### Abstract

Original language | English |
---|---|

Publisher | ArXiv |

Pages | 1-27 |

Number of pages | 27 |

Publication status | Submitted - 16 May 2017 |

### Fingerprint

### Keywords

- characteristic function
- distance correlation coefficient
- distance variance
- Gini’s mean difference
- measure of spread
- dispersive ordering
- stochastic ordering
- U-statistic
- order statistic
- sample spacing
- asymptotic efficiency

### Cite this

*The Distance Standard Deviation*. (pp. 1-27). ArXiv.

**The Distance Standard Deviation.** / Edelmann, Dominic; Richards, Donald ; Vogel, Daniel.

Research output: Working paper

}

TY - UNPB

T1 - The Distance Standard Deviation

AU - Edelmann, Dominic

AU - Richards, Donald

AU - Vogel, Daniel

PY - 2017/5/16

Y1 - 2017/5/16

N2 - The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. New representations for the distance standard deviation are obtained in terms of Gini's mean difference and in terms of the moments of spacings of order statistics. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini's mean difference. Further, it is shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread. Explicit closed-form expressions for the distance variance are obtained for a broad class of parametric distributions. The asymptotic distribution of the sample distance variance is derived.

AB - The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. New representations for the distance standard deviation are obtained in terms of Gini's mean difference and in terms of the moments of spacings of order statistics. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini's mean difference. Further, it is shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread. Explicit closed-form expressions for the distance variance are obtained for a broad class of parametric distributions. The asymptotic distribution of the sample distance variance is derived.

KW - characteristic function

KW - distance correlation coefficient

KW - distance variance

KW - Gini’s mean difference

KW - measure of spread

KW - dispersive ordering

KW - stochastic ordering

KW - U-statistic

KW - order statistic

KW - sample spacing

KW - asymptotic efficiency

UR - https://www.researchgate.net/publication/316985499_The_Distance_Standard_Deviation

M3 - Working paper

SP - 1

EP - 27

BT - The Distance Standard Deviation

PB - ArXiv

ER -