Quadratic prediction problems in multivariate linear models

Abstract

a b s t r a c tLinear and quadratic prediction problems in finite populations have become of great interest to many authors recently. In the present paper, we mainly aim to extend the problem of quadratic prediction from a general linear model, of form y = Xβ + e, e ∼ N (0, σ 2 V), to a multivariate linear model, denoted by. . , y q ). Firstly, the optimal invariant quadratic unbiased (OIQU) predictor and the optimal invariant quadratic (potentially) biased (OIQB) predictor of Y HY for any particular symmetric nonnegative definite matrix H satisfying HX = 0 are derived.Secondly, we consider predicting a Y HYb and tr(Y HY). The corresponding restricted OIQU predictor and restricted OIQB predictor for them are given. In addition, we also offer four concluding remarks. One concerns the generalization of predicting a Y HYb and tr(Y HY), and the others are concerned with three possible extensions from multivariate linear models to growth curve models, to restricted multivariate linear models, and to matrix elliptical linear models.

On Quadratic Forms of MultivariatetDistribution with Applications

“…By reading Liu and Rong (2007a,b); Liu et al (2009) carefully, one can surprisingly find that the resulting conclusions in the mentioned three articles depend extraordinarily on the same formula, (1.1). Hence, we wonder really what it will lead to if the distribution of the population is not normal.…”

“…See Liu and Rong (2007a,b), respectively, for details. As an extension of Liu and Rong (2007b), the multivariate case was considered in Liu et al (2009). The mentioned formula is that Cov u Au u Bu = 2tr A B + 4 A B…”

Research on Computer-Aided Design of Axial Turbine Blades with Ultra-Low Specific Speed

“…However, the geometric parameter and the checking curves on the hub and the flange can't be obtained firsthand, for the hydraulic calculation can't be implemented in their sections. Because the boundary condition of the fitting curves is parabola, according to the trend extrapolation of exciting curves, they are extended to the hub and the flange intelligently using the quadratic curve [6]. The predicting model is shown as format 5.…”

Quadratic prediction and quadratic sufficiency in finite populations

“…matrix satisfying HX = 0, provided that y is observed while Y r is potentially unobservable. See also [2][3][4] for a further study. They imposed some extra and useful restrictions including Invariance, and Minimality and/without Unbiasedness, on the proposed predictor, and derived the representations of the optimal invariant quadratic unbiased predictor (OIQUP) and the optimal invariant quadratic (potentially) biased predictor (OIQBP).…”