| Summary: | Descriptive analysis of sample survey data estimates means, totals and their variances in a design
framework. When analysis is extended to linear models, the standard design-based method for regression parameters
includes inverse selection probabilities as weights, ignoring the joint selection probabilities. When
joint selection probabilities are included to improve estimation, and the error covariance is not a diagonal
matrix, the unbiased sample based estimator of the covariance is the Hadamard product of the population
covariance, the elementwise inverse of selection probabilities and joint selection probabilities, and a sample
selection matrix of rank equal to the sample size. This Hadamard product is however not always positive
definite, which has implications for best linear unbiased estimation. Conditions under which a change in
covariance structure leaves BLUEs and/or BLUPs are known. Interestingly, this class of “equivalent” matrices
for linear models includes non-positive semi-definite matrices. The paper uses these results to explore how
the estimated covariance from the sample can be modified so that it meets necessary conditions to be positive
semidefinite, while retaining the property that fitting a linear model to the sampled data yields the same
BLUEs and/or BLUPs as when the original Hadamard product is used.
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