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Sampling Theory
Published in Marcello Pagano, Kimberlee Gauvreau, Principles of Biostatistics, 2018
Marcello Pagano, Kimberlee Gauvreau
Simple random sampling does not take into consideration any information that is known about the elements of a population and that might affect the characteristic of interest. Under this sampling scheme, it is also possible that a particular subgroup of the population would not be represented in a given sample purely by chance. When selecting teenagers in Massachusetts, for instance, we might not choose any 17‐year‐old males. This could occur merely as the result of sampling variation. If we feel that it is important to include 17‐year‐old males, we can avoid this problem by selecting a stratified random sample. Our strata might consist of various combinations of gender and age: 15‐year‐old males, 15‐year‐old females, 16‐year‐old males, 16‐year‐old females, 17year‐old males, and 17‐year‐old females. Using this method, we divide the population into H distinct subgroups, or strata, such that the hth stratum has size Nh. A separate simple random sample of size nh is then selected from each stratum, resulting in a sampling fraction of nh/Nh for the hth subgroup. This method ensures that each stratum is represented in the overall sample. It does not require that all study units have an equal chance of being selected, however. Certain small subgroups of a population might be oversampled to provide sufficient numbers for more in‐depth analysis.
Robust inference under r-size-biased sampling without replacement from finite population
Published in Journal of Applied Statistics, 2020
P. Economou, G. Tzavelas, A. Batsidis
The results from the above simulation for the Gamma distribution for r = 1 and r = 2 are presented in Table 2. The last columns present the results of the proposed method. The estimation of both parameters appears to have a relatively small bias and does not seem to depend on the sampling fraction. The left and the center columns demonstrate the effect of treating incorrectly the observed sample either as a random sample for the respective weighted distribution or as a random sample from the original distribution, respectively. In the former case, it is true that treating the observed samples as classical r-size-biased samples does not have the same impact on the estimation as incorrect Weibull distribution. Of course, even in this case, as the sampling fraction increases the estimations become less accurate. In the latter case as the sample fraction increases, the accuracy of the estimators increases too since the sample tends to become a random one. It is worth mentioning that the scale parameter for both incorrect approaches is the same since the gamma distribution belongs to the log-exponential family of distributions and so the r-size Gamma distribution is again a Gamma distribution with the same scale parameter but different shape parameter.