Asymptotic properties of Bernstein estimators on the simplex. Part 2: the boundary case

1 Introduction

The $d$ -dimensional simplex and its interior are defined by

(1.1)

where $∥x∥1\vcentcolon=∑di=1|xi|$ . For any cumulative distribution function $F$ on $S$ , define the Bernstein polynomial of order $m$ for $F$ by

F⋆m(x)\vcentcolon=∑k∈Nd0∩mSF(k/m)Pk,m(x),x∈S, m∈N,

(1.2)

where the weights are the following probabilities from the

Multinomial (m, x)

distribution :

Pk,m(x)\vcentcolon=m!(m−∥k∥1)!∏di=1ki!⋅(1−∥x∥1)m−∥k∥1d∏i=1xkii,k∈Nd0∩mS.

(1.3)

The Bernstein estimator of $F$ , denoted by $F_{n, m}^{⋆}$ , is the Bernstein polynomial of order $m$ for the empirical cumulative distribution function $Fn(x)\vcentcolon=n−1∑ni=11(−∞,x](Xi)$

, where the random variables

X_{1}, X_{2}, \dots, X_{n}

are independent and

F

distributed. Precisely,

F⋆n,m(x)\vcentcolon=∑k∈Nd0∩mSFn(k/m)Pk,m(x),x∈S, m,n∈N.

(1.4)

Similarly, if $F$ has a density function $f$ , we define the Bernstein density estimator of $f$ (also called smoothed histogram) by

^fn,m(x)\vcentcolon=∑k∈Nd0∩(m−1)Smdnn∑i=11(km,k+1m](Xi)Pk,m−1(x),x∈S, m,n∈N,

(1.5)

where $m^{d}$ is just a scaling factor, namely the inverse of the volume of the hypercube $(\frac{k}{m}, \frac{k + 1}{m}]$ .

2 Results for the density estimator ${^f}_{n, m}$

For every result stated in this section, we will make the following assumption.

∙ \begin{matrix} f is two times % differentiable and its second order partial derivatives are (uniformly) continuous on S . \end{matrix}

(2.1)

In the first lemma, we obtain a general expression for the bias of the density estimator.

Lemma 2.1 (blueBias of ${^f}_{n, m} (x)$ on $S$ ).

Under assumption (2.1), we have, uniformly for $x \in S$ ,

	$B i a s [{^f}_{n, m} (x)]$	$= m^{- 1} Δ_{1} (x) + m^{- 2} Δ_{2} (x)$		(2.2)
		$+ \frac{1}{m^{2}} d \sum i = 1 o (1 + \sqrt{(m - 1) x_{i} (1 - x_{i}) + x_{i}^{2}} + (m - 1) x_{i} (1 - x_{i}) + x_{i}^{2}),$		(2.2)

as $m \to \infty$ , where

	$Δ_{1} (x)$	$\vcentcolon=d∑i=1(12−xi)∂∂xif(x)+12d∑i,j=1(xi1{i=j}−xixj)∂2∂xi∂xjf(x),$		(2.3)
	$Δ_{2} (x)$	$\vcentcolon=d∑i,j=1(161{i=j}+181{i≠j}−12xi1{i=j}−12xj+xixj)∂2∂xi∂xjf(x).$		(2.3)

By considering points $x \in S$ that are close to the boundary in some components (see the subset of indices $J \subseteq [d]$ below), we get the bias of the density estimator near the boundary.

Theorem 2.2 (blueBias of ${^f}_{n, m} (x)$ near the boundary of $S$ ).

Assume (2.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

	$B i a s ({^f}_{n, m} (x))$


	$+ o_{λ_{J}} (m^{- 2} + 1_{{J \neq [d]}} m^{- 1}),$		(2.4)

as $m, n \to \infty$ , where $λ_{i} > 0$ are fixed, $xJ\vcentcolon=(xi)i∈J$ , $x[d]∖J\vcentcolon=(xi)i∈[d]∖J$ and $λJ\vcentcolon=(λi)i∈J$ .

