Chapter 4 discusses multivariate density estimation, emphasizing the challenges of exploring and identifying structure in multivariate data due to the difficulty of graphical representation and the limited availability of parametric models. It highlights that while scatter plots can show relationships between two variables, they are not effective for identifying high-density regions of observations. Improved methods, such as sunflower plots, can better represent the number of replicates, but direct density estimation methods are more effective for revealing structure. The chapter generalizes histograms to multiple dimensions. For a random sample $ \{x_{1},\ldots,x_{n}\} $ from $ f(\mathbf{x}) $, the histogram estimator is given by $ \hat{f}(\mathbf{x}) = \frac{n_{k}}{n h_{1}\cdots h_{d}} $ for $ x \in B_{k} $, where $ B_{k} $ is a hyperrectangular bin containing $ n_{k} $ observations. The asymptotic mean integrated squared error (AMISE) is derived as $ \frac{1}{n h_{1}\cdots h_{d}} + \frac{1}{12}\sum_{i=1}^{d}h_{i}^{2}R(\dot{f}_{i}) $, where $ R(\dot{f}_{i}) $ is related to the partial derivatives of the density function. The optimal bin widths that minimize AMISE are given by $ h_{j0} = R(\dot{f}_{j})^{-1/2} \left[6\prod_{i=1}^{d}R(\dot{f}_{i})^{1/2}\right]^{1/(d+2)}n^{-1/(d+2)} $. The minimized AMISE is $ \mathrm{AMISE}_{0} = \frac{1}{4}\left[36\prod_{i=1}^{d}R(\dot{f}_{i})\right]^{1/(d+2)}n^{-2/(d+2)} $. These results provide a theoretical foundation for choosing bin widths in multivariate histograms to minimize estimation error.Chapter 4 discusses multivariate density estimation, emphasizing the challenges of exploring and identifying structure in multivariate data due to the difficulty of graphical representation and the limited availability of parametric models. It highlights that while scatter plots can show relationships between two variables, they are not effective for identifying high-density regions of observations. Improved methods, such as sunflower plots, can better represent the number of replicates, but direct density estimation methods are more effective for revealing structure. The chapter generalizes histograms to multiple dimensions. For a random sample $ \{x_{1},\ldots,x_{n}\} $ from $ f(\mathbf{x}) $, the histogram estimator is given by $ \hat{f}(\mathbf{x}) = \frac{n_{k}}{n h_{1}\cdots h_{d}} $ for $ x \in B_{k} $, where $ B_{k} $ is a hyperrectangular bin containing $ n_{k} $ observations. The asymptotic mean integrated squared error (AMISE) is derived as $ \frac{1}{n h_{1}\cdots h_{d}} + \frac{1}{12}\sum_{i=1}^{d}h_{i}^{2}R(\dot{f}_{i}) $, where $ R(\dot{f}_{i}) $ is related to the partial derivatives of the density function. The optimal bin widths that minimize AMISE are given by $ h_{j0} = R(\dot{f}_{j})^{-1/2} \left[6\prod_{i=1}^{d}R(\dot{f}_{i})^{1/2}\right]^{1/(d+2)}n^{-1/(d+2)} $. The minimized AMISE is $ \mathrm{AMISE}_{0} = \frac{1}{4}\left[36\prod_{i=1}^{d}R(\dot{f}_{i})\right]^{1/(d+2)}n^{-2/(d+2)} $. These results provide a theoretical foundation for choosing bin widths in multivariate histograms to minimize estimation error.