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A Note on Bounds for the Supremum Metric for Discrete Random Variables
Author(s) -
Daley D. J.
Publication year - 1980
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19800990112
Subject(s) - mathematics , infimum and supremum , combinatorics , integer (computer science) , random variable , metric (unit) , discrete mathematics , statistics , operations management , computer science , economics , programming language
For positive or negative integer‐valued random variables X and Y with finite second moments the inequality sup \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\sup }\limits_n |\Pr \{ X \le n\} - \Pr \{ Y \le n\} |\, \le \,|EX - EY| + \frac{1}{2}(EX(EX - 1) + (EY(Y - 1)) $\end{document} is established by elementary manipulation, and shown to be tight. Use of generating functions and an inversion formula yields the larger bound with 1/2 replaced by 2/π.