Moment inequalities for positive random variables

Christopher S. Withers; Saralees Nadarajah

doi:10.1016/j.crma.2010.04.014

Statistics

Moment inequalities for positive random variables

Presented by: Paul Deheuvels

Christopher S. Withers ¹; Saralees Nadarajah ²

¹Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand
²School of Mathematics, University of Manchester, Manchester M13 9PL, UK

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 687-690

Abstract (Original language)
Résumé

Given a random variable X, the moments, ${m_{r} = E X^{r}}$ , satisfy

D_{r} = \det (m_{i + j} : 0 ⩽ i, j ⩽ r) ⩾ 0

for

r ⩾ 0

. If

X ⩾ 0

, then for

n ⩾ 1

, there is a random variable

X_{n}

such that

E X_{n}^{r} = m_{r + n} m_{n}^{- 1} / (\begin{matrix} n + r \\ n \end{matrix})

for

r ⩾ 0

. We apply the inequality

D_{r} ⩾ 0

to

X_{n}

to obtain new inequalities for the moments when

X ⩾ 0

. An application is illustrated to obtain tighter bounds for skewness and kurtosis.

Soit X une variable aléatoire, les moments ${m_{r} = E X^{r}}$ vérifient :

D_{r} = \det (m_{i + j} : 0 ⩽ i, j ⩽ r) ⩾ 0, r ⩾ 0 .

Si

X ⩾ 0

, alors pour

n ⩾ 1

, il existe une variable aléatoire

X_{n}

telle que

E X_{n}^{r} = m_{r + n} m_{n}^{- 1} / (\begin{matrix} n + r \\ n \end{matrix}), r ⩾ 0 .

On applique l'inégalité

D_{r} ⩾ 0

à

X_{n}

pour obtenir de nouvelles inégalités sur les moments lorsque

X ⩾ 0

. On applique le résultat à l'obtention de bornes plus précises du coefficient de dissymétrie et du coefficient d'applatissement.

Received: 2010-03-15
Accepted: 2010-04-12
Published online: 2010-05-06

DOI: 10.1016/j.crma.2010.04.014

Author's affiliations:

Christopher S. Withers ¹ ; Saralees Nadarajah ²

¹ Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand
² School of Mathematics, University of Manchester, Manchester M13 9PL, UK

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     author = {Christopher S. Withers and Saralees Nadarajah},
     title = {Moment inequalities for positive random variables},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {687--690},
     year = {2010},
     publisher = {Elsevier},
     volume = {348},
     number = {11-12},
     doi = {10.1016/j.crma.2010.04.014},
     language = {en},
}

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Christopher S. Withers; Saralees Nadarajah. Moment inequalities for positive random variables. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 687-690. doi: 10.1016/j.crma.2010.04.014

References
Cited by

[1] J. Shao Mathematical Statistics, Springer Verlag, New York, 2003

[2] A. Bulinski; A. Shashkin Limit Theorems for Associated Random Fields and Related Systems, World Scientific Publishing Company, Hackensack, New Jersey, 2007

[3] T. Artikis; A. Voudouri Limiting behaviour of certain mixtures of distributions, Analysis Mathematica, Volume 16 (1990), pp. 81-86

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