Creates the quadratic form matrix for a variance estimator for balanced samples, proposed by Deville and Tillé (2005).
make_deville_tille_matrix(probs, aux_vars)A symmetric matrix whose dimension matches the length of probs.
A vector of first-order inclusion probabilities
A matrix of auxiliary variables,
with the number of rows matching the number of elements of probs.
See Section 6.8 of Tillé (2020) for more detail on this estimator, including an explanation of its quadratic form. See Deville and Tillé (2005) for the results of a simulation study comparing this and other alternative estimators for balanced sampling.
The estimator can be written as follows: $$ v(\hat{Y})=\sum_{k \in S} \frac{c_k}{\pi_k^2}\left(y_k-\hat{y}_k^*\right)^2, $$ where $$ \hat{y}_k^*=\mathbf{z}_k^{\top}\left(\sum_{\ell \in S} c_{\ell} \frac{\mathbf{z}_{\ell} \mathbf{z}_{\ell}^{\prime}}{\pi_{\ell}^2}\right)^{-1} \sum_{\ell \in S} c_{\ell} \frac{\mathbf{z}_{\ell} y_{\ell}}{\pi_{\ell}^2} $$ and \(\mathbf{z}_k\) denotes the vector of auxiliary variables for observation \(k\) included in sample \(S\), with inclusion probability \(\pi_k\). The value \(c_k\) is set to \(\frac{n}{n-q}(1-\pi_k)\), where \(n\) is the number of observations and \(q\) is the number of auxiliary variables.
See Li, Chen, and Krenzke (2014) for an example of this estimator's use as the basis for a generalized replication estimator. See Breidt and Chauvet (2011) for a discussion of alternative simulation-based estimators for the specific application of variance estimation for balanced samples selected using the cube method.
- Breidt, F.J. and Chauvet, G. (2011). "Improved variance estimation for balanced samples drawn via the cube method." Journal of Statistical Planning and Inference, 141, 411-425.
- Deville, J.‐C., and Tillé, Y. (2005). "Variance approximation under balanced sampling." Journal of Statistical Planning and Inference, 128, 569–591.
- Li, J., Chen, S., and Krenzke, T. (2014). "Replication Variance Estimation for Balanced Sampling: An Application to the PIAAC Study." Proceedings of the Survey Research Methods Section, 2014: 985–994. Alexandria, VA: American Statistical Association. http://www.asasrms.org/Proceedings/papers/1984_094.pdf.
- Tillé, Y. (2020). "Sampling and estimation from finite populations." (I. Hekimi, Trans.). Wiley.