Density estimation from an individual numerical sequence

Andrew B. Nobel; Gusztáv Morvai; Sanjeev R. Kulkarni

doi:10.1109/18.661503

Density estimation from an individual numerical sequence

Andrew B. Nobel, Gusztáv Morvai, Sanjeev R. Kulkarni

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that 1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and 2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L ₁ consistent when 1) and 2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition 1).

Original language	English (US)
Pages (from-to)	537-541
Number of pages	5
Journal	IEEE Transactions on Information Theory
Volume	44
Issue number	2
DOIs	https://doi.org/10.1109/18.661503
State	Published - 1998

All Science Journal Classification (ASJC) codes

Information Systems
Computer Science Applications
Library and Information Sciences

Keywords

Bounded variation
Density estimation
Ergodic processes
Individual sequences

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10.1109/18.661503

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title = "Density estimation from an individual numerical sequence",

abstract = "This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that 1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and 2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L 1 consistent when 1) and 2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition 1).",

keywords = "Bounded variation, Density estimation, Ergodic processes, Individual sequences",

author = "Nobel, {Andrew B.} and Guszt{\'a}v Morvai and Kulkarni, {Sanjeev R.}",

note = "Funding Information: Manuscript received November 18, 1996; revised August 4, 1997. The research of A. B. Nobel was supported in part by NSF under Grant DMS 9501926. Part of this work was completed while A. B. Nobel was on leave as a Beckman Institute Fellow at the University of Illinois, Urbana-Champaign. A. B. Nobel is with the Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260 USA. G. Morvai is with the Research Group for Informatics and Electronics of the Hungarian Academy of Sciences, Budapest, Hungary. S. R. Kulkarni is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA. Publisher Item Identifier S 0018-9448(98)00836-0.",

year = "1998",

doi = "10.1109/18.661503",

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volume = "44",

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journal = "IEEE Transactions on Information Theory",

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T1 - Density estimation from an individual numerical sequence

AU - Nobel, Andrew B.

AU - Morvai, Gusztáv

AU - Kulkarni, Sanjeev R.

N1 - Funding Information: Manuscript received November 18, 1996; revised August 4, 1997. The research of A. B. Nobel was supported in part by NSF under Grant DMS 9501926. Part of this work was completed while A. B. Nobel was on leave as a Beckman Institute Fellow at the University of Illinois, Urbana-Champaign. A. B. Nobel is with the Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260 USA. G. Morvai is with the Research Group for Informatics and Electronics of the Hungarian Academy of Sciences, Budapest, Hungary. S. R. Kulkarni is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA. Publisher Item Identifier S 0018-9448(98)00836-0.

PY - 1998

Y1 - 1998

N2 - This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that 1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and 2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L 1 consistent when 1) and 2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition 1).

AB - This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that 1) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and 2) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be L 1 consistent when 1) and 2) apply. In addition, it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition 1).

KW - Bounded variation

KW - Density estimation

KW - Ergodic processes

KW - Individual sequences

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JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 2

ER -

Density estimation from an individual numerical sequence

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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