Bounds and approximations for pairwise comparisons of independent multinormal means

Properties and relationships of some commonly used probability bounds, along with other recently developed bounds and approximations, are evaluated for their performance with pairwise comparisons. The comparisons are of independent sample means obtained from normal random variables with a common variance. Computational methods are presented and numerical results are used to further evaluate the performance of the bounds.     

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Let X ₁,X ₂,…,X _n be independent exponential random variables such that X _i has hazard rate λ for i = 1,…,p and X _j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D _i:n (λ, λ*) = X _i:n  ? X _i?1:n the ith spacing of the order statistics X _1:n  ≤ X _2:n  ≤ ··· ≤ X _n:n , i = 1,…,n, where X _0:n ≡ 0. It is shown that the spacings (D _1,n ,D _2,n ,…,D _n:n ) are MTP₂, strengthening one result of Khaledi and Kochar (2000), and that (D _1:n (λ₂, λ*),…,D _n:n (λ₂, λ*)) ≤ _lr (D _1:n (λ₁, λ*),…,D _n:n (λ₁, λ*)) for λ₁ ≤ λ* ≤ λ₂, where ≤ _lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ₁ < λ₂.     

Lower bounds of the wrap-around -discrepancy and relationships between MLHD and uniform design with a large size

The wrap-around (WD) L₂-discrepancy has been commonly used in experimental designs. In this paper, some lower bounds of the WD L₂-discrepancy for asymmetrical U-type designs are given and the expectation and variance of midpoint Latin hypercube designs (LHD) are also obtained. Relationships between midpoint LHD and uniform designs for symmetrical and asymmetrical cases are discussed in the sense of comparing the lower bound and the expectation of squared wrap-around L₂-discrepancy of U-type designs. Some comparisons between simple random sampling and the lower bounds of U-type designs are given.