We propose a new estimator, the quadratic form estimator, of the Kronecker product model for covariance matrices. We show that this estimator has good properties in the large dimensional case (i.e., the cross-sectional dimension n is large relative to the sample size T). In particular, the quadratic form estimator is consistent in a relative Frobenius norm sense provided ${\log }^3n/T\to 0$ . We obtain the limiting distributions of the Lagrange multiplier and Wald tests under both the null and local alternatives concerning the mean vector $\mu $ . Testing linear restrictions of $\mu $ is also investigated. Finally, our methodology is shown to perform well in finite sample situations both when the Kronecker product model is true and when it is not true.

我们提出了协方差矩阵的 Kronecker 乘积模型的新估计器，即二次型估计器。我们表明该估计器在大维情况下具有良好的特性（即，横截面维数n相对于样本大小T较大）。特别是，二次型估计量在提供 ${\log }^3n/T\to 0$ 的相对 Frobenius 范数意义上是一致的。 我们获得了关于均值向量$\mu $ 的零和局部备选方案下的拉格朗日乘数和 Wald 检验的极限分布。 测试$\mu $ 的线性限制也在调查中。最后，我们的方法被证明在 Kronecker 产品模型为真和不为真时在有限样本情况下均表现良好。