Does scaling a central $\chi^2$ distribution produce a non-central $\chi^2$ distribution?

Question

For independent samples from two normal populations, $X_1,\dotsc,X_n\sim N(\mu_X, \sigma_X^2)$ and $Y_1,\dotsc,Y_m\sim N(\mu_Y,\sigma^2_Y)$, the $F$ test for equality of variances uses the statistic \begin{align*} F=\frac{s^2_X}{s^2_Y}\overset{d}{=}\frac{\sigma^2_X}{\sigma^2_Y}\frac{\chi^2_{n-1}/(n-1)}{\chi^2_{m-1}/(m-1)}. \end{align*} When the variances are equal, $F\sim F_{n-1,m-1}$.

Wikipedia states that when the variances are not equal, $F$ "has a non-central $F$-distribution". However, according to the standard definition of the non-central $F$-distriubtion, this would mean that $(\sigma_X^2/\sigma_Y^2)\cdot\chi_{n-1}^2\sim \chi^2_{n-1;\delta}$ for some $\delta$.

Is it true that scaling a central $\chi^2$ distribution results in a non-central $\chi^2$ distribution? Or can the term "non-central distribution" refer to any altered central distribution?

Answer 1

Unfortunately, the Wikipedia article on "F-test of equality of variances" is incorrect. When the variances are unequal, the distribution of $F$ is neither $F$ nor non-central $F$, it is simply scaled $F$.

The non-central chi-squared distribution on k df specifically refers to the distribution of $$\sum_{i=1}^k (Z_i+\delta_i)^2$$ where the $Z_i$ are independent N(0,1) and at least some of the $\delta_i$ are nonzero. The "non-centrality" refers to the fact that the distributions of the normal random variables $(Z_i+\delta_i)$ are not "centered" at zero.

On the other hand, the distribution of $\sigma^2 X^2$ where $X^2$ is chisquare is simply gamma or "scaled chisquared". It is not "non-central chi-squared".

The non-central $F$ distribution specifically refers to a ratio of non-central chi-squared distributions, with each divided by its degrees of freedom. Again, this is something much complex than a mere scaling transformation.

Edit

I have now corrected the Wikipedia page.