I'm working on a problem (using R) which involves estimating the number of failures in a population of components. I have information on the number of components that were added to the population in each month & the number of failures in each month (over a period of 28 months). Of course I also know age of any component at any given time (in days). This diagram gives an idea of the current age-profile of the population. The failures are included in red (they occur rarely).
Here is a histogram with the age-profile of the failed components
It appears that failure is approx. equally likely at any time (the diagram above suggests that most failures occur in the early stages, < 250 days, but the population contains many more "young" components than old)
Here is info on how the population developed over time:
date,new_components,cumul_new_components,failures,cumul_failures
2013-06,1,1,0,0
2013-07,3,4,0,0
2013-11,64,68,0,0
2013-12,154,222,0,0
2014-01,69,291,0,0
2014-02,5,296,0,0
2014-03,110,406,0,0
2014-04,92,498,0,0
2014-05,424,922,1,1
2014-06,355,1277,0,1
2014-07,402,1679,1,2
2014-08,589,2268,0,2
2014-09,763,3031,1,3
2014-10,601,3632,3,6
2014-11,550,4182,3,9
2014-12,762,4944,2,11
2015-01,632,5576,8,19
2015-02,638,6214,1,20
2015-03,883,7097,1,21
2015-04,845,7942,2,23
2015-05,1263,9205,4,27
2015-06,1374,10579,8,35
2015-07,1357,11936,7,42
2015-08,1686,13622,8,50
2015-09,1384,15006,11,61
2015-10,1920,16926,7,68
2015-11,1739,18665,19,87
2015-12,1978,20643,18,105
2016-01,216,20859,0,105
I want to predict how many failures will occur in the next 3 months, with a confidence interval. I tried to fit a Weibull distribution but I'm not sure it's so suitable given how few failures occur. Are there maybe better (non-parametric perhaps?) methods available that might allow me to estimate the number of failures in the next 3 months? I expect the population to grow in the next 3 months also, if I could allow for this in the estimation that would be even better. Thanks in advance :)
EDIT
Adding the full dataset here
EDIT #2
I returned to the Weibull distribution after looking at some survival function graphs & realising that the components showed a slight wearing out behaviour. I also discovered that the Weibull was suitable for handling cases with relatively few failures. Below is the Kaplan-Meier survival curve. It's clear that the curve begins to dip a bit around the 600 day mark.
I then fit a Weibull distribution to the data using the survival
package in R.
srFit_weibull <- survreg(Surv(age, delta) ~ 1, dist="weibull")
Where delta is 1 for events, i.e failures & 0 otherwise (I've added delta to my original dataset).
Inspecting srFit_weibull
I get the parameters.
Coefficients:
(Intercept)
9.822805
Scale= 0.8101698
Now, from this answer I need to convert these parameters in order to get the cumulative distribution function (cdf), i.e. F(t)
$\beta = 1/0.8101698 = 1.23$
$\eta = exp(9.822805) = 18449.73$
And the cdf
F(t) = $1 - \exp(-(\frac{t}{\eta})^\beta)$
Now my question is
(i) with F(t) how do I get an estimate, with confidence interval, of how many failures will occur in the next 3 months within this current population? I've read that the number of failures is simply $n*F(t)$ but I'm not sure if this holds in my case (as the population contains components of varying ages & the components exhibit a wearing out behaviour)
(ii) If I add 2000, 2200 & 2500 new components to the population over the next 3 months how can I get an estimate, with confidence interval, of how many components will break over the coming 3 months?