3-d probability density with a line of maxima I am dealing with some data for which one can define a probability density $f(x,y)$. If I then look at cross sections of $f(x,y)$, defined by $g_a(x):=f(x,a-bx)$ where $a$ is a parameter that allows to explore the region $x,y$ of relevance, each $g_a$ has a maximum $g_a(x^*)$ for $x^*(a)$ (mode). 
Interestingly, over the plane $xy$, the set of points $\mathcal{L}(a)=(x^*(a),a-bx^*(a))$ forms approximately a straight line. The family of points $\mathcal{C}(a)=(x^*(a),a-bx^*(a),f(x^*(a),a-bx^*(a)))$ forms a curve embedded in 3-dimensions that has the shape of a ridge over the overall $f(x,y)$ distribution.
Is there an established statistical test that somebody may be able to suggest that may help me show that the hypothesis that there is straight line for the curve $\mathcal{L}$ defined above cannot be rejected? Thanks
 A: I know of no test that would allow you to test a linear model vs all other nonlinear models (doesn't mean it doesn't exist though). To test whether your $x^*(a)$ is linear in $a$, the best I can come up with is fitting a second or third degree polynomial to your $x^*(a)$ points via OLS:
$$x^*(a)=\alpha+\beta_1*a+\beta_2*a^2+\beta_3*a^3+\varepsilon,$$
then test whether $\beta_2$ and $\beta_3$ jointly equal zero. You then essentially test a linear model against a second and third degree polynomial. Fit higher or lower order polynomials as desired. Or, using nonlinear regression, you can test linear models against any other nonlinear model (e.g., exponential), provided the linear model is nested within the nonlinear one, so that your null hypothesis is the linear model.
Instead of testing the coefficients, you could use Bayesian/Akaike information criteria or similar to select the best among different models (punishing models with more parameters). But in each of these cases you have to make a pre-selection in terms of which models are compared with the linear one.
