Positive semidefiniteness of estimated covariance matrices in linear models for sample survey data
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.
-  D. Basu. An essay on the logical foundations of survey sampling, Part One, In Foundations of Statistical Inference, V.P. Godambe and D.A. Sprott (eds). Holt, Rinehart & Winston, 203-242, 1971.
-  C-M. Cassel, C.E Särndal, J.H. Wretman. Foundations of Inference in Survey Sampling, Wiley, 1977.
-  R.L. Chambers, C.J. Skinner (eds.). Analysis of Survey Data, Wiley, 2003.
-  W.G. Cochran. Sampling Techniques, Wiley, 1953.
-  W.G. Cochran. Sampling Techniques, 3rd edition, Wiley, 1977.
-  S. Fisk. A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices, arXiv:math/0502408v1
- [math.CA] 18 Feb 2005, http://arxiv.org/pdf/math/0502408v1.pdf, 2005.
-  W.A. Fuller. Sampling Statistics, Wiley, 2009.
-  M.H. Hansen, W.N. Hurwitz, W.G. Madow. Sample Survey Methods and Theory, Volumes 1 and 2, Wiley, 1953.
-  S. Haslett. The linear non-homogeneous estimator in sample surveys, Sankhyä Ser. B, 47(1985), 101-117.
-  S. Haslett, S. Puntanen. Equality of the BLUEs and/or BLUPs under two linear models using stochastic restrictions, Statistical Papers, 51(2010), 465-475. [WoS][Crossref]
-  D.G. Horvitz, D.J. Thompson. A generalization of sampling without replacement from a finite universe, Journal of the American Statistical Association, 47(1952) 663-685. [Crossref]
-  R. Lehtonen, E. Pahkinen. Practical Methods for Design and Analysis of Complex Surveys, 2nd Edition, Wiley, 2003.
-  A. Mercer, P. Mercer. Cauchy’s interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences, 23(2000), 563-566.
-  C.R. Rao. A note on a previous lemma in the theory of least squares and some further results, Sankhyä Ser. A 30(1968), 259-266.
-  R.M. Royal, W.G.Cumberland. Variance estimation in finite population sampling, Journal of the American Statistical Association, 73(1978) 351-358. [Crossref]
-  C-E. Särndal, B. Swensson, J. Wretman. Model Assisted Survey Sampling, Springer, 1992.
-  C.J. Skinner, D. Holt, T.M.F. Smith (eds.). Analysis of Complex Surveys, Wiley, 1989.