Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

Let X = (X₁, X₂,..., X_n) 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,...,X_n) is said to be strongly MRR₂ if for any set of PF₂ functions {φ{symbol}_v} (i.e., φ{symbol}_v(ξ - η) is totally positive of order 2 on -∞ < ξ, η < ∞) each marginal g(x_ν1,x x_ν2,..., x_νk) = ∫ ... ∫ f(x₁,..., x_n) φ{symbol}₁(x_μ1)φ{symbol}₂(x_μ2) ... φ{symbol}_{n - k}(x_{μn - k}) dx_μ1 ... dx_{μn - k} is MRR₂ in the variables (x_ν1, x_ν2,..., x_νk), where (ν₁,..., ν_k) and (μ₁, μ₂,..., μ_{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{a_i ≤ X_i ≤ b_i, i ∈ I, X_ν ≤ b_ν, ν ∈ J ∪ K}P{a_i ≤ X_i ≤ b_i, i ∈ I} ≤ P{a_i ≤ X_i ≤ b_i, i ∈ I, X_ν ≤ b_ν, v ∈ J}P{a_i ≤ X_i ≤ b_i, i ∈ I, X_ν ≤ b_ν, ν ∈ 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.