Asymptotic expansion for distribution of the trace of a covariance matrix under a two-step monotone incomplete sample

The covariance matrix is embedded in several statistics (such as the trace and general variance) of multivariate statistical analysis. We investigate the trace of the covariance matrix in the context of a two-step monotone incomplete sample drawn from Np+q(μ,Σ), a multivariate normal population with mean μ and covariance matrix Σ. Since there are the maximum likelihood estimator (MLE) and the unbiased estimator (UBE) for the covariance matrix, we uniformly deal with these and derive the asymptotic distribution and the asymptotic expansion of the distribution of the trace using them. The accuracy of these results is investigated by numerical simulation, which may be applicable to multivariate analysis under the two-step monotone incomplete sample.