f . See Appendix (8.5).

The Bernstein-type estimator B( )_ms ( ; x )g is defined by (11), where s ! 1/ 2, and m (m₁,!,m_k)T are positive integers. We denote by V2 the asymptotic variance of n1/ 2g( x ), as n o f . That is,

1 1 2 1 _x 1 _x 1 1 ( ) ( , , , , ) ( ) ( , , , , ) [( )( )] . k k i i k j j k i j i j x g x x x x g x x x E X X P _P V P P w w w w

¦¦

! ! ! !

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We suppose that m_i 2s 1dn1/ 2, for every i ! . Then, the distribution of 1, ,k

the Bernstein-type estimator B_m( )s ( ; x )g is asymptotically normal,

1/ 2 ( ) 2

m

{ s ( ; x ) ( )} d (0, )

n B g g P oN V , (17)

where s ! 1/ 2, as m o f_i , i ! , and n o f . See Appendix (8.5). 1, ,k

6.A SIMULATION STUDY

In Figure 3, we report on a small Monte Carlo experiment concerning with the effectiveness of the Bernstein-type estimator B_m( )s ( ; )g x , given by (10), where

3, 4, 4,5

m , and s 0.5, 0.5, 0.6, 2, for approximating the smooth function of means g x( ) x2 . Independent samples of size x n 4,6,10,16, from the uni-form distribution on the interval (0,1) , were simulated. The values of the Bern-stein-type estimator B_m( )s ( ; )g x were practically indistinguishable from the values of the natural estimator ( )g x . We denote by Vˆ_n2 the Monte Carlo variance of

(a) 5 10 15 20 -0.01 0.0 0.01 0.02 0.03 0.04 (b) 5 10 15 20 -0.01 0.0 0.01 0.02 0.03 0.04 (c) 5 10 15 20 -0.01 0.0 0.01 0.02 0.03 0.04 (d) 5 10 15 20 -0.01 0.0 0.01 0.02 0.03 0.04

Figure 3 – The difference B( )_ms ( ; )g x g x( ), (symbol ʀ), where the Bernstein-type estimates B( )_ms ( ; )g x are obtained as (10), and g x( ) x2 , for 20 samples of size n , from the uniform distribution on x the interval (0,1) ; s 0.5, m 3, and n 4, in panel (a), s 0.5, m 4, and n 6, in panel (b),

0.6

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the Bernstein-type estimator B_m( )s ( ; )g x , given by (10). The variance of

( 0.5) 2

3 ( ; )

B x x x was Vˆ₄2 0.070565, the variance of B₄( 0.5)(x2 x x; ) was

2 6

ˆ 0.056010

V , the variance of B( 0.6)₄ (x2 x x; ) was Vˆ₁₀2 0.038841, and the variance of B₅( 2 )(x2 x x; ) was Vˆ₁₆2 0.018470.

In Figure 4, we also report on an equivalent Monte Carlo experiment concern-ing with the effectiveness of the multivariate Bernstein-type estimator B_m( )s ( ; x )g , given by (11), where m₁ m₂ ,3 m1 m2 ,4 m1 m2 , and 4 m1 m2 ,5

and s 0.5, 0.5, 0.6, 2, for approximating the ratio of means g( x ) (x₂)1x₁. In-dependent samples, of size n 4,6,10,16, from a bivariate distribution with in-dependent uniform marginals on the interval (0,1) , were simulated. The multi-variate Bernstein-type estimator B( )_ms ( ; x )g practically took the same values as its natural counterpart g( x ). We denote by Vˆ_n2 the Monte Carlo variance of the multivariate Bernstein-type estimator B_m( )s ( ; x )g , given by (11). The variance of ( 0.5) ₂ 1 ₁

( 3,3)T(( ) ; x )

B x x was Vˆ₄2 0.629584, the variance of

(a) 5 10 15 20 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (b) 5 10 15 20 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (c) 5 10 15 20 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (d) 5 10 15 20 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Figure 4 – The difference B_m( )s ( ; x )g g( x ), (symbol ʀ), where the Bernstein-type estimates ( )

m( ; x )

s

B g are obtained as (11), and g( x ) (x₂)1x₁, for 20 samples of size n , from independent uniform distributions on the interval (0,1) ; s 0.5, m₁ m₂ , and 3 n 4, in panel (a), s 0.5,

