Optimization of an objective function with noise

I want to estimate the parameters that minimize a multivariate function with noise based on realized observations.

As a simple example, say I have observed the univariate sample below:

The samples were generated from a quadratic function plus noise, but in general the function will be unknown and I don't want to make any assumptions about it other than it is "smooth". In this case, the function at $x \approx 5.8$ happens to be the minimum, but I want to make a statistical argument that the function is minimized at $x=5$.

How should I approach this? Should I fit some kind of nonparametric model to the sample data, and then optimize the fitted, smoothed values? Or are there algorithms that tackle the optimization of noisy function directly?

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A starting point may be to simply regress $y$ on $x$ and $x^2$ including a constant. You could then apply robust regression techniques to this approach if is necessary. –  John Oct 3 '12 at 18:38
The data in the question in the question was only an example. I am looking for a general method to minimize a function without making model assumptions about the relationship between the dependent and the independent variables. –  user2303 Oct 3 '12 at 18:46
You might be interested in the approach described by Jones et al. (1998). Although it focuses on functions that are expensive to evaluate, its approach is the right one in general because it aims to find the optimal point rather than to achieve some kind of good average fit overall. –  whuber Oct 4 '12 at 17:27

If you truly have a smooth function with noise there are many ways to fit it. Parametric models linear and nonlinear are possibilities, but based on how you ask the question it sounds like you are looking for some sort of a nonparametric regression. I think splines and loess are options. In the end you are searching for a minimum though. For parametric models local and global minima can be estimated and confidence intervals constructed for it. In the nonparametric situation Bill Huber described a procedure in his answer to a similar question on CV involving time series data. I think the idea can apply to your problem as well.

But the way you pose the problem is strange and contradictory. If you have a model for multivariate data with noise without having a way to perfectly filter out the noise you cannot know exactly where the minimum is. No matter what fitting technique you use you can only approximate the point where the minimum occurs. If your estimate is 5.8 the confidence interval could in a certain statistical sense rule out 5 given sufficient data but it could never prove that the minimum is at 5 and your data really seems to be telling you otherwise. Maybe there is an assumption you are making that you haven't told us about. Now that the picture is clear and you have edited your question,

I see that since the data is simulated you know the truth. The true model is a quadratic function that has a U shape and hence has a unique global minimum at 5. Even a statistical technique that assumes a quadratic model with that much noise will give you an estimate different from 5. Nonparametric models that make less restrictive assumptions will probably do worse.

For real data as I said you can't perfectly separate the signal from the noise and you will never get an 'exact right" answer. There may be models that can estimate the minimum and be consistent in a probabilistic sense but not perfect for any finite sample size.

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Thanks for your answer. I do want to use a nonparametric model and not make any model assumptions other than its smoothness. So if I understand correctly, you are suggesting just fitting with something like LOESS and minimizing the fitted values? –  user2303 Oct 3 '12 at 18:56
That is one way. Huber's suggestion was something like that but I don't remember all the details. For the time series problem we were actually looking for a minimum usingg seasonal data and I recommended harmonic regression and pointed out some advantages. At the time I hadn't read Bill's answer thoroughly and didn't contrast the advantages and disadvantages of each properly. His approach was a little different because of the time series context and amounted to being efficient because it went through the series successively keeping track of the current global minimum. –  Michael Chernick Oct 3 '12 at 19:06