Two surveys were independently conducted to estimate a

Chapter , Problem 37

(choose chapter or problem)

Two surveys were independently conducted to estimate a population mean, \(\mu\). Denote the estimates and their standard errors by \(\bar{X}_{1}\) and \(\bar{X}_{2}\) and \(\sigma_{\bar{x}_{1}} \text { and } \sigma_{\bar{x}_{2}}\). Assume that \(\bar{X}_{1}\) and \(\bar{X}_{2}\) are unbiased. For some \(\alpha\) and \(\beta\), the two estimates can be combined to give a better estimator:

                                  \(X=\alpha \bar{X}_{1}+\beta \bar{X}_{2}\)

a. Find the conditions on \(\alpha\) and \(\beta\) that make the combined estimate unbiased.

b. What choice of \(\alpha\) and \(\beta\) minimizes the variances, subject to the condition of unbiasedness?