1 2 4

m m , and n 6, in panel (b), s 0.6, m₁ m₂ , and 4 n 10, in panel (c), and s 2,

1 2 5

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( 0.5) 1

2 1

( 4 ,4 )T(( ) ; x )

B x x was Vˆ₆2 0.106994, the variance of ( 0.6) ₂ 1 ₁

( 4 ,4 )T(( ) ; x )

B x x was

2 10

ˆ 0.101995

V , and the variance of ( 2 ) ₂ 1 ₁

( 5,5)T(( ) ; x )

B x x was Vˆ₁₆2 0.064750. 7.CONCLUDING REMARKS

1) Alternative definitions for the multivariate Bernstein polynomial B_m( ; x)g , given by (4), where x [0,1] k, and the multivariate Bernstein-type approximation

( ) m ( ; x)

s

B g , given by (5), where s ! 1/ 2, and x [0,1] k, can be obtained by replacing the multivariate binomial distribution by other multivariate discrete dis-tributions. The multinomial distribution, the multinomial-related distributions, and the other multivariate discrete distributions are reviewed in Johnson, Kotz and Balakrishnan (1997), chapters 35 to 42. See also Feller (1968), chapter 6, Johnson and Kotz (1977), chapter 1 and 2, and Sveshnikov (1978), chapters 1 and 2.

2) In the multivariate Bernstein-type approximation B( )_ms ( ; x)g , given by (5), where s ! 1/ 2, and x [0,1] k, we can use a different approximation coefficient for each component. More precisely, we can use s ( ,s₁ !,s_k)T in a straight-forward generalization B_m(s)( ; x)g of the multivariate Bernstein polynomial

m( ; x)

B g given by (4), where x [0,1] k, following the definition (5) of B_m( )s ( ; x)g , where s ! 1/ 2, and x [0,1] k.

3) The Bernstein-type approximations B_m( )s ( ; )g x and B( )_ms ( ; x)g , given by (3) and (5), respectively, where s ! 1/ 2, x [0,1], and x [0,1] k, can be studied in order to approximate derivatives of functions, monotone functions, convex func-tions, functions of bounded variation, discontinuous funcfunc-tions, and integrable functions. See Lorentz (1986), chapters 1 and 2.

4) The Bernstein polynomials B_m( ; )g x and B_m( ; x)g and the Bernstein-type approximations B( )_ms ( ; )g x and B_m( )s ( ; x)g , where s ! 1/ 2, given by (1) and (4), and (3) and (5), admit extensions on bounded intervals x[ , ]D E , and bounded regions x [ D E₁, ₁]u u" [D E_k, _k], and extensions on unbounded intervals

[ , )

x D f , and unbounded regions x [ D1,f u u) " [Dk,f). See Lorentz

(1986), chapter 2.

5) In the Bernstein-type approximations B( )_ms ( ; )g x and B( )_ms ( ; x)g , given by (3) and (5), where s ! 1/ 2, and x [0,1] k, the linear kernels ms(m v1 x)x

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and h( )s (m v x_i, ;_i _i), where s ! 1/2, x [0,1], and v_i 0,1,!,m_i , i ! ,1, ,k

1

x (x ,!,x_k)T [0,1]k.

6) Following Babu, Canty and Chaubey (2002), the Bernstein-type approxima-tion B_m( )s ( ; )g x , given by (3), where s ! 1/ 2, and x [0,1], can be used for dis-tribution and density estimation. Let F_n be the empirical distribution function on a random sample of n i.i.d. observations X_i , i ! , from a distribution 1, ,n

function F , 1 1 ( ) ( ) n n i i

F x n

¦

I X dx , x [0,1], where (I A) denotes the indi-cator function of the set A . The Bernstein-type estimator of F can be defined as B_m( )s (F x_n; ), s ! 1/ 2, x [0,1]. Let f be the density of F . The Bernstein-type estimator of f can be defined as m B_m( )s₁( f x_n; ), s ! 1/ 2, x [0,1], where

1 1 1

( ) ( ( 1)) ( )

n n n

f m v F m v F m v , v 0,1,! , and (0) 0,m f_n .

