Nonparametric Density Estimation
This chapter concerns estimation of a density function f using observations of random variables X1, . . ., Xn sampled independently from f. Initially, this chapter focuses on univariate density estimation. Section 10.4 introduces some methods for estimating a multivariate density function.
In exploratory data analysis, an estimate of the density function can be used to assess multimodality, skew, tail behavior, and so forth. For inference, density estimates are useful for decision making, classification, and summarizing Bayesian posteriors. Density estimation is also a useful presentational tool since it provides a simple, attractive summary of a distribution. Finally, density estimation can serve as a tool in other computational methods, including some simulation algorithms and Markov chain Monte Carlo approaches. Comprehensive monographs on density estimation include [581, 598, 651].
The parametric solution to a density estimation problem begins by assuming a parametric model, X1, . . ., Xn ~ i.i.d. fX|θ, where θ is a very low-dimensional para-meter vector. Parameter estimates are found using some estimation paradigm, such as maximum likelihood, Bayesian, or method-of-moments estimation. The resulting density estimate at x is . The danger ...