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Given a sequence $y = [y_1 \pm \Delta y_1, y_2 \pm \Delta y_2, ..., y_N \pm \Delta y_N]$, how can you compute the maximum of $y$ with its associated uncertainty, and the value of the index $i$ at which the maximum happens and its associated uncertainty?

In other words, how can you compute max and argmax values and uncertainties?

For example, in this case, it's not clear where the maximum occurs and also it's not clear the maximum value.

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Finally, I've decided to use a Monte-Carlo approach to solve that problem. This is, using the mean and the uncertainty values in $y$, generate $M$ experiments of the form $\tilde{y}^i = [\tilde{y}_1, ..., \tilde{y}_N]$. Then, compute the max and argmax for each experiment $\tilde{y}^i$, and finally report the average and standard deviation of the $N$ max and argmax values.

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    $\begingroup$ This probably is the only practical method for such a problem. It is otherwise known as a parametric bootstrap. The implicit issue, well worth analyzing, concerns how sensitive the answer is to the assumed error distributions. Upon discovering that it can be very sensitive, many people might find a Bayesian approach to be appealing. Another approach is to fit a suitable regression model to the data, assuming some underlying theory (or body of experience) suggests a definite model form. $\endgroup$
    – whuber
    Mar 16, 2022 at 14:17

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