7) The Bernstein-type approximations B_m( )s ( ; )g x and B_m( )s ( ; x )g , given by (10) and (11), where s ! 1/ 2, respectively, can be regarded as examples of random functions. See Sveshnikov (1978), chapter 7, and Gikhman and Skorokhod (1996), chapter 4.

8. APPENDIX

8.1. Basic properties of the Bernstein-type approximations (3) and (5)

The Bernstein-type approximations B( )_ms ( ; )g x and B_m( )s ( ; x)g , given by (3) and (5), where s ! 1/ 2, and x [0,1] and x [0,1] k, are linear positive operators. Let G be a finite constant. Let g , g₁, and g₂ be functions, ( )g x , g x₁( ), and

2( ) g x , x [0,1]. We have ( ) ( ) ( ; ) ( ; ) s s m m B G g x GB g x , ( ) ( ) ( ) 1 2 1 2 ( ; ) ( ; ) ( ; ) s s s m m m B g g x B g x B g x , [0,1]

x  . If g x₁( )d g x₂( ), for all x [0,1], we have

( ) ( ) 1 2 ( ; ) ( ; ) s s m m B g x dB g x , [0,1]

x  . Multivariate versions of these properties hold for B( )_ms ( ; x)g , where 1/ 2

s !  , g(x) , and x [0,1] k. The corresponding properties for the Bernstein polynomials B_m( ; )g x and B_m( ; x)g , given by (1) and (4), where x [0,1] and

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8.2. Uniform convergence of the Bernstein-type approximations (3) and (5)

The uniform norm g of the function ( )g x , where x [0,1], is defined as

> @0,1

max ( )

x

g g x

.

The Bernstein-type approximation B_m( )s ( ; )g x , where s ! 1/ 2, and x [0,1], is given by (3). We want to show that, given any constant H ! , there exists a posi-0 tive integer m₀, such that

0 ( ) ( ; ) ( ) s m B g x g x , H (18) for every x [0,1].

For every x [0,1], the Bernstein-type approximation B_m( )s (1; )x is

( )

(1; ) 1

s m

B x . (19)

We define the function P₁( )x as P₁( )x , and the function x P₂( )x as x

2 2( )x x

P . The Bernstein-type approximation B_m( )s ( ( ) ; )P₁ x x is

( ) 1 ( ( ) ; ) s m B P x x . x (20)

The Bernstein-type approximation B_m( )s (P₂( ); )x x is

2 ( ) 1 2 0 ( ( ); ) { ( ) } (1 ) m m v s s v m v m B x x m m v x x x x v P § ·_{¨ ¸} © ¹

¦

2 1 2 (1 ) s m x x x . (21)

Suppose that g M . We take x ₀ [0,1]. We have

0

2M g x( ) g x( ) 2M

d d , (22)

where x x ₀, [0,1]. The function g is continuous; given H₁ ! , there exists a 0 constant G ! , such that 0

1 g x( 0) g x( ) 1

H H

, (23)

for x₀ x , and G x x ₀, [0,1]. From (22) and (23), it follows that

2 2 1 2 0 0 1 2 0 2 2 ( ) ( ) ( ) ( ) M M x x g x g x x x H H G G d d , (24)

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for x x ₀, [0,1]. If x₀ x , (23) implies (24), G x x ₀, [0,1]. If

0

x x t , then G G2( x₀ x )2 t , and (22) implies (24), 1 x x ₀, [0,1]. Fol-lowing Appendix (8.1), (24) becomes

2 ( ) ( ) 1 2 0 1 2 ( ) 0 2 2 ( ( ) ; ) ( ; ) ( ) 2 ( ( ) ; ) s s m m s m M B x x x B g x g x M B x x x H H G G d d , (25)

for x x ₀, [0,1]. We observe that ( x₀ x )2 x₀2 x2 2x x₀ , x x ₀, [0,1]. The Bernstein-type approximations B( )_ms (1; )x , B_m( )s ( ( ); )P₁ x x , and

