# Can the numerical variance be lower than Cramer-Rao Lower Bound?

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.

• "I plotted the variance": how did you do that? where are these values coming from? – Marcus Müller Jan 9 at 16:47
• @Marcus Muller...basically the way I plotted the variance is: 1. add white gaussian noise to a constant $N_1$ times, so $N_1$ realizations of the white gaussian noise with a fixed value of $\sigma^2$. 2. Chopped it up into 10 pieces. Calculated the sample mean estimate (10 values of the estimate). 3. Calculated the variance of these values. 4 .Repeated this for $N_2$ different $\sigma^2$ values. – Zero Jan 9 at 17:07
• you mean, you made a statement about a variance of less than $10^{-10}$ by doing 10 runs? I have bad news for you. How large is $N_1$? – Marcus Müller Jan 9 at 17:43
• Empirical variance isn't the same as theoretical, obviously for finite trials it could be less in some runs. – FourierFlux Jan 9 at 18:07
• @Zero think about that. If your error was always on the same side of the curve, how can that estimator be unbiased anymore? – Marcus Müller Jan 9 at 20:27