Optimal Rates of Convergence for Deconvolving a Density

Abstract

Suppose we observe the sum of two independent random variables X and Z, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. It is shown that if Z is normally distributed and if f has k bounded derivatives, then the fastest attainable convergence rate of any nonparametric estimator of f is only (log n)-k/1. Therefore deconvolution with normal errors may not be a practical proposition. Other error distributions are also treated. Stefanski-Carroll (1978b) estimators achieve the optimal rates. Our results have versions for multiplicative errors, where they imply that even optimal rates are exceptionally slow.

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Document Details

Document Type
Technical Report
Publication Date
Aug 01, 1988
Accession Number
ADA197748

Entities

People

  • Peter Hall
  • Raymond J. Carroll

Organizations

  • Texas A&M University

Tags

Communities of Interest

  • C4I

DTIC Thesaurus Topics

  • Air Force
  • Algorithms
  • Availability
  • Convergence
  • Data Science
  • Distribution Functions
  • Estimators
  • Information Science
  • Measurement
  • Probability
  • Random Variables
  • Statistical Algorithms
  • Statistical Analysis
  • Statistical Decision Theory
  • Statistics
  • Theorems
  • Universities

Fields of Study

  • Mathematics

Readers

  • Adaptive Control and Estimation with Uncertainty in Dynamic Systems.
  • Statistical inference.