On the histogram as a density estimator:L2 theory

@article{Freedman1981OnTH,
  title={On the histogram as a density estimator:L2 theory},
  author={David Freedman and Persi Diaconis},
  journal={Zeitschrift f{\"u}r Wahrscheinlichkeitstheorie und Verwandte Gebiete},
  year={1981},
  volume={57},
  pages={453-476}
}
  • D. Freedman, P. Diaconis
  • Published 1981
  • Mathematics
  • Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Let f be a probability density on an interval I, finite or infinite: I includes its finite endpoints, if any; and f vanishes outside of I. Let X1, . . . ,X k be independent random variables, with common density f The empirical histogram for the X's is often used to estimate f To define this object, choose a reference point xosI and a cell width h. Let Nj be the number of X's falling in the j th class interval: 
AN ASYMPTOTICALLY OPTIMAL HISTOGRAM SELECTION RULE
A random sample is available from a multivariate distribution having a bounded density, which is assumed to satisfy a mild additional condition. A finite collection of histogram estimates of theExpand
How many bins should be put in a regular histogram
Given an n-sample from some unknown density f on [0,1], it is easy to construct an histogram of the data based on some given partition of [0,1], but not so much is known about an optimal choice ofExpand
What Is the Optimal Bin Size of a Histogram: An Informal Description
A natural way to estimate the probability density function of an unknown distribution from the sample of data points is to use histograms. The accuracy of the estimate depends on the size of theExpand
An optimal variable cell histogram
A simple procedure for specifying a histogram with variable cell sizes is proposed. The procedure chooses a set of cutpoints that maximizes a criterion function based on the sample spacings:UnderExpand
Oversmoothed Nonparametric Density Estimates
Abstract The optimal histogram for a sample of size n from a density defined on the entire real line requires at least (2n)1/3 bins, under mild smoothness conditions. Similar bounds exist for theExpand
Minimum local distance density estimation
ABSTRACT We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sampleExpand
Optimal Data-Based Binning for Histograms
Histograms are convenient non-parametric density estimators, which continue to be used ubiquitously. Summary quantities estimated from histogram-based probability density models depend on the choiceExpand
On stochastic complexity and nonparametric density estimation
SUMMARY We use the concepts of stochastic complexity, description length, and model selection to develop data-based methods for choosing smoothing parameters in nonparametric density estimation. InExpand
Are Few Bins Enough: Testing Histogram Distributions
TLDR
A sample and time-efficient algorithm for this problem is obtained, complemented by a nearly-matching information-theoretic lower bound on the number of samples required for this task. Expand
Probability Density Estimation
The probability density function is useful for summarizing and exploring a set of data. When the parametric form is unknown, then a nonparametric estimator such as the histogram is appropriate. ThisExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 16 REFERENCES
Nonparametric Probability Density Estimation: I. A Summary of Available Methods
The time-honored histogram is widely used among scientists and others to represent, at least approximately, the shape of the probability density function. Recent literature in statistical theory hasExpand
On optimal and data based histograms
SUMMARY In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify theExpand
A HIERARCHY OF PROBABILITY DENSITY FUNCTION ESTIMATES
1. Abstract and Summary The purpose of this paper is to consider a hierarchy of probability density function estimation procedures. The discussion will culminate in a speculation about a universalExpand
An Introduction to the Implementation and Theory of Nonparametric Density Estimation
Since the pioneering papers of M. Rosenblatt [28] and E. Parzen [25] numerous publications dealing with nonparametric density estimation have appeared in the mathematical statistical literature.Expand
An Introduction to Probability Theory and Its Applications, Vol. 2
TLDR
This is an introduction to probability theory and its applications vol 2, where people cope with some infectious bugs inside their desktop computer. Expand
An introduction to probability theory
  • P. Moran
  • Computer Science, Mathematics
  • 1968
This classic text and reference introduces probability theory for both advanced undergraduate students of statistics and scientists in related fields, drawing on real applications in the physical andExpand
Frontiers of Pattern Recognition
TLDR
Reading frontiers of pattern recognition is also a way as one of the collective books that gives many advantages, not only for you, but for the other peoples with those meaningful benefits. Expand
The Art of Computer Programming
TLDR
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid. Expand
Statistical density estimation: A bibliography
  • Internat. Statist Rev
  • 1972
Nonparametric Probability Density Estimation
...
1
2
...