Next, we obtain a general expression for the variance of the density estimator.

Lemma 2.3 (blueVariance of ${^f}_{n, m} (x)$ on $S$ ).

Assume (2.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

	$V a r ({^f}_{n, m} (x))$	$= \frac{m^{d}}{n} f (x) \sum k \in N_{0}^{d} \cap (m - 1) S P_{k, m - 1}^{2} (x)$		(2.5)
				(2.5)

as $m, n \to \infty$ .

By combining Lemma 2.3 and the technical estimate in Lemma A.1, we get the asymptotics of the variance of the density estimator near the boundary.

Theorem 2.4 (blueVariance of ${^f}_{n, m} (x)$ near the boundary of $S$ ).

Assume (2.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

(2.6)

as $m, n \to \infty$ .

By combining Theorem 2.2 and Theorem 2.4, we get the asymptotics of the mean squared error of the density estimator near the boundary. In particular, the optimal smoothing parameter $m$ will depend on the number of components of $x$ that are close to the boundary.

Corollary 2.5 (blueMean squared error of ${^f}_{n, m} (x)$ near the boundary of $S$ ).

Assume (2.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

$M S E ({^f}_{n, m} (x))$	$= n^{- 1} m^{(d + \| J \|) / 2} f (x) {∣ ∣}_{x_{J} = 0} ψ_{[d] ∖ J} (x) \prod i \in J e^{- 2 λ_{i}} I_{0} (2 λ_{i})$	(2.7)

	$+ n^{- 1} m^{(d + \| J \|) / 2} O_{λ_{J}, x_{[d] ∖ J}} (m^{- 1} + 1_{{J \neq [d]}} m^{- 1 / 2})$
	$+ O_{λ_{J}} (m^{- 3}) + o_{λ_{J}} (1_{{J \neq [d]}} m^{- 2}),$

as $m, n \to \infty$ . If the quantity inside the big bracket is non-zero in (2.7), the asymptotically optimal choice of $m$ , with respect to MSE, is

(2.8)

in which case

		$M S E ({^f}_{n, m_{o p t}} (x))$		(2.9)

as $n \to \infty$ .

By adding conditions on the partial derivatives of $f$ , we can remove terms from the bias in Theorem 2.2 and obtain another expression for the mean squared error of the density estimator near the boundary, and the corresponding optimal smoothing parameter $m$ when $J = [d]$ .

Corollary 2.6 (blueMean squared error of ${^f}_{n, m} (x)$ near the boundary of $S$ ).

Assume (2.1) and also

(2.10)

(in particular, the first bracket in (2.2) is zero). Then, for any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ independent of $m$ for all $i \in [d] ∖ J$ , we have

		$M S E ({^f}_{n, m} (x))$		(2.11)

as $m, n \to \infty$ . Note that the last error term is bigger than the main term except when $J = [d]$ . Therefore, if $J = [d]$ and we assume that the quantity inside the big bracket is non-zero in (2.11), the asymptotically optimal choice of $m$ , with respect to MSE, is

(2.12)

in which case

		$M S E ({^f}_{n, m_{o p t}} (x))$		(2.13)

as $n \to \infty$ .

Remark 2.7.

In order to optimize $m$ when $J \neq [d]$ in Corollary 2.6, we would need an even more precise expression for the bias in Theorem 2.2 by assuming more regularity conditions on $f$ than we did in (2.1). We have not tried to do so because the number of terms to manage in the proof of Theorem 2.2 is already barely trackable.

3 Results for the c.d.f. estimator $F_{n, m}^{⋆}$

For every result stated in this section, we will make the following assumption.

∙ \begin{matrix} F is three times % differentiable and its third order partial derivatives are (uniformly) continuous on S . \end{matrix}

(3.1)

Below, we obtain a general expression for the bias of the c.d.f. estimator on the simplex, and then near the boundary.

Lemma 3.1 (blueBias of $F_{n, m}^{⋆} (x)$ on $S$ ).