( ) 2

( ( ); )

s m

B P x x , given by (19), (20) and (21), imply in (25) that

2 ( ) 2 1 0 ( ( ) ; ) (1 ) s s m B x x x m x x , 0, [0,1] x x  . That is, 2 ( ) 2 1 0 (( ) ; ) ( ) s s m B x x x O m ,

as m o f , x [0,1]. The condition s ! 1/ 2 is required for the uniform con-vergence. In fact, we can observe that 0dx(1x) 1/4d , x [0,1]. Then, (25) becomes ( ) 2 1 1 2 ( ; ) ( ) 2 s s m M B g x g x H m G d , (26) [0,1]

x  . Setting H₁ H/ 2, for any m₀ !(G H2 )1M , the uniform convergence (18) is proved.

The convergence B_m( )s ( ; )g x o g x( ), where s ! 1/ 2, is uniform, at any point of continuity x [0,1], as m o f , in the sense that the upper bound (26) for the uniform norm does not depend on x , x [0,1].

The multivariate Bernstein-type approximation B( )_ms ( ; x)g , where s ! 1/ 2, and x [0,1] k, is given by (5). We observe that k is fixed and does not depend on m . Considering the uniform norm g of the function (x)g , x [0,1] k, defined as

> @ x 0,1 max (x) k g g .

we want to show that, given any constant H ! , there exist positive integers 0

0 01 0

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0 ( ) m ( ; x) (x) s B g g , H (27) for every x [0,1] k.

For every x [0,1] k, the multivariate Bernstein-type approximation B_m( )s (1; x) is

( )

m (1; x) 1 s

B , (28)

where s ! 1/ 2. We define the functions ₁

1 (x) k i i x P

¦

and ₂ 2 1 (x) k i i x P

¦

. The multivariate Bernstein-type approximation B_m( )s (P₁(x);x) is

( ) m 1 1 ( (x); x) k s i i B P

¦

x , (29)

and the multivariate Bernstein-type approximation B_m( )s (P₂(x); x) is 1 1 2 ( ) 1 m 2 1 1 1 ( (x); x) { ( ) } k k m m k s s i i i i i v v i B P m m v x x

¦ ¦ ¦

" 1 1 1 1 1 1 1 (1 ) k(1 ) k k k v m v v m v k k k m m x x x x v v § · § · _¨ _¸ _¨ _¸ © ¹"© ¹ " 2 1 2 1 (1 ) k s i i i i i m x x x

¦

. (30)

Suppose that g M . We take x₀ (x₀₁,!,x₀_k)T [0,1]k. We observe that

2 2 2 0 0 0 1 ( x x ) ( 2 ) k i i i i i x x x x

¦

, 0

x ,x [0,1] k. The uniform convergence (27) follows from the result

2 ( ) 2 1 m 0 1 (( x -x ) ; x) (1 ) k s s i i i i B

¦

m x x , 0 x ,x [0,1] k. That is,

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( ) 2 2 1 m 0 1 (( x -x ) ; x) ( ) k s s i i B

¦

O m , as m o f_i , where i ! ,1, ,k x ,x [0,1]0 k

. Under the condition s ! 1/ 2, the convergence B( )_ms ( g;x)o g(x) is uniform at any point of continuity x [0,1] k, as

i

m o f, where i ! .1, ,k

The uniform convergence of the Bernstein polynomials B_m( ; )g x and

m( ; x)

B g , given by (1) and (4), where x [0,1] and x ,x [0,1]₀ k, can also be ob-tained by setting s 0 above.

8.3. Degrees (8) and (9) of approximation by the Bernstein-type approximations (3) and (5) For every G ! , we denote by 0 O(x x₀, ; )G the maximum integer less than or equal to G1 x₀ , where x x x ₀, [0,1]. We recall the definition of modulus of continuity ( )Z G , where G ! . We have 0

0 0

( ) ( ) ( ){1 ( , ; )}

g x g x dZ G O x x G , (31)

0, [0,1]

x x  .

