Nonparametric Estimation of a Regression Function: Limiting Distribution

Kuang‐Fu ‐F Cheng; Pi‐Erh ‐E Lin

doi:10.1111/j.1467-842X.1981.tb00776.x

Nonparametric Estimation of a Regression Function: Limiting Distribution

Kuang‐Fu ‐F Cheng, Pi‐Erh ‐E Lin

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

12 Citations (Scopus)

Abstract

Consider the regression model Y_i= g(x_i) + e_i, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x₀ < x₁ < … < x_n≤ 1 are chosen so that max_1≤i≤n(x_i‐x_{i‐ 1}) = 0(n^‐1), and where {e_i} are i.i.d. with Ee₁= 0 and Var e₁ ‐ s̀². In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g_1n of Cheng & Lin (1979), g_2n of Clark (1977), and g_3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &_in, i = 1, 2, 3, and that of the isotonic estimator g**_n are considered.

Original language	English
Pages (from-to)	186-195
Number of pages	10
Journal	Australian Journal of Statistics
Volume	23
Issue number	2
DOIs	https://doi.org/10.1111/j.1467-842X.1981.tb00776.x
Publication status	Published - Jun 1981
Externally published	Yes

Keywords

Asymptotic normality
Berry‐Esséen bound
Isotonic
Kernel function
Liapunov's theorem
Lipschitz
phrases

ASJC Scopus subject areas

Statistics and Probability

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10.1111/j.1467-842X.1981.tb00776.x

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@article{01ad74cbd7594e83805675bcfe95c614,

title = "Nonparametric Estimation of a Regression Function: Limiting Distribution",

abstract = "Consider the regression model Yi= g(xi) + ei, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x0 < x1 < … < xn≤ 1 are chosen so that max1≤i≤n(xi‐xi‐ 1) = 0(n‐1), and where {ei} are i.i.d. with Ee1= 0 and Var e1 ‐ {\`s}2. In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g1n of Cheng & Lin (1979), g2n of Clark (1977), and g3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &in, i = 1, 2, 3, and that of the isotonic estimator g**n are considered.",

keywords = "Asymptotic normality, Berry‐Ess{\'e}en bound, Isotonic, Kernel function, Liapunov's theorem, Lipschitz, phrases",

author = "Cheng, {Kuang‐Fu ‐F} and Lin, {Pi‐Erh ‐E}",

year = "1981",

month = jun,

doi = "10.1111/j.1467-842X.1981.tb00776.x",

language = "English",

volume = "23",

pages = "186--195",

journal = "Australian Journal of Statistics",

issn = "0004-9581",

publisher = "Statistical Society of Australia",

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T1 - Nonparametric Estimation of a Regression Function

T2 - Limiting Distribution

AU - Cheng, Kuang‐Fu ‐F

AU - Lin, Pi‐Erh ‐E

PY - 1981/6

Y1 - 1981/6

N2 - Consider the regression model Yi= g(xi) + ei, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x0 < x1 < … < xn≤ 1 are chosen so that max1≤i≤n(xi‐xi‐ 1) = 0(n‐1), and where {ei} are i.i.d. with Ee1= 0 and Var e1 ‐ s̀2. In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g1n of Cheng & Lin (1979), g2n of Clark (1977), and g3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &in, i = 1, 2, 3, and that of the isotonic estimator g**n are considered.

AB - Consider the regression model Yi= g(xi) + ei, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x0 < x1 < … < xn≤ 1 are chosen so that max1≤i≤n(xi‐xi‐ 1) = 0(n‐1), and where {ei} are i.i.d. with Ee1= 0 and Var e1 ‐ s̀2. In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g1n of Cheng & Lin (1979), g2n of Clark (1977), and g3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &in, i = 1, 2, 3, and that of the isotonic estimator g**n are considered.

KW - Asymptotic normality

KW - Berry‐Esséen bound

KW - Isotonic

KW - Kernel function

KW - Liapunov's theorem

KW - Lipschitz

KW - phrases

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SN - 0004-9581

VL - 23

SP - 186

EP - 195

JO - Australian Journal of Statistics

JF - Australian Journal of Statistics

IS - 2

ER -

Nonparametric Estimation of a Regression Function: Limiting Distribution

Abstract

Keywords

ASJC Scopus subject areas

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