Sampling design proportional to order statistic of auxiliary variable |
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Authors: | Janusz L Wywia? |
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Institution: | (1) Department of Statistics, Katowice University of Economics, 14 Bogucicka, 40-226 Katowice, Poland |
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Abstract: | The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140,
1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika
67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87,
1999) 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. in Wywiał (Some contributions to multivariate methods in,
survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some
sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations
of the auxiliary variable. Moreover, they can be much too sensitive to outliers 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. Its particular cases 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. |
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Keywords: | Sampling design Order statistic Sample quantile Auxiliary variable Horvitz– Thompson statistic Ratio estimator Inclusion probabilities Sampling scheme |
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