Probability sample encounters the problems of increasing cost and nonresponse. The cost has rapidly been increasing in executing a large probability sample survey, and, for some surveys, response rate can be below the 10 percent level. Therefore, statisticians seek some alternative methods. One of them is to use a large nonprobability sample (S_1 ) supplemented by a small probability sample (S_2 ). Both samples are taken from the same population and they include common covariates, and a third sample (S_3 ) is created by combining these two samples; S_1 can be biased and S_2 may have large sample variance. These two problems are reduced by survey weights and combining the two samples. Although S_2 is a small sample, it provides good properties of unbiasedness in estimation and of survey weights. With these known weights, we obtain adjusted sample weights (ASW), and create a sample model from a finite population model. We fit the sample model to obtain its parameters and generate values from the population model. Similarly, we repeat these processes for other two samples, S_1 and S_3 and for different statistical methods. We show reduced biases of the finite population means and reduced variances.as the combined sample size becomes large. We analyze sample data to show the reduction of these two errors.
Bayesian estimation for finite population proportion under selection bias via surrogate samples