Are there amplitude and phase characteristics of harmonic signal components?
If the amplifier is memoryless (i.e. there is only AM/AM conversion and no AM/PM conversion) and the input signal is bandpass (centered around some carrier frequency $F$), for input signal:

$$
x(t) = a(t)\cos(2\pi Ft + \alpha(t))
$$
the output signal
$$
y(t) = v(x(t)) = v(a\cos(2\pi F t)) = v(a \cos \theta)
$$

can be expanded using a Fourier series:
$$
v(a\cos\theta) = v_0(a) + v_1(a)\cos(\theta) + v_2(a)\cos(2\theta) \ldots
$$

If the amplifier is memoryless, the coefficients $v_m(a)$ are real and determine the harmonic amplitudes, and there is only AM/AM conversion and no phase shift on the harmonics. 

The coefficients are known as the "Chebyshev Transform" of the nonlinearity $v(x)$ and are given by:
$$
v_m(a) = \frac{2}{\pi} \int_0^\pi v(a \cos\theta)\cos(m\theta)d\theta
$$
If the amplifier has some memory, but "not much", it is known as a "quasi memoryless" system, and the coefficients are complex (giving AM/PM conversion). 

To characterize the amplifier, you would need to measure $v_m(a)$ for each harmonic $m$. However, since usually all you need is the first harmonic because others will get filtered out later, you can measure $v_1(a)$ from standard power out/power in curves. (The same ones used to measure 1dB compression and 3rd order intercept). The power gain measured in this curve is just $(v_1(a)/a)^2$. 

Source: 
appendix C of:
https://smartech.gatech.edu/bitstream/handle/1853/5327/ku_hyunchul_200312_phd.pdf?sequence=1&amp;isAllowed=y