The Bernstein-type approximation B( )_ms ( ; )g x is given by (3), where s ! 1/ 2, and x [0,1]. Then, we have

¦

[0,1] x  . It follows that ( ) 2 2 1 ( ; ) ( ) ( ){1 (1 )} s s m B g x g x dZ G G m x x , [0,1]

x  . We observe that 0dx(1x) 1/4d , x [0,1]. Setting G m1/ 2, we finally have the degree of approximation (8).

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For every G ! , we denote by 0 O(x ,x ; )₀ G the maximum integer less than or equal to G1 x₀ , where x x , x [0,1]0 k . We have 0 0 (x ) (x) ( ){1 (x ,x ; )} g g dZ G O G ,

where ( )Z G is the modulus of continuity, G ! , and 0 x , x [0,1]₀ k.

The multivariate Bernstein-type approximation B( )_ms ( ; x)g is given by (5), where 1/ 2 s !  , and x [0,1] k. We have 1 1 1 1 1 1 1 1 1 ( ) m 0 0 ₁ ( ) ( ; x) (x) ( ) k k s m m s v v _s k k k k k k m m v x x x B g g g g x m m v x x § · § · ¨ ¸ _¨ _¸ d _¨ _¸ _¨ _¸ ¨ ¸ ¨ ¸ _© _¹ © ¹

¦ ¦

" # # 1 1 1 1 1 1 1 (1 ) k(1 ) k k k v m v v m v k k k m m x x x x v v § · § · _¨ _¸ _¨ _¸ © ¹"© ¹ " 1 1 2 2 2 2 0 0 1 ( ) 1 ( ) k k m m k s i i i i v v i m v m x Z G G ½ d _® _¾ ¯ ¿

¦ ¦

"

¦

1 1 1 1 1 1 1 (1 ) k(1 ) k k k v m v v m v k k k m m x x x x v v § · § · _¨ _¸ _¨ _¸ © ¹"© ¹ " , x [0,1] k. Thus, we have ( ) 2 2 1 m 1 ( ; x) (x) ( ) 1 (1 ) k s s i i i i B g g dZ G ® G m x x ½¾ ¯

¦

¿,

x [0,1] k. We can observe that 0dxi(1xi) 1/4d , i ! , x [0,1]1, ,k

k . Set-ting G m1/ 2, where 1 k i i

m

¦

m , we finally have the degree of approximation (9). Degrees (6) and (7) of approximation by the Bernstein polynomials B_m( ; )g x

and B_m( ; x)g , given by (1) and (4), where x [0,1] and x [0,1] k, can be proved by setting s 0 above.

8.4. Orders of error in (12) and (13), and orders of error in probability in (14) and (15) The Bernstein-type approximation B_m( )s ( ; )g x is given by (10), where

1/ 2

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2 2

''( ) ( ) ( )

g x dx d g x be the first two derivatives of the function g x( ), [0,1]

x  . By Taylor expanding the function g m( s(m v1 x)x) around x , for every v 0,1,! , we have ,m ( ) 1 0 2 2 1 0 2 1 3 2 ( ; ) ( ) 1 '( ) ( ) (1 ) 2 1 ''( ) ( ) (1 ) 2 1 ( ) ''( ) (1 ) ( ), 2 s m m v m v s v m v m v s v s s B g x g x m g x m m v x x x v m g x m m v x x x v g x g x m x x O m § · _{¨ ¸} © ¹ § · _{¨ ¸} © ¹

¦

"

where s ! 1/ 2, as m o f . Order O m( 2s 1) of error in (12), s ! 1/ 2, as

m o f, with n fixed, is thus proved.

The Bernstein-type approximation B_m( )s ( ; x )g is given by (11), where s ! 1/ 2. We consider n as fixed. By Taylor expanding the function

around x (x₁,!,x_k)T, for every v_i !1, ,m_i , i ! , we can prove the 1, ,k

order 2 1 1 ( ) k s i i O m

¦

of error in (13), s ! 1/ 2, as m o f_i , where i ! ,1, ,k with n fixed.

