I have a "continuous" sample, and I want to fit it to particular model using MLE. For this I need to be able to compute its likelihood function, and I am not sure what the correct definition should be. To be specific, here is a very simple version of my situation. (The actual situation has different number of bins with different bin sizes.)
I have collected 20ng of air particles, of which 5ng have particle diameter less than 1$\mu$m, 12ng have particle diameter 1-10$\mu$m, and 3ng have particle diameter bigger than 10$\mu$m. I want to fit a particle size distribution function from this sample.
For this, given an actual size distribution with cdf F, I need to compute the log-likelihood function. Let $F_1=F(1\mu m)$, $F_2=F(10\mu m)-F(1\mu m)$, and $F_3=1-F(10\mu m)$ (the probabilites of landing in each bin.)
Now, instead of this "continuous" sample of 20ng of particles, if I actually had a "discrete" sample of 20N particles, distributed 5N, 12N, 3N in the three bins, the log-likehood function would have been $$5N\log(F_1)+12N\log(F_2)+3N\log(F_3),$$ which scales linearly with N. (In reality, N is close to the Avogardo's number.) So, my first question is:
Q1) What is a correct log-likelihood function in this case? It seems I should divide the above expression by 20N. Is that correct, and is there a reference for this?
Of course, for fitting a particular model (say a lognormal distribution, which has 2 parameters), we will maximize the above formula, and dividing by N doesn't change anything. The main reason is the second question:
Q2) If I am comparing two different models (say a lognormal distribution (2 parameters) vs a sum of 2 lognormal distributions (5 parameters)) using AIC or BIC, what should the formula be? AIC has a term $2kn/(n-k-1)$, while BIC has a term $k\ln(n)$, where n is the sample size. What should n be in my case? (The issue is that, the log-likelihood formula above is linear in N, while the above two terms in AIC and BIC are not linear in n.)