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I am trying to fit a gamma distribution to the failure time of a kind of bulb. I have 40 data. However only half of them are actually the failure time. The result 20 are times those bulbs being used (but they haven't failed yet).

How can I fit a gamma distribution to all the data I have?

The Laconic
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Harold
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    Hi, welcome to SE. It sounds like the best approach is to use Survival Analysis techniques because some of your bulbs have *censored* lifetimes. Essentially, if you want to fit a Gamma distribution, the likelihood function needs to be adjusted for those censored observations. – epp Feb 11 '18 at 21:36
  • @StatsPlease Could you please provide more about the second way? It's coursework so I cannot choose other models to fit. – Harold Feb 11 '18 at 21:54
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    Many questions on site relate to [estimation via maximum likelihood](https://en.wikipedia.org/wiki/Survival_analysis#Fitting_parameters_to_data) under [censoring](https://en.wikipedia.org/wiki/Censoring_(statistics)). A search should turn some of them up. While it's easy enough (as you can see at the 2nd wikipedia link) to write the log-likelihood fo the censored and uncensored observations (and to use a good optimization routine to maximize it), I'd use a survival analysis routine (like `survreg` in R) to fit a gamma to censored data myself - it takes care of a lot of the effort automatically – Glen_b Feb 11 '18 at 22:08
  • Look at https://stats.stackexchange.com/questions/133347/ml-estimate-of-exponential-distribution-with-censored-data/133360#133360 for instance – kjetil b halvorsen May 12 '18 at 20:15

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Numerically solving the likelihood equation remains possible when some or all of the observations are censored.

For instance, suppose we have observations of failure times $\boldsymbol x = (x_1, \ldots, x_n)$, and observations of censoring times $\boldsymbol y = (c_1, \ldots, c_m)$, for a total sample of $m+n$ bulbs, where observations are IID gamma with shape $a$ and rate $b$. Then the likelihood is simply $$\mathcal L(a, b \mid \boldsymbol x, \boldsymbol c) = \prod_{i=1}^n \frac{b^a x_i^{a-1} e^{-b x_i}}{\Gamma(a)} \prod_{j=1}^m S_X(c_j),$$ where $S$ is the survival function of the lifetime; i.e. $$S_X(c_j) = \Pr[X > c_j] = \int_{x = c_j}^\infty \frac{b^a x^{a-1} e^{-b x}}{\Gamma(a)} \, dx = \Gamma(a;c_j).$$ A closed-form solution in the general case is not possible. Software exists to calculate the solution when the data are provided.

heropup
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