Modes and moments of unimodal distributions
type: Item
instance of: scholarly article
title: Modes and moments of unimodal distributions
zbMATH Open document ID: 0633.60023
DOI: 10.1007/BF02491478
author: Ken-Iti Sato
published in: Annals of the Institute of Statistical Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The paper considers the author's [(*) Nagoja Math. J. 104, 29-42 (1986; Zbl 0613.60065)] inequality relationships between the mode \(\alpha\) of a unimodal distribution and its absolute moment of order \(p,\beta_ p=\int | x|^ p\mu (dx)\) or its absolute central moment of order \(p,\gamma_ p=\int | x-m|^ p\mu (dx)\). (Here m is the mean of the distribution which is considered finite). It was shown by the author (*) that, for some constants \(A_ p\) and \(B_ p\), \[ | \alpha | \leq A_ p\beta_ p^{1/p}\quad and\quad | \alpha -m| \leq B_ p\gamma_ p^{1/p}. \] The present paper provides an alternative proof of these inequalities as well as explicit expressions for \(B_ p\), \(A_{1/2}\), \(A_ 1\), \(A_ 2\) and a lower bound for \(A_ p\), \(p>0.\) The paper also constructs inequalities referring to the mode \(\alpha\) of a unimodal distribution and what it defines to be the g-moment \((\int g(x)\mu (dx))\) of the distribution and the central g-moment \((\int g(x)\mu (dx))\) of the distribution.
Mathematics Subject Classification ID: 60E05
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4030596
zbMATH Keywords: unimodal distribution
zbMATH Keywords: absolute central moment
zbMATH Keywords: inequalities
reviewed by: Evdokia Xekalaki
MaRDI profile type: Publication
cites work: Q5537802
cites work: The Moment Problem for Unimodal Distributions
cites work: Bounds of modes and unimodal processes with independent increments
cites work: On a Gauss Inequality for Unimodal Distributions
Recommended article: On Bounds for Moments of Unimodal Distributions
Recommended article: Q3817363
Recommended article: A subclassification of unimodal distributions
Recommended article: Some results on unimodality of distributions
Recommended article: On a Mixture of Unimodal Distributions
Recommended article: Odd central moments of unimodal distributions
Recommended article: Q3732640
Recommended article: Estimation of a unimodal distribution function
Recommended article: Discrete moment problems with distributions known to be unimodal
description: scientific article; zbMATH DE number 4030596