I don't have any background in statistics, so maybe I may say things that are incorrect.

I have to model the production of waste deposition at each container of a set of containers and I have access to the values of the fill-up rates for certain intervals of time. I need to generate future values based on the historical past values.

First I thought that a Normal Distribution could be adopted, but since there's no negative waste deposition rates I thought on using the Log-Normal Distribution as it starts at the zero value.

Imagine that I have the following ten values: 5; 0; 9; 2; 6; 4; 1; 0; 5; 3

My question is: How can I obtain the parameters μ and σ that are needed to model the probability distribution?

I see that an approach is to use: enter image description here

But then, since I have values of xk that are 0, I have ln (0) which is a problem that I don't know how to work around.

From the wikipedia page I see also that when the individual values x1,x2,...,xn are not available I can calculate the parameters using the following formulas: enter image description here

So, from my sample, $\bar{x}$ = 3.5 and $\hat{\sigma}$ = 2.8771. So, can I say that μ and σ are equal to 0.9946 and to 0.7185, respectively?

If yes, then what bugs me the most is that I have the individual values but some of them happen to be 0. How could it be that I could use those first formulas only if the values are different than 0?

Any help would be great! Thank you in advance!

  • 3
    $\begingroup$ The support of the lognormal distribution is $(0, +\infty)$. The fact that you have values of $0$ thus immediately rules out the lognormal distribution as a suitable model for these data. What exactly do these fill-up rates mean (e.g. what does a value of $5$ mean)? $\endgroup$ – COOLSerdash Nov 29 '19 at 13:54
  • $\begingroup$ A value of 5 means that for a specific day a container was filled up by 5%. A value of 0 represents that no waste was deposited during that day. If there are 0 values but no negative ones, what suitable models could be used to characterize this? $\endgroup$ – Talochas Nov 29 '19 at 14:17
  • $\begingroup$ So it's always a percentage between 0% and 100%? Am I understanding this correctly? $\endgroup$ – COOLSerdash Nov 29 '19 at 14:24
  • $\begingroup$ Yes. It could get overfilled, but let's neglect that $\endgroup$ – Talochas Nov 29 '19 at 14:31
  • 1
    $\begingroup$ @COOLSerdash given we know that there are exact 0's, the beta itself would be unsuitable for the same reason the lognormal is. A 0- & 1- inflated beta might be a better choice. $\endgroup$ – Glen_b -Reinstate Monica Nov 30 '19 at 3:27

Pad it with an arbitrary small number, e.g. 0.00001; basically your minimum precision. It will yield a highly negative value of the logarithm, but that's fine.

Assuming the production was continuous in time, you can never actually measure a point where it's exactly zero in reality anyway, it's asymptotic too, so it's not entirely unprincipled.

  • 1
    $\begingroup$ The "yield a highly negative value" looks like it's going to bias any estimation procedure strongly, so please explain what you think this is "fine." $\endgroup$ – whuber Nov 29 '19 at 17:32

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