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Suppose there are two unbiased estimators T1 and T2 to estimate the average age at marriage, say, μ of the married females based on two independent random samples. One uses simple random sampling without replacement and other uses stratified random sampling. Suppose further variances of T1 and T2 are known to be 70 and 30 respectively. Two new estimators of μ are proposed and are given by T = (T1 + T2)/2 and T* = (0.3T1+0.7T2) Prove that T and T* are consistent for μ given that T1 and T2 are consistent for μ and their variances tend to 0 as sample size tends to infinity (You may use the sufficient conditions for consistency, i.e., as the sample size tends to infinity, the expected value of the estimator approaches the true population value and the variance approaches zero).

Note by Sourav Agarwal
2 years, 9 months ago