In this chapter, the authors consider the problem of estimating the population means of two sensitive variables by making use ranked set sampling. The final estimators are unbiased and the variance expressions that they derive show that ranked set sampling is more efficient than simple random sampling. A convex combination of the variance expressions of the resultant estimators is minimized in order to suggest optimal sample sizes for both sampling schemes. The relative efficiency of the proposed estimators is then compared to the corresponding estimators for simple random sampling based on simulation study and real data applications. SAS codes utilized in the simulation to collect the empirical evidence and application are included.
Top1. Introduction
The theory of ratio estimation is recommendable when the sampler has a complete knowledge of an auxiliary variable X which is highly correlated with the studied variable Y. The theory appears in text books as Cochran (1977), Wu and Thompson (2020). Some basic notation is needed. The sampler has all the information on the auxiliary variable and is able to compute
the population mean and variance of the auxiliary variable
X;
The population mean and variance of the study variable Y are unknown. The correlation coefficient between X and Y
plays a key role in the theory of ratio estimation.
Commonly is used sampling design simple random sampling (SRS). The differences between using replacement or not is not important when the sampling fraction is small. Note that are unbiased estimators of the means and variances, of a sample of size n selected from a population U={u1,…,uN},
,
zi=xi,yi.
The sampling fraction is f=n/N. In many inquires is acceptable that f is not far from zero.
Ratio type estimators appear as good alternatives for improving the niceties of the usual ratio estimators. Some recent contributions are Al-Omari et al. (2008), Singh et al. (2010, 2014), Bouza- Al-Omari (2011), Al-Omari and Al-Nasser (2018). Singh et al . (2010) proposed using as auxiliary variable a function of the sampling and population means and Xm (the minimum of {X1,…,XN}) or XM (the maximum of {X1,…,XN}). The sampling design considered was SRS.
Section 2 is concerned with introducing basic elements on Stratified Simple Random Sampling (SSRS). We extend Singh et al. (2010) results to sampling a stratified population. The bias and mean squared error (MSE) are derived. SRS is used independently for selecting the samples from the strata.
Section 3 is devoted to introducing basic elements on Ranked Set Sampling (RSS). We extend Singh et al. (2010) results under the RSS design. They are used for developing the estimation in a stratified population. RSS is used for drawing the samples independently from the strata. The bias and mean squared error (MSE) of the developed estimators are derived. A comparison between the biases and MSE´s obtained for the sampling designs SRS and RSS is made. Under mild conditions the comparisons sustained that each RSS model is better than its SRS alternative
Section 4 presents numerical experiments for illustrating the behavior of the proposals. Data provided by a real-life study on the emission of contaminants are used. The results sustain the ideas obtained from the theoretical comparisons. Simulated data, where the sufficient conditions do not hold, generated the preference for RSS too.
Top2. Extending Srswr Estimators To Stratification
We are considering that the population is divided into K strata. Formally
Kadilar and Cingi (2003, 2005) developed some ratio estimators for stratified random .