Asymptotic Properties for the Eigenvalue and Eigenvector of a Covariance Matrix under Two-Step Monotone Incomplete Multivariate Normal Data

SYNOPTIC ABSTRACT

We derive the asymptotic properties for the eigenvalue and eigenvector of a covariance matrix in the context of two-step monotone incomplete data drawn from $N_{p+q}(\mathit{\mu },\mathbf{\Sigma })$, which is a multivariate normal population with mean $\mathit{\mu }$ and covariance matrix $\mathbf{\Sigma }$. Our data consist of n observations of all p + q variables and an additional N – n observations of the first p variables; all observations are mutually independent. We use a maximum likelihood estimator (MLE) and an unbiased estimator (UBE) for a covariance matrix $\mathbf{\Sigma }$. Furthermore, we correct for bias of the eigenvalue and eigenvector by evaluating asymptotic expectations. Finally, we investigate the accuracy of our results using numerical simulations.

KEY WORDS AND PHRASES:

• Asymptotic distribution
• eigenvalue
• eigenvector
• monotone incomplete data
• missing value
• perturbation method