Under assumption (3.1), we have, uniformly for $x \in S$ ,

$E [F_{n, m}^{⋆} (x)]$	$= F (x) + \frac{1}{2 m} d \sum i, j = 1 (x_{i} 1_{{i = j}} - x_{i} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x)$	(3.2)

as $m \to \infty$ .

Theorem 3.2 (blueBias of $F_{n, m}^{⋆} (x)$ near the boundary of $S$ ).

Assume (3.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

	$B i a s (F_{n, m}^{⋆} (x))$
	$= \frac{1}{m} \sum i, j \in [d] ∖ J \frac{1}{2} (x_{i} 1_{{i = j}} - x_{i} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x) {∣ ∣}_{x_{J} = 0}$

	$+ o_{λ_{J}} (m^{- 2} + 1_{{J \neq [d]}} m^{- 3 / 2}),$		(3.3)

as $m, n \to \infty$ .

Next, we obtain a general expression for the variance of the c.d.f. estimator on the simplex.

Lemma 3.3 (blueVariance of $F_{n, m}^{⋆} (x)$ on $S$ ).

Under assumption (3.1), we have, uniformly for $x \in S$ ,

$V a r (F_{n, m}^{⋆} (x))$	$= n^{- 1} F (x) (1 - F (x))$	(3.4)
	$- n^{- 1} d \sum i = 1 \frac{\partial}{\partial x_{i}} F (x) \sum k, ℓ \in N_{0}^{d} \cap m S ((k_{i} \land ℓ_{i}) / m - x_{i}) P_{k, m} (x) P_{ℓ, m} (x)$
	$+ n^{- 1} d \sum i = 1 O (E [\| ξ_{i} / m - x_{i} \|^{2}]),$

as $m, n \to \infty$ .

By combining Lemma A.2 and Lemma 3.3, we get the asymptotics of the variance of the c.d.f. estimator near the boundary.

Theorem 3.4 (blueVariance of $F_{n, m}^{⋆} (x)$ near the boundary of $S$ ).

Assume (3.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

		$V a r (F_{n, m}^{⋆} (x))$		(3.5)
		$= n^{- 1} F (x) (1 - F (x)) 1_{{J = \emptyset}}$
		$- n^{- 1} m^{- 1} d \sum i = 1 \frac{\partial}{\partial x_{i}} F (x) {∣ ∣}_{x_{J} = 0} {\begin{matrix} - (λ_{i} - e^{- 2 λ_{i}} (I_{0} (2 λ_{i}) + λ_{i} I_{1} (2 λ_{i}))) \cdot 1_{{i \in J}} m^{1 / 2} \sqrt{π^{- 1} x_{i} (1 - x_{i})} \cdot 1_{{i \in [d] ∖ J}} \end{matrix}}$

as $m, n \to \infty$ .

By combining Theorem 2.2 and Theorem 2.4, we get the asymptotics of the mean squared error of the c.d.f. estimator near the boundary.

Corollary 3.5 (blueMean squared error of $F_{n, m}^{⋆} (x)$ near the boundary of $S$ ).

Assume (3.1). For any $x \in S$ such that $x_{i} = \frac{λ_{i}}{m}$ for all $i \in J \subseteq [d]$ and $x_{i} \in (0, 1)$ is independent of $m$ for all $i \in [d] ∖ J$ , we have

		$M S E (F_{n, m}^{⋆} (x))$		(3.6)
		$= n^{- 1} F (x) (1 - F (x)) 1_{{J = \emptyset}}$
		$- n^{- 1} m^{- 1} d \sum i = 1 \frac{\partial}{\partial x_{i}} F (x) {∣ ∣}_{x_{J} = 0} {\begin{matrix} - (λ_{i} - e^{- 2 λ_{i}} (I_{0} (2 λ_{i}) + λ_{i} I_{1} (2 λ_{i}))) \cdot 1_{{i \in J}} m^{1 / 2} \sqrt{π^{- 1} x_{i} (1 - x_{i})} \cdot 1_{{i \in [d] ∖ J}} \end{matrix}}$

as $m, n \to \infty$ . As pointed out in (Leblanc, 2012b, p.2772) for $d = 1$ , there is no optimal $m$ with respect to the MSE when $J \neq \emptyset$ . This is also true here. The remaining case $J = \emptyset$ (when $x$ is far from the boundary in every component) was already treated in Corollary 2.4 of Ouimet (2020a).

