Suppose I have data of a certain quantity $X$, given by $x_1,...,x_n$. Now, I take the first digit $d_i$ of each quantity $x_i$, and I want to study the relationship between the empirical distribution of first digits $\hat{p}=(\hat{p}_1,...,\hat{p}_n)$ where $\hat{p_i}$ is the normalized frequency of ocurrence of $i$ as the first digit, and benford's law $$ p_i = \log_{10} (1 + 1/i) $$ Now, I've read this paper on the subject of comparing empirical frequencies of first digits vs benford's law. However, they don't mention whether the methods they mention can be used to be able to reject benford's law with a certain confidence in real time, where data arrive with a certain frequency (like, say, 50 data per second).
I think these methods can be applied to real time comparison with benford law in the following way: given a (small) interval of time (say, 3 seconds), we compute the empirical frecuencies of first digits $\hat{p}=(\hat{p}_1,...,\hat{p}_n)$ and then we compute the simultaneous confidence intervals and $p$-values of the statistics shown in the reference I mentioned previously (we have to make sure to have a sample size of at least 60 data, so that the distribution of the statistics should be relatively close to the asymptotic distributions, so the computed $p$-values should be reliable).
My question is, is this a valid procedure? Does it make sense? If not, is there some sound method for comparing empirical first digit distribution with benford law in real time?
One potential problem I see is that the underlying distribution of first digits may change in a given time window (perhaps even more than once). Which is why I think it is a good idea to use relatively small time windows, so as to have a decent sample size, while reducing the odds that the underlying distribution of first digits may change.
