Some optimization problems with applications to canonical correlations and sphericity tests

AbstractOptimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)−1ΣY(Y′ΣY)−1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h1 + ϵ1hp, …, hr + ϵrhp−r+1) and Y = (h1 − ϵ1hp, …, hr − ϵrhp−r+1, Yr+1, …, Ys), where h1, …, hp are the eigenvectors of Σ corresponding to the eigenvalues λ1 ≥ λ2 ≥ … ≥ λp > 0, ϵj = +1 or −1 for j = 1,2,…, r and Yr+1, …, Ys, are linear functions of hr+1,…, hp−r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ1,…,θr and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13)

A note on idempotent matrices

AbstractLet H be an n × n matrix, and let the trace, the rank, the conjugate transpose, the Moore-Penrose inverse, and a g-inverse (or an inner inverse) of H be respectively denoted by trH, ρ(H), H∗, H†, and H−. This note develops two results: (i) the class of idempotent g-inverse of an idempotent matrix, and (ii) if H is an n × n matrix and ρ(H) = trH, then tr(H2H†H∗) ⩾ ρ(H), and the equality holds iff H is idempotent. This result is compared with the previous result of Khatri (1983), and some consequences of (i) and (ii) are given

Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

Correcting remarks on "characterization of normality within the class of elliptical contoured distributions"

Let x have a spherical distribution (SD), with characteristic function φ(t′t), t ε{lunate} Tp, to be denoted by x ≈ Sp(φ). Let x′Ax be a quadratic form, with A = A′. A correction is given to Theorem 1 in Khatri and Mukerjee (1987), wrongly asserting that if Q = x′Ax is chi-square distributed with r degrees of freedom (χ2r), then x is necessarily normal and r = Rank(A), the rank of A. A related characterization of the gamma-type SD (1.1) by a gamma-distributed Q is also given. © 1991