4 Proof of the results for the density estimator ${^f}_{n, m}$

4.1 Proof of Lemma 2.1

Using Taylor expansions for any $k$ such that $∥ k / m - x ∥_{1} = o (1)$ , we obtain

	$m^{d} \int_{(\frac{k}{m}, \frac{k + 1}{m}]} f (y) d y - f (x)$
	$= f (k / m) - f (x) + \frac{1}{2 m} d \sum i = 1 \frac{\partial}{\partial x_{i}} f (k / m)$
	$+ \frac{1}{m^{2}} d \sum i, j = 1 (\frac{1}{6} 1_{{i = j}} + \frac{1}{8} 1_{{i \neq j}}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (k / m) \cdot (1 + o (1))$
	$= \frac{1}{m} d \sum i = 1 (k_{i} - m x_{i}) \frac{\partial}{\partial x_{i}} f (x)$
	$+ \frac{1}{2 m^{2}} d \sum i, j = 1 (k_{i} - m x_{i}) (k_{j} - m x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) \cdot (1 + o (1))$
	$+ \frac{1}{2 m} d \sum i = 1 \frac{\partial}{\partial x_{i}} f (x) + \frac{1}{2 m^{2}} d \sum i, j = 1 (k_{j} - m x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) \cdot (1 + o (1))$
	$+ \frac{1}{m^{2}} d \sum i, j = 1 (\frac{1}{6} 1_{{i = j}} + \frac{1}{8} 1_{{i \neq j}}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) \cdot (1 + o (1))$
	$= \frac{1}{m} d \sum i = 1 (k_{i} - (m - 1) x_{i}) \frac{\partial}{\partial x_{i}} f (x) + \frac{1}{m} d \sum i = 1 (\frac{1}{2} - x_{i}) \frac{\partial}{\partial x_{i}} f (x)$
	$+ \frac{1}{2 m^{2}} d \sum i, j = 1 (k_{i} - (m - 1) x_{i}) (k_{j} - (m - 1) x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$- \frac{1}{2 m^{2}} d \sum i, j = 1 x_{i} (k_{j} - (m - 1) x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$- \frac{1}{2 m^{2}} d \sum i, j = 1 x_{j} (k_{i} - (m - 1) x_{i}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$+ \frac{1}{2 m^{2}} d \sum i, j = 1 x_{i} x_{j} \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) + \frac{1}{2 m^{2}} d \sum i, j = 1 (k_{j} - (m - 1) x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$+ \frac{1}{m^{2}} d \sum i, j = 1 (\frac{1}{6} 1_{{i = j}} + \frac{1}{8} 1_{{i \neq j}} - \frac{1}{2} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$+ \frac{1}{m^{2}} d \sum i = 1 o (1 + \| k_{i} - m x_{i} \| + \| k_{i} - m x_{i} \|^{2}) .$

If we multiply the last expression by $P_{k, m - 1} (x)$ and sum over $k \in N_{0}^{d} \cap (m - 1) S$

, then the joint moments from Lemma

B.1, the notation for

Δ_{1} (x)

and

Δ_{2} (x)

in (2.3), Jensen’s inequality and

Vi\vcentcolon=∑k∈Nd0∩(m−1)S|ki−mxi|2Pk,m−1(x),

(4.1)

yield

		$E [{^f}_{n, m} (x)] - f (x)$		(4.2)
		$= \frac{1}{m} d \sum i = 1 (\frac{1}{2} - x_{i}) \frac{\partial}{\partial x_{i}} f (x) + \frac{m - 1}{2 m^{2}} d \sum i, j = 1 (x_{i} 1_{{i = j}} - x_{i} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$

		$= m^{- 1} Δ_{1} (x) + m^{- 2} Δ_{2} (x)$
		$+ \frac{1}{m^{2}} d \sum i = 1 o (1 + \sqrt{(m - 1) x_{i} (1 - x_{i}) + x_{i}^{2}} + (m - 1) x_{i} (1 - x_{i}) + x_{i}^{2}) .$

This ends the proof.

