## On the recovery of discrete probability densities from imperfect measurements (1979)

Venue: | J. Franklin Inst |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Devroye79onthe,

author = {Luc P. Devroye and Luc P. Devroye and Gary L. Wise},

title = {On the recovery of discrete probability densities from imperfect measurements},

journal = {J. Franklin Inst},

year = {1979},

volume = {307},

pages = {20}

}

### Years of Citing Articles

### OpenURL

### Abstract

ABSTRACT The problem of the estimation of a discrete probability density from indepen-dent observations is considered. For a wide class of noises, a method is given for estimating a probability density when the measurements are corrupted by additive noise. This method is shown to be consistent, and several bounds on the error are given. An application to the detection of a (nonparametric) random signal is discussed. Finally, the estimation of a probability density is considered where the measurements are noisy and some of the measurements are incorrect. This situation may arise when a machine collecting the data fails part of the time. 1. Zntroduction The need for considering discrete data is often encountered in data com-munications, digital signal processing, and other areas. In this paper we consider discrete valued random variables, and we are concerned with estimat-ing the discrete probability density function. Measurements are taken, and from these measurements a density function is obtained. However, we assume that the measurements are imperfect. We derive the estimators, establish the appropriate forms of convergence, and supply an abundance of bounds on the errors. Assume that we can observe X1, X2,..., x, a sequence of independent identically distributed random variables with the unknown discrete probability density f. An obvious way of estimating f(x) is to use the empirical density based on the n observations. However, the estimation problem is complicated if we can only observe X1 +z,, x*+z2,...,xn+zn, where XI, Z,, X,,Z*,..., X,, Z,, are independent random variables and the Zi’s, commonly referred to as noise, have a common known discrete probability density function g. For a wide class of densities g, a method is given to recover

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