This is why epidemiologists (ref: https://pdfs.semanticscholar.org/ebf2/48c9fc0a1a23d1778b9408... or https://news.ycombinator.com/item?id=22195827 for my comment on the paper) prefer instead this formula which only counts patients for which the eventual outcome is already known:
deaths / (deaths + recoveries)
With this formula, there is no need to guess the "patient timeline". If you do the math right now for the largest case study known to date (published yesterday: https://www.thelancet.com/journals/lancet/article/PIIS0140-6... covering 99 patients, 11 dead, 31 recovered) you get:
11 / (11 + 31) = 26% case fatality rate
If you do the math on ALL data from mainland China (170 deads, 124 recovered as of 2020-01-29 http://www.nhc.gov.cn/xcs/yqtb/202001/e71bd2e7a0824ca69f87bb... or in english: https://en.wikipedia.org/wiki/Timeline_of_the_2019%E2%80%932...), you get:
170 / (170 + 124) = 58% case fatality rate
This formula gives a pretty good estimate of the eventual observed CFR according to the paper I referenced above. Of course, there are caveats, for one deaths or recoveries may not be properly tracked: there could be many patients with mild cases that are not even detected/confirmed and end up recovering on their own.
PS: I am not stating the lower bound CFR is 26%! Another method I applied (described in twitter thread https://twitter.com/zorinaq/status/1222427708723879936) suggest it could be as low as 9%.
I'm not an epidemiologist, but I believe I see flaws in your application of the methods from the paper.
[1] At the bottom page 481 it clearly say the virus peaked on March 27th. The first date they use data for the e2 calculation is April 6th. Two full weeks past the peak.
Citation needed. I am not aware of data supporting this (or the reverse.)
How does that produce a meaningful number? If you got 1000s of infected people, and 1 dies... The formula would be 1 / (1 + 0) = 100%. What does that mean?
The way you used it "1 / (1 + 0) = 100%" means just "the knowledge used in the formula is that one patient died, zero recovered." In that case, yes, the fatality rate is "everybody dies." But using the sample of 1 is of course not meaningful for any statistics, not only for that formula.
In reality, though, people who are going to die don't die at the moment of infection confirmation and people who are going to live do lag in the log because it takes time to confirm if someone has recovered and virus-free.
Some will suggest to simply compare the deaths to the confirmed case from a week? ago but this assumes that the patients are admitted at the same stage of their infection and all cases at the time are accounted for(deaths are probably much better tracked than the infection and the publicity has a huge impact on the reporting). It would be a guesstimate at best.
I would say, let's stick to official numbers since the calculation requires each patients timeline. Surely an educated guess can be constructed by modelling the patient prognosis and virus behaviour but that also requires much more data than what the public has.