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There is a variable Y with exponential distribution with parameter theta. The prior distribution is gamma distribution with parameters alpha and beta. If we don't have an actual observation of Y, but only observations that are equal to the minimum between the value of Y and some constant a. How in this case can we find the posterior distribution?

crab13
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  • I like the title... – SacAndSac Apr 02 '22 at 11:41
  • What about only using observations that are larger than $a$? – Lei Hao Apr 02 '22 at 13:23
  • @LeiHao The task is to use observations Z where $Z_i = min(Y, a)$, and we have only information regarding Z (a is some fixed constant). For example, there is a warehouse and we have an order of 5 boxes of smth, but we have only 4 in our warehouse, so the observation available is 4, but if we have 6 boxes but the retailer needs only 5 this day, then the value is 5. Could you help me with that? – crab13 Apr 02 '22 at 13:58

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The tail probability of $Z$ is $P(Z > s) = P(Y > a) = e^{-\lambda a}$ if $s < a$ and $= P(Y > s) = e^{-\lambda s}$ if $s >=a$.

The pdf of $Z$ is $f(z) = \lambda e^{-\lambda z} / e^{-\lambda a} = \lambda e^{-\lambda (z-a)}$ for $z \ge a$, and $0$ otherwise.

The posterior $p(\lambda) \propto \lambda^{\alpha - 1} e^{-\beta \lambda} * \prod_{z_i > a} \lambda e^{-\lambda(z_i - a)} = \lambda^\alpha e^{-\lambda \big (\beta + \sum_{z_i \ge a} (z_i -a) \big) } $, which implies the posterior follows a $Gamma\big(\alpha + 1, \beta + \sum_{z_i \ge a} (z_i -a)\big)$ distribution.

Lei Hao
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