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Lifetimes with censoring

StatisticsHard

Component lifetimes are independent and exponential with unknown rate $λ$ . $4$ components failed at the times $2$ , $3$ , $5$ and $6$ hours. $2$ more components were still working when observation stopped at $10$ hours, so each is only known to last longer than $10$ hours. What is the maximum likelihood estimate of $λ$ ?

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Hint

A failure contributes the density and a survivor contributes the survival probability.

Solution

A failure at time $t$ contributes $λ e^{- λ t}$ , and a survivor past $10$ hours contributes $e^{- 10 λ}$ .
The likelihood is $λ^{4} e^{- λ (2 + 3 + 5 + 6 + 10 + 10)}$ , which is $λ^{4} e^{- 36 λ}$ .
Setting the derivative of $4 ln λ - 36 λ$ to zero gives $4/ λ = 36$ .
So the estimate is $λ = \frac{1}{9}$ .

Answer: $\frac{1}{9}$

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