Conferences

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (1980) proposed the sampling design proportionate to a sample variance while Wywiał (1999, 1999a) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. by Brewer and Hanif (1983), Tillé (2006) and Wywiał (2000, 2003). These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be too sensitive to outli observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Let a fixed population be of the size N. An observation of a variable under study (an auxiliary variable) attached to the i-th population element will be denoted by yi (xi>0), i=1,...,N. The sample of size n, drawn without replacement from the population, will be denoted by s. The sampling design is denoted by P(s). Let X® be the r-th order statistic from a simple sample drawn without replacement. Let G(i,r)={s: X®=xi} be the set of all samples s whose r-th order statistic of the auxiliary variable is equal to xi where r ≤ i ≤ N-n+r.

Wywiał (2008) proposed the following sampling design proportional to the xi value of the X® statistic.

$$
P_2(s|r) = \frac{x_i}{\sum^{N-n+r}_{j=r} {j-1 \choose r-1} {N-j \choose n-r} x_j }
$$
for s in G(i,r), i=r ,..., N-n+r.

Particularly, if xi = c>0 for all i=1,...,N, the above sampling design is reduced to the P0(s) simple sampling design.

The particular cases of the sampling design as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz-Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.

Wywiał (2007) deals with an analysis of the accuracy of the strategies for estimating the mean as well as the total value of a variable under study. Three strategies called quantile types are compared with a simple sample mean, with an order ratio estimator from the simple sample mean as well as with the order ratio estimator from a sample drawn according to the sampling design proportional to the sample mean of an auxiliary variable. The comparison of the strategies’ accuracy has been based on a computer simulation. In the case of a small size population, the mean square errors have been evaluated on the basis of all possible samples which can be selected. In the case of a larger population, the samples have been drawn according to the considered sampling schemes. Finally, appropriate conclusions have been formulated. The results of the comparison of the introduced quantile type strategies with the well known moment-based ratio strategies are equivocal. Generally, the quantile type strategies can be more precise than the simple sample means when the degree of the order statistic is large and the sample size is small. Hence, it can be eventually useful as sampling design on the second stage of two-stage sampling design.

Key words: sampling design, order statistic, auxiliary variable, sampling scheme, estimation, strategy, accuracy comparison, relative efficiency

References

Brewer K. R.W., Hanif M. (1983). Sampling with unequal probabilities. Springer Verlag, New York-Heidelberg-Berlin 1983.

Lahiri G. W. (1951): A method for sample selection providing unbiased ratio estimator. Bulletin of the International Statistical Institute, vol. 33, s. 133-140.

Singh P., Srivastava A.K. (1980). Sampling schemes providing unbiased regression estimators. Biometrika vol. 67, 1, pp. 205-9.

Tillé Y.(2006). Sampling algorithms. Springer.

Wywiał J. L. (1999). Sampling designs dependent on the sample generalised variance of auxiliary variables. Journal of the Indian Statistical Association vol. 37, pp. 73-87.

Wywiał J. L. (1999a). Generalisation of Singh and Srivastava’s schemes providing unbiased regression estimatiors. Statistics in Transition vol. 4, nr 2, pp. 259-281.

Wywiał J. (2003). Some Contributions to Multivariate Methods in Survey Sampling. Katowice University of Economics, Katowice.

Wywiał J. (2000). On precision of Horvitz-Thompson strategies. Statistics in Transition vol. 4, nr 5, s. 779-798.

Wywiał J. L. Simulation analysis of accuracy estimation of population mean on the basis of strategy dependent on sampling design proportionate to the order statistic of an auxiliary variable. Statistics in Transition-new series vol. 8, no. 1, 2007, pp. 125-137.

Wywiał J. L. (2008). Sampling design proportional to order statistic of auxiliary variable. Statistical Papers vol. 49, Nr. 2/April, 2008, s. 277-289.