Minimise computational cost for given level of MSE

Question

I am trying to understand how to minimise cost of a Monte Carlo implementation for a given value of MSE/RMSE.

Please see the notes attached...I do not follow the second line.

I would be grateful if someone could explain.

Thanks, and best wishes, HP

Answer 1

As is stated, we have $$MSE(h)=aC^{-1}h^{-\gamma}+bh^{2\alpha}$$to find the minimum, we need to find the roots of $${\partial MSE(h)\over \partial h}=0$$which leads to $$-\gamma C^{-1}h^{-\gamma-1}+2\alpha bh^{2\alpha-1}=0$$and by simplification $$\gamma C^{-1}h^{-\gamma}=2\alpha bh^{2\alpha}$$