关于密度估计的好书
Table of Contents
INTRODUCTION
What is density estimation?
Density estimates in the exploration and presentation of data
Further reading
SURVEY OF EXISTING METHODS
Introduction
Histograms
The naive estimator
The kernel estimator
The nearest neighbour method
The variable kernel method
Orthogonal series estimators
Maximum penalized likelihood estimators
General weight function estimators
Bounded domains and directional data
Discussion and bibliography
1. INTROUCTION
1.1. What is density estimation?
The probability density function is a fundamental concept in statistics. Consider any random quantity X that has probability
density function f . Specifying the function f gives a natural description of the distribution of X , and allows probabilities
associated with X to be found from the relation
Suppose, now, that we have a set of observed data points assumed to be a sample from an unknown probability density function.
Density estimation, as discussed in this book, is the construction of an estimate of the density function from the observed data.
The two main aims of the book are to explain how to estimate a density from a given data set and to explore how density
estimates can be used, both in their own right and as an ingredient of other statistical procedures.
One approach to density estimation is parametric. Assume that the data are drawn from one of a known parametric family of
distributions, for example the normal distribution with mean μ and variance 2. The density f underlying the data could then be
estimated by finding estimates of μ and 2 from the data and substituting these estimates into the formula for the normal
density. In this book we shall not be considering parametric estimates of this kind; the approach will be more non parametric in
that less rigid assumptions will be made about the distribution of the observed data. Although it will be assumed that the
distribution has a probability density f, the data will be allowed to speak for themselves in determining the estimate of f more
than would be the case if f were constrained to fall in a given parametric family.
Density estimates of the kind discussed in this book were first proposed by Fix and