Checking for a statistically significant peak I have a set of data, $y$ and $x$. I would like to test the following hypothesis: There is a peak in $y$; that is as $x$ increases, $y$ first increases and then decreases.
My first idea was fitting $x$ and $x^2$ in a SLR. That is, if I find that the coefficient before $x$ is significantly positive and the coefficient before $x^2$ is significantly negative, then I have support for the hypothesis. However, this only checks for one type of relationship (quadratic) and may not necessarily capture the existence of the peak.
Then I thought of finding $b$, such a region of (sorted values of) $x$, that $b$ is between $a$ and $c$, two other regions of $x$ that contain at least as many points as $b$, and that  $\bar{y_b}>\bar{y_a}$ and $\bar{y_b}>\bar{y_c}$ significantly. If the hypothesis is true, we should expect many such regions $b$. Thus, if the number of $b$ is sufficiently large, there should be support for the hypothesis.
Do you think I am on the right track to find a suitable test for my hypothesis? Or am I inventing the wheel and there is an established method for this problem? I will greatly appreciate your input.
UPDATE. My dependent variable $y$ is count (non-negative integer).
 A: I was thinking of the smoothing idea also.  But there is a whole area called response surface methodology that searches for peaks in noisy data (it does primarily involve using local quadratic fits to the data) and there was a famous paper I recall with "Bump hunting" in the title.  Here are some links to books on response surface methodology.  Ray Myer's books are particularly well-written.  I will try to find the bump hunting paper.
Response Surface Methodology: Process and Product Optimization Using Designed Experiments 
Response Surface Methodology And Related Topics
Response surface methodology 
Empirical Model-Building and Response Surfaces
Although not the article I was looking for, here is a very relevant article by Jerry Friedman and Nick Fisher that deals with these ideas applied to high-dimensional data.
Here is an article with some online comments.
So I hope you at least appreciate my response.  I think your ideas are good and on the right track but yes I do think you might be reinventing the wheel and I hope you and others will look at these excellent references.
A: Even though you have not answered my question, if my guess is right you are looking for a test of white noise which amounts in the frequency domain to show that spectrum is flat.  So Fisher's periodogram test which in this reference is called Fisher's kappa could be used.  See the link.
http://www4.stat.ncsu.edu/~dickey/Spain/pdf_Notes/Spectral2.pdf
Bartlett's test is also mentioned in the reference.
Now rejecting the null hypothesis amounts to finding a significant peak in the periodogram. This would mean that a periodic component exists in the time series.
Because the test is in the frequency domain and involves periodogram ordinates the ordinate have a chi square 2 distribution under the null hypothesis and are independent.  This special distribution comes about only because of the transformation to the frequency domain.  If x were time this would not work in the time domain or in general the distribution for the ys would not be independent chi square.
But take the model y=constant independent of x.  Use the y$_m$, the mean of the ys as the estimate for the constant.  Then testing for the existence of a peak would amount to rejecting that the residuals form a white noise sequence.
