I have a quick question on design effects due to survey weights. I would like to ask help since I am stuck in some particular parts though. Here it is:
Show that the following expressions for design effect due to weights are equivalent: $$\left(\sum_hW_hk_h\right)\left(\sum_h\frac{W_h}{k_h}\right)=1+[cv(_h)]^2$$ where $cv$ is the coefficient of variation.
So here is what I have so far:
$$1+[cv(W_h)]^2=1+\frac{\frac{\sum_hnk_hW_h^2}{\sum_hnk_h}-\left(\frac{\sum_hnk_hW_h}{\sum_hnk_h}\right)^2}{\left(\frac{\sum_hnk_hW_h}{\sum_hnk_h}\right)^2}$$ $$=\frac{\sum_hk_hW_h^2}{\left(\sum_hk_hW_h\right)^2}$$
Then I got stuck. I do not know what to do next. I hope someone can help. By the way, if I may just add, it might be helpful to know that $$\sum_hk_h=1$$