Li-Wen Xu’s research while affiliated with North China University of Technology and other places

In this article, we consider the two-factor unbalanced nested design model without the assumption of equal error variance. For the problem of testing ‘main effects’ of both factors, we propose a parametric bootstrap (PB) approach and compare it with the existing generalized F (GF) test. The Type I error rates of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the GF test. The PB test performs very satisfactorily even for small samples while the GF test exhibit poor Type I error properties when the number of factorial combinations or treatments goes up. It is also noted that the same tests can be used to test the significance of the random effect variance component in a two-factor mixed effects nested model under unequal error variances.

We propose and study parametric bootstrap (PB) tests for heteroscedastic two-factor MANOVA with nested designs. For the problem of testing “main effects” of both factors, we develop a flexible test based on a parametric bootstrap approach. The PB test is shown to be invariant under affine-transformations. Moreover, the PB test does not depend on the chosen weights used to define the parameters uniquely. The proposed test is compared with the approximate Hotelling T 2 (AHT) test by the simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations and generally outperforms the AHT test in terms of controlling the nominal size. For the heteroscedastic cases, the PB test outperforms the AHT test in terms of power. In addition, the PB test does not lose too much power when the homogeneity assumption is actually valid.

In this article, we propose a parametric bootstrap (PB) test for testing main, simple and interaction effects in heteroscedastic two-way MANOVA models under multivariate normality. The PB test is shown to be invariant under permutation-transformations, and affine-transformations, respectively. Moreover, the PB test is independent of the choice of weights used to define the parameters uniquely. The proposed test is compared with existing Lawley-Hotelling trace (LHT) and approximate Hotelling T2T2 (AHT) tests by the invariance and the intensive simulations. Simulation results indicate that the PB test performs satisfactorily for various cell sizes and parameter configurations when the homogeneity assumption is seriously violated, and tends to outperform the LHT and AHT tests for moderate or larger samples in terms of power and controlling size. In addition, simulation results also indicate that the PB test does not lose too much power when the homogeneity assumption is actually valid or the model assumptions are approximately correct.

In this article we consider the Two-Way ANOVA model with unequal cell frequencies without the assumption of equal error variances. For the problem of testing no interaction effects and equal main effects, we propose a parametric bootstrap (PB) approach and compare it with existing the generalized F (GF) test. The Type I error rates and powers of the tests are evaluated using Monte Carlo simulation. Our studies show that the PB test performs better than the generalized F-test. The PB test performs very satisfactorily even for small samples while the GF test exhibits poor Type I error properties when the number of factorial combinations or treatments goes up.

Admissible prediction problems in finite populations with arbitrary rank under matrix loss function are investigated. For the general random effects linear model, we obtained the necessary and sufficient conditions for a linear predictor of the linearly predictable variable to be admissible in the two classes of homogeneous linear predictors and all linear predictors and the class that contains all predictors, respectively. Moreover, we prove that the best linear unbiased predictors (BLUPs) of the population total and the finite population regression coefficient are admissible under different assumptions of superpopulation models respectively.

Optimal prediction problems in finite population are investigated. Under matrix loss, we provide necessary and sufficient conditions for the linear predictor of a general linearly predictable variable to be the best linear unbiased predictor (BLUP). The essentially unique BLUP of a linearly predictable variable is obtained in the general superpopulation model. Surprisingly, the both BLUPs under matrix and quadratic loss functions are equivalent to each other. Next, we prove that the BLUP is admissible in the class of linear predictors. Conditions for optimality of the simple projection predictor (SPP) are given. Furthermore, the robust SPP and the robust BLUP are characterized on the misspecification of the covariance matrix.

The notion of linear sufficiency for the whole set of estimable functions in the general Gauss–Markov model is extended to the estimation of any special set of estimable functions in a general growth curve model. Some general results with respect to the concept of linear sufficiency are obtained, from which a necessary and sufficient condition is established for a linear transformation, {F1,F2}, of the observation matrix Y to have the property that there exists a linear function of which is the BLUE of the estimable functions .

In this paper, the estimation of variance components in the linear mixed model with two random effects is investigated. The class of combination estimates based on the quadratic invariant statistics and consistent nonnegative estimates are obtained. Furthermore, it is shown that the consistent nonnegative estimate dominates ANOVA estimate under some conditions. © 2010 Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer Berlin Heidelberg.

For the problem of comparing the means of k populations with unequal population variances, a new generalized test variable is defined and the generalized p-value based on this generalized test variable is given. It is shown that the proposed generalized p-value is invariant under the group of scale transformations. Numerical results show that the proposed generalized p-value test performs better than a generalized F-test.