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Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case
2.	Creator	Author's name, affiliation, country	Michael J Klass; University of California Departments of Mathematics and Statistics Berkeley, CA
2.	Creator	Author's name, affiliation, country	Krzysztof Nowicki; Lund University Department of Statistics Box 743 S-220 07 Lund, Sweden
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Sum of independent rv's, tail distributions, tail distributions,tail probabilities, quantile approximation, Hoffmann-Jo rgensen/Klass-Nowicki Inequality
3.	Subject	Subject classification	Primary 60G50, 60E15, 46E30; secondary 46B09
4.	Description	Abstract	Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $\underline q_y$ for which we prove that $$\frac 1 2 \underline q_{y/2} < s_y^* < 2 \underline q_{2y} \quad \text{ and } \quad \frac 1 2 \underline q_{ (y/4) ( 1 + \sqrt{ 1 - 8/y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $y \geq 2$ and the LHS's for $y \geq 94$ and $y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	NSF
7.	Date	(YYYY-MM-DD)	2007-10-16
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/452
10.	Identifier	Digital Object Identifier	10.1214/EJP.v12-452
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 12
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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