The Bernstein-type approximation B_m( )s ( , )g x is given by (10), where 1/ 2

s !  . We observe that x P O n_p( 1/2), as n o f, and

2 1

(x P) O n_p( ), as n o f. By Taylor expanding the function

1

( s( ) )

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( ) 1 0 2 2 1 0 2 1 1/ 2 2 1/ 2 2 ( ; ) ( ) '( ) { ( ) ( )} (1 ) 1 ''( ) { ( ) ( )} (1 ) 2 ( ) '( ) ( ) 1 ''( ){ ( (1 ) ( )) ( ) } 2 ( ) ( ) ( s m m v m v s v m v m v s v s p s p B g x g m g m m v x x x x v m g m m v x x x x v g g x g m O n x g O n O m P P P P P P P P P P P P P § · _{¨ ¸} © ¹ § · _{¨ ¸} © ¹

¦

" " 1 2 1 1/ 2 1 2 1 1/ 2 ) ( ) ( ) ( ) ( ) ( ), s p p s p O m n O n g P O m O n

where s ! 1/ 2, as m o f , and n o f . Order O m( 2s 1)O n_p( 1/2) of error in probability in (14), as m o f , and n o f , is thus proved.

The Bernstein-type approximation B_m( )s ( ; x )g is given by (11), where s ! 1/ 2. We consider n as fixed. By Taylor expanding the function

around P (P₁,!,P_k)T , for every v_i !1, ,m_i , i ! , we can prove the 1, ,k

order 2 1 1/ 2 1 ( ) ( ) k s i p i O m O n

¦

of error in probability in (15), s ! 1/ 2, as i m o f, where i ! , and n o f .1, ,k

8.5. Asymptotic normality in (16) and (17) Following (12) and (14), we have

1/ 2 ( ) 1/2

{ _ms ( ; ) ( )} { ( ) ( )}

n B g x g P n g x g P ,

where s ! 1/ 2, as m o f , and n o f . We consider the Taylor expansion of ( )

g x around P . An application of the Central Limit Theorem then shows the asymptotic normality in (16), as m o f , and n o f .

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Following (13) and (15), we have

1/ 2 ( ) 1/ 2

m

{ s ( ; x ) ( )} { ( x ) ( )}

n B g g P n g g P ,

where s ! 1/ 2, m (m₁,!,m_k)T, as m o f_i , where i ! , and n o f .1, ,k

We consider the Taylor expansion of ( x )g around P (P₁,!,P_k)T . An applica-tion of the Central Limit Theorem then shows the asymptotic normality in (17), as m o f_i , where i ! , and n o f .1, ,k

Dipartimento di Statistica e Matematica Applicata all’Economia ANDREA PALLINI

Università di Pisa

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RIASSUNTO

Approssimazioni del tipo di Bernstein di funzioni regolari

Viene proposta e studiata un’approssimazione del tipo di Bernstein per funzioni rego-lari. Proponiamo un’approssimazione del tipo di Bernstein con definizioni che diretta-mente applicano la distribuzione binomiale e la distribuzione binomiale multivariata. Le approssimazioni del tipo di Bernstein generalizzano i corrispondenti polinomi di Ber-nstein, considerando definizioni che dipendono da un conveniente coefficiente di appros-simazione in nuclei lineari. Nelle approssimazioni del tipo di Bernstein, studiamo la con-vergenza uniforme ed il grado di approssimazione. Vengono anche proposti e studiati stimatori del tipo di Bernstein per funzioni regolari di medie nella popolazione.

SUMMARY

Bernstein-type approximations of smooth functions

The Bernstein-type approximation for smooth functions is proposed and studied. We propose the Bernstein-type approximation with definitions that directly apply the bino-mial distribution and the multivariate binobino-mial distribution. The Bernstein-type approxi-mations generalize the corresponding Bernstein polynomials, by considering definitions that depend on a convenient approximation coefficient in linear kernels. In the Bernstein-type approximations, we study the uniform convergence and the degree of approxima-tion. The Bernstein-type estimators of smooth functions of population means are also proposed and studied.