In Chapter 4, Section 4.1, the focus is on multivariate density estimation, particularly simple density estimation methods. The chapter highlights the importance of exploring and identifying structures in multivariate data, which is more challenging than univariate data due to the difficulties in visually representing and modeling multivariate data. Scatter plots are commonly used to present bivariate data, but they are not effective for representing the distribution of observations or high-density regions. To address this, methods like sunflower plots can be used to represent the number of (near) replicates, but a direct density estimation representation is more effective. The histogram can be generalized to multiple dimensions by dividing the region of interest into hyperrectangles. The histogram estimator is given by: $$ \hat{f}(\mathbf{x}) = \frac{n_k}{n h_1 \cdots h_d} $$ where \( n_k \) is the number of observations in the hyperrectangular bin \( B_k \). For a smooth function \( f \), the asymptotic mean integrated squared error (AMISE) is minimized by choosing bin widths \( h_{j0} \) such that: $$ h_{j0} = R(\hat{f}_j)^{-1/2} [ 6 \prod_{i=1}^{d} R(\hat{f}_i)^{1/2} ]^{1/(d+2)} n^{-1/(d+2)}, $$ with the minimized AMISE equaling: $$ \text{AMISE}_0 = \frac{1}{4} [ 36 \prod_{i=1}^{d} R(\hat{f}_i) ]^{1/(d+2)} n^{-2/(d+2)}. $$ This section provides a theoretical foundation for understanding how to choose optimal bin widths to minimize the error in density estimation.In Chapter 4, Section 4.1, the focus is on multivariate density estimation, particularly simple density estimation methods. The chapter highlights the importance of exploring and identifying structures in multivariate data, which is more challenging than univariate data due to the difficulties in visually representing and modeling multivariate data. Scatter plots are commonly used to present bivariate data, but they are not effective for representing the distribution of observations or high-density regions. To address this, methods like sunflower plots can be used to represent the number of (near) replicates, but a direct density estimation representation is more effective. The histogram can be generalized to multiple dimensions by dividing the region of interest into hyperrectangles. The histogram estimator is given by: $$ \hat{f}(\mathbf{x}) = \frac{n_k}{n h_1 \cdots h_d} $$ where \( n_k \) is the number of observations in the hyperrectangular bin \( B_k \). For a smooth function \( f \), the asymptotic mean integrated squared error (AMISE) is minimized by choosing bin widths \( h_{j0} \) such that: $$ h_{j0} = R(\hat{f}_j)^{-1/2} [ 6 \prod_{i=1}^{d} R(\hat{f}_i)^{1/2} ]^{1/(d+2)} n^{-1/(d+2)}, $$ with the minimized AMISE equaling: $$ \text{AMISE}_0 = \frac{1}{4} [ 36 \prod_{i=1}^{d} R(\hat{f}_i) ]^{1/(d+2)} n^{-2/(d+2)}. $$ This section provides a theoretical foundation for understanding how to choose optimal bin widths to minimize the error in density estimation.