Survey Methodology
Exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement

by Kees van Berkel and Erwin VondenhoffNote 1

  • Release date: June 29, 2026
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Abstract

Probability-proportional-to-size sampling is widely used by national statistical offices. Here population units are selected with probabilities proportional to an auxiliary variable. Variance formulas in such designs require both first- and second-order inclusion probabilities. The computation of second-order inclusion probabilities is particularly challenging for large populations, and has been the subject of extensive research. This article presents some new exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement.

Key Words: Horvitz-Thompson estimator; Probability-proportional-to-size (PPS) sampling; Variance estimation.

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How to cite

van Berkel, K. and Vondenhoff, E. (2026). Exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement. Survey Methodology, 52(1), 235-258. Available at: http://www.statcan.gc.ca/pub/12-001-x/2026001/article/00003-eng.pdf.

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ISSN : 1492-0921

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Published by authority of the Minister responsible for Statistics Canada.

© His Majesty the King in Right of Canada, as represented by the Minister of Industry, 2026

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Catalogue No. 12-001-X

Frequency: Semi-annual

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