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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:

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

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    $\begingroup$ 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 $V^\prime(x)$. $\endgroup$
    – Xi'an
    Jun 27 at 7:56
  • $\begingroup$ I see, so would you say it's meaningless to consider the CI for $V'(x)$ if I compute it this way? $\endgroup$ Jun 27 at 8:41
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    $\begingroup$ 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. $\endgroup$
    – whuber
    Jun 27 at 11:08

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