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3-P Sampling and Some Alternatives, I

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The relationship of 3-P sampling to some other forms of unequal probability sampling is given first. The true variance and an unbiased sample-based estimator for the variance of the "unadjusted estimate" are then derived. For the "adjusted estimate" there are theoretical expressions for the bias, approximations for the true variance, and a sample-based estimator of the variance. Comparisons based on repeated Monte Carlo sampling indicate that the bias of the adjusted estimate is small for these populations of trees. The true variance of the adjusted estimate is much smaller than that of the unadjusted estimate. But the approximations for the true variance and the sample-based estimator of variance for the adjusted estimate appear somewhat unreliable. For the true variance the best approximation is based on successive modifications of the ordinary formula for p.p.s. sampling with replacement and fixed sample size. Numerical results contrast the unconditional and conditional properties (bias and variance) of the 3-P estimates, illustrate the variability of sample size and the consequent difficulty of planning such a survey, and provide data for planning 3-P sampling in populations like these. Logical considerations indicate that 3-P sampling should be more fully compared with alternative selection rules and estimators that may use concomitant information more effectively in certain circumstances. Some alternative methods for obtaining appropriate probability samples and a numerical investigation of their properties are to be reported in part II, a sequel to this paper.

Document Type: Journal Article

Affiliations: Professor of Forestry, Iowa State Univ., Ames.

Publication date: December 1, 1968

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