A ratio of amplitude quantities $A$ and $B$ can be expressed in decibels as:

$$20 \log_{10}\left(\frac{A}{B}\right) \text{ dB}.$$

A ratio of power quantities $A^2$ and $B^2$ can be expressed in decibels as:

$$10 \log_{10}\left(\frac{A^2}{B^2}\right) \text{ dB}.$$

The ratio of the squares of the errors of approximating an amplitude quantity $C$ by two approximations $A$ and $B$ can be expressed in decibels as:

$$10 \log_{10}\left(\frac{(A - C)^2}{(B - C)^2}\right)\text{ dB},$$

How should the ratio of the squares of the errors of approximating a power quantity $C^2$ by two approximations $A^2$ and $B^2$ be expressed in decibels?

$$10 \log_{10}\left(\frac{(A^2 - C^2)^2}{(B^2 - C^2)^2}\right)\text{ dB}$$

or perhaps:

$$5 \log_{10}\left(\frac{(A^2 - C^2)^2}{(B^2 - C^2)^2}\right)\text{ dB}?$$

Or is there no common convention?

An amplitude quantity $a$ is estimated by two estimators $\hat a_1$ and $\hat a_2.$ The error of estimator $\hat a_2$ is compared to that of $\hat a_1$. One possible comparison is the ratio of expected values of absolute error:

$$\frac{\operatorname{E}[|a - \hat a_2|]}{\operatorname{E}[|a - \hat a_1|]}.$$

This is a ratio of amplitude quantitities and can be expressed in decibels as:

$$20 \log_{10}\left(\frac{\operatorname{E}[|a - \hat a_2|]}{\operatorname{E}[|a - \hat a_1|]}\right) \text{ dB.}$$

Another possible comparison is the ratio of expected values of square error:

$$\frac{\operatorname{E}[(a - \hat a_2)^2]}{\operatorname{E}[(a - \hat a_1)^2]}.$$

This is a ratio of power quantities and can be expressed in decibels as:

$$10 \log_{10}\left(\frac{\operatorname{E}[(a - \hat a_2)^2]}{\operatorname{E}[(a - \hat a_1)^2]}\right) \text{ dB.}$$

The squares of the estimators can be used to estimate $a^2.$ The estimators $\hat a_1^2$ and $\hat a_2^2$ can be compared by the ratio of expected values of square error:

$$\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}.$$

This is a ratio of squares of power quantities. How should it be expressed in decibels?

$$10 \log_{10}\left(\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}\right) \text{ dB,}$$

or perhaps:

$$5 \log_{10}\left(\frac{\operatorname{E}[(a^2 - \hat a_2^2)^2]}{\operatorname{E}[(a^2 - \hat a_1^2)^2]}\right) \text{ dB} = 10 \log_{10}\left(\frac{\sqrt{\operatorname{E}[(a^2 - \hat a_2^2)^2]}}{\sqrt{\operatorname{E}[(a^2 - \hat a_1^2)^2]}}\right) \text{ dB},$$

where the square roots are power quantities, or is there no commonly agreed convention?