當考慮不等機率抽樣設計時，若存在一輔助訊息與我們欲估計之未知量成比例，則機率比例抽樣設計(probability proportional to size sampling design)，記為PPS機率抽樣設計，是一個不錯的選擇。Horvitz-Thompson估計量在 PPS 機率抽樣設計之下變異數較小。換句話說，可以更精確地估計母體總和。本文提出一個由Hedayat-Lin方法推廣而得之新演算法，利用本演算法可建構出一些PPS機率抽樣設計。此種設計不僅能不偏的估計Horvitz-Thompson估計量之變異數，且能確保此估計量為非負值。

When unequal probability sampling design is under consideration, and when there exists auxiliary information which is proportional to the unknown quantity that we want to estimate, probability proportional to size sampling design, abbreviated as PPS sampling design, is a good choice.

Under PPS sampling design, Horvitz-Thompson estimator has small variance. In this paper, a new algorithm, which is a generalization of Hedayat-Lin's procedure, is given. Using this algorithm, a PPS(N, n) sampling design can be constructed. This design not only provides unbiased estimate of the variance of Horvitz-Thompson estimator but also assures its non-negativity.

[1] Brewer, K.R.W., A Model of Systematic Sampling with Unequal Probabilities. Australia Journal of Statistics, Vol. 5, 5-13 (1963).
[2] Durbin, J., Design of Multi-stage Surveys for the Estimation of Sampling Errors. Applied Statistics, Vol. 16, 152-164 (1967).
[3] Goodman, R. and Kish, L., Controlled Selection: A Technique in Probability Sampling. Journal of American Statistical Association, Vol. 45, 350-372 (1950).
[4] Hedayat A. and B.Y. Lin, Controlled Probability Proportional to Size Sampling Designs. Technical Report, University Illinois at Chicago (1980).
[5] Hedayat A., B.Y. Lin and Stufken J., The Construction of PS Sampling Designs through a Method of Emptying Boxes. The Annals of Statistics, Vol. 17, No. 4, 1886-1905 (1989).
[6] Horvitz, D.G. and Thompson, D. J., A Generalization of Sampling without Replacement from a Finite Universe. Journal of American Statistical Association, Vol. 47, 663-685 (1952).
[7] Rao, J.N.K., On Two Simple Schemes of Unequal Probability Smapling without Replacement. Journal of Indian Statistical Association, Vol. 3, 173-180 (1965).
[8] Sampford, M.R., On Sampling without Replacement with Unequal Probabilities of Selection. Biometrika, Vol. 54, 499-513 (1967).
[9] Sen, A.R., On the Estimate of the Variance in Sampling with Varying Probabilities. Journal of Indian Society Agricultural Statistics, Vol. 5, 119-127 (1953).
[10] Yates, F. and Grundy, P. M., Selection without replacement from within strata with probability proportional to size. Journal of Royal Statistical Society, Series B, Vol. 15, 253-261 (1953).