4.2 Proof of Theorem 2.2

Take $x$ as in the statement of the theorem. Using the notation from (2.3), we have

$Δ_{1} (x)$	$= \sum i \in J (\frac{1}{2} - \frac{λ_{i}}{m}) \frac{\partial}{\partial x_{i}} f (x) + \sum i \in J \frac{λ_{i}}{2 m} \cdot \frac{\partial^{2}}{\partial x_{i}^{2}} f (x)$	(4.3)
	$+ \sum i \in [d] ∖ J (\frac{1}{2} - x_{i}) \frac{\partial}{\partial x_{i}} f (x) - \sum \begin{matrix} i \in J j \in [d] ∖ J \end{matrix} \frac{λ_{i} x_{j}}{m} \cdot \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$
	$+ \sum i, j \in [d] ∖ J \frac{1}{2} (x_{i} 1_{{i = j}} - x_{i} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) + O_{λ_{J}} (m^{- 2}),$

and

	$Δ_{2} (x)$	$= \sum (i, j) \in [d]^{2} ∖ ([d] ∖ J)^{2} (\frac{1}{6} 1_{{i = j}} + \frac{1}{8} 1_{{i \neq j}}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x)$		(4.4)
				(4.4)

Using the fact that, for all $i, j \in [d]$ , note that

\begin{matrix} \frac{\partial}{\partial x_{i}} f (x) = \frac{\partial}{\partial x_{i}} f (x) {∣ ∣}_{x_{J} = 0} + \sum j \in J \frac{λ_{j}}{m} \cdot \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) {∣ ∣}_{x_{J} = 0} \cdot (1 + o_{λ_{J}} (1)), \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) = \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} f (x) {∣ ∣}_{x_{J} = 0} \cdot (1 + o_{λ_{J}} (1)), \end{matrix}

(4.5)

Then, from (4.3), (4.4) (4.5) and Lemma 2.1, we can deduce the conclusion.

4.3 Proof of Lemma 2.3

By the independence of the observations $X_{1}, X_{2}, \dots, X_{n}$ , we have

(4.6)

From Lemma 2.1, we already know that $E [{^f}_{n, m} (x)] = f (x) + O (m^{- 1})$ , uniformly for $x \in S$ . We can also expand the integral using a Taylor expansion :

		$m^{d} \sum k \in N_{0}^{d} \cap (m - 1) S \int_{(\frac{k}{m}, \frac{k + 1}{m}]} f (y) d y P_{k, m - 1}^{2} (x)$		(4.7)
		$= f (x) \sum k \in N_{0}^{d} \cap (m - 1) S P_{k, m - 1}^{2} (x)$
		$+ \frac{1}{m} d \sum i = 1 O (\sum k \in N_{0}^{d} \cap (m - 1) S \| k_{i} - m x_{i} \| P_{k, m - 1}^{2} (x)) + O (m^{- 1}) .$

Since $P_{k, m - 1}^{3} (x) \leq 1$ , the Cauchy-Schwarz inequality yields

		$\sum k \in N_{0}^{d} \cap (m - 1) S \| k_{i} - m x_{i} \| P_{k, m - 1}^{2} (x)$		(4.8)
		$\leq \sqrt{\sum k \in N_{0}^{d} \cap (m - 1) S \| k_{i} - m x_{i} \|^{2} P_{k, m - 1} (x)} \sqrt{\sum k \in N_{0}^{d} \cap (m - 1) S P_{k, m - 1}^{2} (x)} .$		(4.8)

Putting (4.6), (4.7) and (4.8) together, we get the conclusion.

