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Let's say I observe a variable $F_{obs}$ and I have a model to predict this variable through the function $F_{pred} = f(x_1,x_2,x_3,x_4)$.

I want to find the possible range of each predictor covariate $x_i$, so that the combination of all $x_i$'s yields values of $F_{pred}$ that are close to $F_{obs}$. I assume a uniform distribution on each $x_i$.

My approach is to draw $N_1$ samples for each $x_i$ and keep the value of $x_i$ that minimize $F_{obs} - F_{pred}$. I repeat this $N_2$ times to obtain a range of possible values of $x_i$ as well as relative frequencies for bins within each $x_i$.

Is there a name for this approach, or is it just plain Monte Carlo? Does it resemble any other common method?

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  • $\begingroup$ It seems highly inefficient (or worse) to consider the $x_i$'s separately as you want a joint realisation of $(x_1,\ldots,x_n)$ that brings your predictor and your observed value close together. $\endgroup$
    – Xi'an
    Commented Feb 14, 2018 at 16:00
  • $\begingroup$ @Xi'an thanks for your answer. Do you have a recommendation on how to improve the approach? $\endgroup$
    – ye-ti-800
    Commented Feb 14, 2018 at 18:16
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    $\begingroup$ There are many ways to seek optima by pseudo-random simulation, from stochastic optimisation to simulated annealing, to stochastic EM, &tc. One chapter of both our books, Monte Carlo Statistical Methods and Introducing Monte Carlo Methods with R, among many others. $\endgroup$
    – Xi'an
    Commented Feb 14, 2018 at 18:56
  • $\begingroup$ @Xi'an many thanks for the recommendations. My primary aim is to sample the ranges of each parameter, and only the secondary aim is to find an optimum. That's why I went for the simpler approach $\endgroup$
    – ye-ti-800
    Commented Feb 15, 2018 at 14:47

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From your description, the method seems to belong to the category of $M$ estimators: for a sample $(x_1,…,x_n)$ with expectation $F(\theta)$, the estimate of $\theta$ is produced by$$\arg\min_\theta \sum_{i=1}^n d(x_i,F(\theta))$$where $d(\cdot,\cdot)$ is a distance (or a deviance), as for instance in the least squares estimator$$\arg\min_\theta \sum_{i=1}^n (x_i-F(\theta))^2$$The proposed exploration technique is a (rudimentary) type of Monte Carlo optimisation.

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