Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative

Andrzej Kozek

doi:10.2307/3314864

Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative

Andrzej Kozek^*

^*Corresponding author for this work

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

1 Citation (Scopus)

Abstract

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) ‐ _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov‐Lévy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x) ≤ G(x + αϵ) + ϵ for all x ℝ}.

Original language	English
Pages (from-to)	77-85
Number of pages	9
Journal	Canadian Journal of Statistics
Volume	15
Issue number	1
DOIs	https://doi.org/10.2307/3314864
Publication status	Published - 1987
Externally published	Yes

Keywords

Asymptotic distribution
empirical distribution function
Kolmogorov‐Smirnov metric
Prohorov‐Lévy metric

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10.2307/3314864

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title = "Limiting distributions of Kolmogorov‐L{\'e}vy‐type statistics under the alternative",

abstract = "Let Xi, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρα(Fn, G) ‐ α(F, G)}, where Fn is the empirical distribution function based on X1,…,Xn and α is a Kolmogorov‐L{\'e}vy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric pα is given by pα(F, G) = inf {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x) ≤ G(x + αϵ) + ϵ for all x ℝ}.",

keywords = "Asymptotic distribution, empirical distribution function, Kolmogorov‐Smirnov metric, Prohorov‐L{\'e}vy metric",

author = "Andrzej Kozek",

year = "1987",

doi = "10.2307/3314864",

language = "English",

volume = "15",

pages = "77--85",

journal = "Canadian Journal of Statistics",

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publisher = "Statistical Society of Canada",

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T1 - Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative

AU - Kozek, Andrzej

PY - 1987

Y1 - 1987

N2 - Let Xi, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρα(Fn, G) ‐ α(F, G)}, where Fn is the empirical distribution function based on X1,…,Xn and α is a Kolmogorov‐Lévy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric pα is given by pα(F, G) = inf {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x) ≤ G(x + αϵ) + ϵ for all x ℝ}.

AB - Let Xi, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρα(Fn, G) ‐ α(F, G)}, where Fn is the empirical distribution function based on X1,…,Xn and α is a Kolmogorov‐Lévy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric pα is given by pα(F, G) = inf {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x) ≤ G(x + αϵ) + ϵ for all x ℝ}.

KW - Asymptotic distribution

KW - empirical distribution function

KW - Kolmogorov‐Smirnov metric

KW - Prohorov‐Lévy metric

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SP - 77

EP - 85

JO - Canadian Journal of Statistics

JF - Canadian Journal of Statistics

IS - 1

ER -

Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative

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Keywords

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