4.4 Proof of Theorem 2.4

By Lemma 2.3, and Lemma A.1 and (B.2) in Lemma B.1, we have

	$V a r ({^f}_{n, m} (x))$			(4.9)
				(4.9)

Using the Taylor expansion $f (x) = f (x) {∣ ∣}_{x_{J} = 0} + O_{λ_{J}} (m^{- 1})$ , we get

$V a r ({^f}_{n, m} (x))$	$= \frac{m^{(d + \| J \|) / 2}}{n} (f (x) {∣ ∣}_{x_{J} = 0} + O_{λ_{J}} (m^{- 1}))$	(4.10)

The conclusion follows.

5 Proof of the results for the c.d.f. estimator $F_{n, m}^{⋆}$

5.1 Proof of Lemma 3.1

By a Taylor expansion,

$F (k / m)$	$= F (x) + d \sum i = 1 (k_{i} / m - x_{i}) \frac{\partial}{} \partial x_{i} F (x)$	(5.1)
	$+ \frac{1}{2} d \sum i, j = 1 (k_{i} / m - x_{i}) (k_{j} / m - x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x)$
	$+ \frac{1}{6} d \sum i, j, ℓ = 1 (k_{i} / m - x_{i}) (k_{j} / m - x_{j}) (k_{ℓ} / m - x_{ℓ}) \frac{\partial^{3}}{\partial x_{i} \partial x_{j} \partial ℓ} F (x)$
	$+ o (∥ k / m - x ∥_{1}^{3}) .$

If we multiply by $P_{k, m} (x)$ , sum over $k \in N_{0}^{d} \cap m S$ , and then take the expectation on both sides, we get

$E [F_{n, m}^{⋆} (x)]$	$= \sum k \in N_{0}^{d} \cap m S F (k / m) P_{k, m} (x)$	(5.2)
	$= F (x) + d \sum i = 1 E [ξ_{i} / m - x_{i}] \frac{\partial}{\partial x_{i}} F (x)$
	$+ \frac{1}{2} d \sum i, j = 1 E [(ξ_{i} / m - x_{i}) (ξ_{j} / m - x_{j})] \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x)$
	$+ \frac{1}{6} d \sum i, j, ℓ = 1 E [(ξ_{i} / m - x_{i}) (ξ_{j} / m - x_{j}) (ξ_{ℓ} / m - x_{ℓ})] \frac{\partial^{3}}{\partial x_{i} \partial x_{j} \partial ℓ} F (x)$
	$+ d \sum i, j, ℓ = 1 o (E [\| ξ_{i} / m - x_{i} \| \| ξ_{j} / m - x_{j} \| \| ξ_{ℓ} / m - x_{ℓ} \|]) .$

From the multinomial joint central moments in Lemma B.1, we get

$E [F_{n, m}^{⋆} (x)]$	$= F (x) + \frac{1}{2 m} d \sum i, j = 1 (x_{i} 1_{{i = j}} - x_{i} x_{j}) \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x)$

	$+ d \sum i, j, ℓ = 1 o (E [\| ξ_{i} / m - x_{i} \| \| ξ_{j} / m - x_{j} \| \| ξ_{ℓ} / m - x_{ℓ} \|]) .$	(5.3)

We apply the Cauchy-Schwarz inequality on the error term to get the conclusion.

5.2 Proof of Theorem 3.2

Take $x$ as in the statement of the theorem. For all $i, j, ℓ \in [d]$ , note that

	$\frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x) = \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} F (x) {∣ ∣}_{x_{J} = 0} + \sum ℓ \in J \frac{λ_{ℓ}}{m} \cdot \frac{\partial^{3}}{\partial x_{i} \partial x_{j} \partial x_{ℓ}} F (x) {∣ ∣}_{x_{J} = 0} \cdot (1 + o_{λ_{J}} (1)),$		(5.4)
	$\frac{\partial^{3}}{\partial x_{i} \partial x_{j} \partial x_{ℓ}} F (x) = \frac{\partial^{3}}{\partial x_{i} \partial x_{j} \partial x_{ℓ}} F (x) ∣$

Asymptotic properties of Bernstein estimators on the simplex. Part 2: the boundary case