Apply band pass filter (BPF) before calculating the power of a signal (PSD)? From my understanding PSD tries to give a good estimation on the power each frequency attributes to the overall signal power. 

If I am only interested in a frequency range $(\omega_0 \leq \omega \leq \omega_1)$, could I put the signal through a band pass filter for that same frequency range then calculate the signal power using this formula:

$$P_x = \frac{1}{N_1 - N_0 + 1}\sum_{n = N_0}^{n=N_1} \left|x(n)\right|^2  $$

Would this technique give me what I am looking for, (the power of the signal from $N_0$ to $N_1$ for the specific frequency range)?

How does this compare to PSD estimation using any of the popular techniques?

Pros, Cons?

Thank you!
You can switch to the frequency domain with regards to Parseval's theorem. I guess you can transform your problem via discrete-time fourier transformation.