# Standard errors of Monte Carlo plus linear combination

I'm using Monte Carlo to estimate some quantity $$V(x)$$. To get an approximation of $$V'(x)$$ I would use the following $$V'(x)\approx\frac{V(x+h)-V(x-h)}{2h}$$ so I can simply evaluate it with two estimations and get the approximation. My question is, given the standard errors of $$V(x+h)$$ and $$V(x-h)$$, what should be the SE of $$V'(x)$$?

What about the SE for the second derivative:

$$V''(x)\approx\frac{V(x+h)-2V(x)+V(x-h)}{h^2}$$

• The main issue is that the finite difference approximation to the derivative can be quite a poor approximation to $V^\prime(x)$. One can thus find the SE of the RHS but this does not tell anything about $V^\prime(x)$. For instance, confidence intervals have no guarantee to include $V^\prime(x)$. Jun 27 at 7:56
• I see, so would you say it's meaningless to consider the CI for $V'(x)$ if I compute it this way? Jun 27 at 8:41
• It might be far better to estimate $V^\prime$ in each MC iteration: that will correctly account for the (strong possibility) that $V(x+h)$ and $V(x-h)$ are highly positively correlated. This also reveals that, at a minimum, you need to have information about the covariance of $V(x+h)$ and $V(x-h)$ in addition to their standard errors.
– whuber
Jun 27 at 11:08