A characterization of multivariate normal distribution and its application

Zhen Hai Yang; Kai Tai Fang; Jia Juan Liang

doi:10.1016/S0167-7152(95)00238-3

A characterization of multivariate normal distribution and its application

Zhen Hai Yang, Kai Tai Fang^*, Jia Juan Liang

^*Corresponding author for this work

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

10 Citations (Scopus)

Abstract

Let X₁, X₂, ⋯ ,X_n be i.i.d. d-dimensional random vectors with a continuous density. Let S_k = ∑^k_i = ₁ X_iX^T_i, Y_k = S_k^-1/2X_k and Z_k = Y_k/(√1 - Y^T_k Y_k) (k ≥ d). In this paper we find that the distribution of Z_k (or Y_k) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.

Original language	English
Pages (from-to)	347-352
Number of pages	6
Journal	Statistics and Probability Letters
Volume	30
Issue number	4
DOIs	https://doi.org/10.1016/S0167-7152(95)00238-3
Publication status	Published - 15 Nov 1996
Externally published	Yes

Scopus Subject Areas

Statistics and Probability
Statistics, Probability and Uncertainty

User-Defined Keywords

Characterization of multinormality
Multivariate normal distribution
Spherical distribution

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10.1016/S0167-7152(95)00238-3

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title = "A characterization of multivariate normal distribution and its application",

abstract = "Let X1, X2, ⋯ ,Xn be i.i.d. d-dimensional random vectors with a continuous density. Let Sk = ∑ki = 1 XiXTi, Yk = Sk-1/2Xk and Zk = Yk/(√1 - YTk Yk) (k ≥ d). In this paper we find that the distribution of Zk (or Yk) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.",

keywords = "Characterization of multinormality, Multivariate normal distribution, Spherical distribution",

author = "Yang, {Zhen Hai} and Fang, {Kai Tai} and Liang, {Jia Juan}",

note = "Funding Information: * Corresponding author. 1 This work was supported by Hong Kong UGC grant, the Statistics Research and Consultancy Unit, HKBU and the NSF of China and the NSF of Beijing.",

year = "1996",

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doi = "10.1016/S0167-7152(95)00238-3",

language = "English",

volume = "30",

pages = "347--352",

journal = "Statistics and Probability Letters",

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T1 - A characterization of multivariate normal distribution and its application

AU - Yang, Zhen Hai

AU - Fang, Kai Tai

AU - Liang, Jia Juan

N1 - Funding Information: * Corresponding author. 1 This work was supported by Hong Kong UGC grant, the Statistics Research and Consultancy Unit, HKBU and the NSF of China and the NSF of Beijing.

PY - 1996/11/15

Y1 - 1996/11/15

N2 - Let X1, X2, ⋯ ,Xn be i.i.d. d-dimensional random vectors with a continuous density. Let Sk = ∑ki = 1 XiXTi, Yk = Sk-1/2Xk and Zk = Yk/(√1 - YTk Yk) (k ≥ d). In this paper we find that the distribution of Zk (or Yk) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.

AB - Let X1, X2, ⋯ ,Xn be i.i.d. d-dimensional random vectors with a continuous density. Let Sk = ∑ki = 1 XiXTi, Yk = Sk-1/2Xk and Zk = Yk/(√1 - YTk Yk) (k ≥ d). In this paper we find that the distribution of Zk (or Yk) can be used for characterizing multivariate normal distribution. This characterization can be employed for testing multivariate normality in terms of the so-called transformation method.

KW - Characterization of multinormality

KW - Multivariate normal distribution

KW - Spherical distribution

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U2 - 10.1016/S0167-7152(95)00238-3

DO - 10.1016/S0167-7152(95)00238-3

M3 - Article

AN - SCOPUS:0030588768

SN - 0167-7152

VL - 30

SP - 347

EP - 352

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 4

ER -

A characterization of multivariate normal distribution and its application

Abstract

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