Assume I have some measurements $\{(y_i, t_i)\}^n_{i=1}$, where $y_i$ is a real number (measured quantity) from some unknown random function $y_i = f(x_i)$, $x_i$ are inputs to that function and $t_i$ is the time needed to obtain the measurement (measuring time) associated with measurement $i$.

I want to characterize the black-box function $f$ by the time $T$ it takes to get a output larger than some threshold, i.e. $y \geq \theta$, when I naively evaluate $f$ on uniform random inputs. So $T = \sum^m_{j=1} t_j$, where $y_m$ is the first measurement exceeding my threshold.

Now I could make multiple runs, e.g. $K$, and average the $T_k$ to get the empirical mean estimator $\bar{T} = \frac{1}{K} \sum_{k=1}^K T_k$ and variance estimator.

However, measurements are costly and I want to minimize the number of measurements needed to get an estimate. Therefore I would like to use Bootstrapping. Can I apply the basic bootstrap algorithm in that case? Simply resampling with replacement from an initial set of measurements, calculating the time $T$ on each bootstrap sample.


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