Positive semidefiniteness of estimated covariance matrices in linear models for sample survey data

• Autorzy: Stephen Haslett
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2016
• Tom: 4
• Numer: 1
• Strony: 218-224

Daty

otrzymano

2015-07-16

zaakceptowano

2016-04-09

online

2016-04-29

Twórcy

autor

Stephen Haslett

• Massey University, New Zealand and The Australian National University, Australia

Bibliografia

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Identyfikatory

DOI

10.1515/spma-2016-0020