make_deville_tille_matrix: Create a quadratic form's matrix for a Deville-Tillé variance...

In bschneidr/svrep: Tools for Creating, Updating, and Analyzing Survey Replicate Weights

Create a quadratic form's matrix for a Deville-Tillé variance estimator for balanced samples

Description

Creates the quadratic form matrix for a variance estimator for balanced samples, proposed by Deville and Tillé (2005).

Usage

make_deville_tille_matrix(probs, aux_vars)

Arguments

probs	A vector of first-order inclusion probabilities
aux_vars	A matrix of auxiliary variables, with the number of rows matching the number of elements of probs.

Details

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.

Value

A symmetric matrix whose dimension matches the length of probs.

References

- 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.

bschneidr/svrep documentation built on Feb. 11, 2025, 4:24 a.m.