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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?

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    $\begingroup$ 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. $\endgroup$ – StatsPlease Feb 11 '18 at 21:36
  • $\begingroup$ @StatsPlease Could you please provide more about the second way? It's coursework so I cannot choose other models to fit. $\endgroup$ – Harold Feb 11 '18 at 21:54
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    $\begingroup$ Many questions on site relate to estimation via maximum likelihood under censoring. 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 $\endgroup$ – Glen_b -Reinstate Monica Feb 11 '18 at 22:08
  • $\begingroup$ Look at stats.stackexchange.com/questions/133347/… for instance $\endgroup$ – 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.

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