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Coupling methods in approximations
Author(s) -
Wang Y. H.
Publication year - 1986
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315038
Subject(s) - measure (data warehouse) , coupling (piping) , mathematics , metric (unit) , upper and lower bounds , combinatorics , discrete mathematics , physics , mathematical analysis , computer science , mechanical engineering , operations management , database , engineering , economics
Let X and Y be two arbitrary k ‐dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P ( X ≠ Y ). This result is used to find the least upper bound for the metric d ( X, Y ) = sup A | P ( X ∈ A ) − P ( Y ∈ A )| and to derive the inequality d(Σ i =1 nX i , Σ i =1 nY i ) ≤ Σ i =1 n d( X i , Y i ). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.