For a sample mean unbiased estimator, which achieves the CRLB, I plotted the variance and compared it to the theoretical CRLB. The plot below shows what I got. I have small number of samples to calculate this variance.
The black line is the theoretical bound and the red line is the variance calculated using 10 samples.
Why is the calculation lower than CRLB? If CRLB gives the lowest possible value, should it not mean that the red line should always be on or above the black one, even considering low sample size and numerical errors? An insightful answer will be welcome.