Let X = (X1, X2,..., Xn) be a random vector in Rn (Euclidean n-space) with density f(x). X or f(x) is said to be multivariate reverse rule of order 2 (MRR2) if f(x [curly logical or] y) f(x [curly logical and] y) <= f(x) f(y) where the lattice operations x [curly logical and] y and x [curly logical or] y refer to the usual ordering of Rn. A density f(x) of X = (X1,...,Xn) is said to be strongly MRR2 if for any set of PF2 functions {[phi]v} (i.e., [phi]v([xi] - [eta]) is totally positive of order 2 on -[infinity] < [xi], [eta] < [infinity]) each marginal g(x[nu]1,x x[nu]2,..., x[nu]k) = [integral operator] ... [integral operator] f(x1,..., xn) [phi]1(x[mu]1)[phi]2(x[mu]2) ... [phi]n - k(x[mu]n - k) dx[mu]1 ... dx[mu]n - k is MRR2 in the variables (x[nu]1, x[nu]2,..., x[nu]k), where ([nu]1,..., [nu]k) and ([mu]1, [mu]2,..., [mu]n - k) are complementary sets of indices. The property of strongly MRR2 prevails for the multinormal, multivariate hypergeometric, Dirichlet, and many other densities. For a strong MRR2 density we establish the reverse generalized correlation inequality P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], [nu] [set membership, variant] J [union or logical sum] K}P{ai <= Xi <= bi, i [set membership, variant] I} <= P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], v [set membership, variant] J}P{ai <= Xi <= bi, i [set membership, variant] I, X[nu] <= b[nu], [nu] [set membership, variant] K}, where I, J and K denote the set of indices {1,..., k}, {k + 1,..., k + l}, {k + l + 1,..., n}, respectively. Other inequalities and applications are given.
Download Info
To download:
If you experience problems downloading a file, check if you have the
proper application to
view it first. Information about this may be contained
in the File-Format links below. In case of further problems read
the IDEAS help
page. Note that these files are not on the IDEAS
site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Volume (Year): 10 (1980) Issue (Month): 4 (December) Pages: 499-516 Download reference. The following formats are available: HTML
(with abstract),
plain text
(with abstract),
BibTeX,
RIS (EndNote, RefMan, ProCite),
ReDIF
For technical questions regarding this item, or to correct its listing, contact: (Heidi Boesdal).
Related research
Keywords:
Cited by: (explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)