# Techniques to Calculate Exact Inclusion Probabilities for Conditional Poisson Sampling and Pareto .pi.ps Sampling Designs

[Doktorsavhandling]

This thesis consists of five papers and treats two finite population sampling methods, viz. the Conditional Poisson and the Pareto .pi.ps sampling schemes. Both methods belong to a class of sampling schemes with unequal inclusion probabilities, commonly used in approximate probability to size sampling schemes.

Paper A addresses the problem of determining first and second order inclusion probabilities for both methods, which is a vital element in deriving linear estimators. Tools which consist of dynamic programming algorithms, are created to calculate these exact inclusion probabilities. They make it possible to compute the inclusion probabilities in reasonable execution time for small and moderate samples. Algorithms to adjust the parameters so that arbitrary desirable exact inclusion probabilities are achieved are also given. The results in this paper opened the possibility to use the exact Horvitz-Thompson and Yates-Grundy-Sen variance estimators for quite large samples for Conditional Poisson and for moderate samples in the Pareto case.

In Paper B, using those algorithms, a computer system was developed to compare and contrast the Conditional Poisson and Pareto .pi.ps sampling designs in terms of estimators of population totals, biases and variances. The computer programme produced an empirical comparison of both methods to check the convergence to the asymptotical inclusions. It also enabled adjustment of the parameters to obtain exact variances and make comparisons in these terms. The results from these studies show that the Pareto scheme approaches asymptotical inclusions faster than the Conditional Poisson, and that both methods are very similar in terms of second order inclusions for the adjusted procedures for moderate samples.

The second order inclusion algorithms for the Conditional Poisson Sampling design are generalised in Paper C to a recursive fast procedure to derive higher order inclusion probabilities of arbitrary order.

Paper D proves the existence and partial uniqueness of a set of scale parameters when exact inclusion probabilities are required for any order sampling of fixed distribution shape, a class of schemes of which Pareto .pi.ps sampling is a special case.

Lastly, Paper E reports on a thorough study of approximation accuracy for Pareto inclusion probabilities, aiming at practical use recommendations, for using asymptotically motivated approximations. Numerical results in this study are presented in Appendix 1 and 2.

**Nyckelord: **sampling theory, conditional Poisson sampling, Pareto pips sampling, order sampling, numerical integration, algorithms

Denna post skapades 2006-08-25. Senast ändrad 2013-09-25.

CPL Pubid: 789