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Is there any method to test if the mean function, $f(x)$, of a regression model $y=f(x)+\epsilon$ is even or not?

bankrip
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  • Do you mean $f(x) = f(-x)$? – user2566092 Sep 08 '14 at 19:36
  • Shouldn't it follow from the definition of $f(x)$ given at the beginning of the regression problem? – Avitus Sep 08 '14 at 19:36
  • @Avitus : No, it should not. That is silly. The function is found at the end, not at the beginning. – Michael Hardy Sep 08 '14 at 19:43
  • but then there is something I do not get; if the regression problem is tackled by specifying a function $f=f(x)$ s.t. in mean $y=f(x)$, then is the OP simply trying to check if $f(x)=-f(-x)?$. If this is the case, then the problem is easy. If the OP is trying to check symmetry of the function $\hat{f}$ (with the estimators of the regression coefficients) then it is more complex. – Avitus Sep 08 '14 at 20:10
  • I should have mentioned this is a non-parametric problem, so we do not know the form of $f(x)$. – bankrip Sep 08 '14 at 20:53

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$\newcommand{\m}{\begin{bmatrix}}\newcommand{\em}{\end{bmatrix}}$ You're not saying what your data look like, so any answer must be fairly limited.

Suppose you fit a polynomial $y=a+bx+cx^2$ by least squares. You could then consider the null hypothesis $b=0$ versus the alternative hypothesis $b\ne 0$. You have $$ \m y_1 \\ \vdots \\ y_n \em = \m 1 & x_1^2 & x_1 \\ \vdots & \vdots & \vdots \\ 1 & x_n & x_n^2 \em \m a \\ c \\ b \em + \m \varepsilon_1 \\ \vdots \\ \varepsilon_n \em, $$ and write this as $$ Y = X A + \varepsilon. $$ The least-squares estimates $\hat A=\m \hat a \\ \hat c \\ \hat b \em$ are given by $$ \hat A = (X^\top X)^{-1}X^\top Y $$ Now write $X$ as $$ X = \m X_1 & X_2 \em \text{ where }X_1 = \m 1 & x_1^2 \\ \vdots & \vdots \\ 1 & x_n \em $$ and $X_1$ is the last column.

Take the residuals from regression of $Y$ on these two "even" columns, and regress that vector on the third column $X_2$. Do the usual two-sided $t$-test of whether the slope $b$ is $0$, with $n-3$ degrees of freedom. (Or, since the alternative hypothesis $b\ne0$ is two-sides, do the $F$-test with one degree of freedom in the numerator and $n-3$ in the denominator, and it will be equivalent.)

I won't go into higher-degree terms before knowing more about the problem.

Michael Hardy
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