## 1 Introduction

The -dimensional simplex and its interior are defined by

(1.1) |

where . For any cumulative distribution function on , define the Bernstein polynomial of order for by

(1.2) |

where the weights are the following probabilities from the

distribution :(1.3) |

The Bernstein estimator of , denoted by , is the Bernstein polynomial of order for the empirical cumulative distribution function

, where the random variables

are independent and distributed. Precisely,(1.4) |

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

(1.5) |

where is just a scaling factor, namely the inverse of the volume of the hypercube .

## 2 Results for the density estimator

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

(2.1) |

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

###### Lemma 2.1 (blueBias of on ).

By considering points that are close to the boundary in some components (see the subset of indices below), we get the bias of the density estimator near the boundary.

###### Theorem 2.2 (blueBias of near the boundary of ).

Assume (2.1). For any such that for all and is independent of for all , we have

(2.4) |

as , where are fixed, , and .

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

###### Lemma 2.3 (blueVariance of on ).

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 near the boundary of ).

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 will depend on the number of components of that are close to the boundary.

###### Corollary 2.5 (blueMean squared error of near the boundary of ).

By adding conditions on the partial derivatives of , 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 when .

###### Corollary 2.6 (blueMean squared error of near the boundary of ).

Assume (2.1) and also

(2.10) |

(in particular, the first bracket in (2.2) is zero). Then, for any such that for all and independent of for all , we have

(2.11) | ||||

as . Note that the last error term is bigger than the main term except when . Therefore, if and we assume that the quantity inside the big bracket is non-zero in (2.11), the asymptotically optimal choice of , with respect to MSE, is

(2.12) |

in which case

(2.13) | ||||

as .

###### Remark 2.7.

In order to optimize when 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 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

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

(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 on ).

###### Theorem 3.2 (blueBias of near the boundary of ).

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

###### Lemma 3.3 (blueVariance of on ).

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 near the boundary of ).

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 near the boundary of ).

Assume (3.1). For any such that for all and is independent of for all , we have

(3.6) | ||||

as . As pointed out in (Leblanc, 2012b, p.2772) for , there is no optimal with respect to the MSE when . This is also true here. The remaining case (when 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

### 4.1 Proof of Lemma 2.1

Using Taylor expansions for any such that , we obtain

If we multiply the last expression by and sum over

, then the joint moments from Lemma

B.1, the notation for and in (2.3), Jensen’s inequality and(4.1) |

yield

(4.2) | ||||

This ends the proof.

### 4.2 Proof of Theorem 2.2

### 4.3 Proof of Lemma 2.3

### 4.4 Proof of Theorem 2.4

## 5 Proof of the results for the c.d.f. estimator

### 5.1 Proof of Lemma 3.1

By a Taylor expansion,

(5.1) | ||||

If we multiply by , sum over , and then take the expectation on both sides, we get

(5.2) | ||||

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

(5.3) |

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

### 5.2 Proof of Theorem 3.2

Take as in the statement of the theorem. For all , note that

(5.4) | |||