Solution Found!
Hartley and Ross (1954) derived the following exact bound
Chapter , Problem 50(choose chapter or problem)
Hartley and Ross (1954) derived the following exact bound on the relative size of the bias and standard error of a ratio estimate:
\(\frac{|E(R)-r|}{\sigma_{R}} \leq \frac{\sigma_{\bar{X}}}{\mu_{x}}=\frac{\sigma_{x}}{\mu_{x}} \sqrt{\frac{1}{n}\left(1-\frac{n-1}{N-1}\right)}\)
a. Derive this bound from the relation
\(\operatorname{Cov}(R, \bar{X})=E\left(\frac{\bar{Y}}{\bar{X}} \bar{X}\right)-E\left(\frac{\bar{Y}}{\bar{X}}\right) E(\bar{X})\)
b. Apply the bound to using sample estimates in place of the given population parameters.
Questions & Answers
QUESTION:
Hartley and Ross (1954) derived the following exact bound on the relative size of the bias and standard error of a ratio estimate:
\(\frac{|E(R)-r|}{\sigma_{R}} \leq \frac{\sigma_{\bar{X}}}{\mu_{x}}=\frac{\sigma_{x}}{\mu_{x}} \sqrt{\frac{1}{n}\left(1-\frac{n-1}{N-1}\right)}\)
a. Derive this bound from the relation
\(\operatorname{Cov}(R, \bar{X})=E\left(\frac{\bar{Y}}{\bar{X}} \bar{X}\right)-E\left(\frac{\bar{Y}}{\bar{X}}\right) E(\bar{X})\)
b. Apply the bound to using sample estimates in place of the given population parameters.
ANSWER:Step 1 of 5
a.
The relation is provided as,
Solving the RHS, we have:
