Discussion
Let \(X_{1}, \ldots, X_{n}\) be a random sample from a lifetime distribution F with parameter \(\theta\). Let f denote the corresponding probability density function and let \(T_{0}\) be a fixed time. We observe \(X_{i} = x_{i}\) only if \(x_{i} \le T_{0}\), i = 1, 2, ....., n. Let \(n_{u}\) denote the number of observed lifetimes and \(x_{(1)} \lt x_{(2)} \lt \ldots\) be the ordered observations. Which of the following is the most appropriate 'likelihood' for estimating \(\theta\) ?
Answer: D
The appropriate likelihood is \(\prod_{i=1}^{n_{u}} f(x_{(i)})\,(1 - F(T_{0}))^{n - n_{u}}\).
This is Type I censoring at the fixed time \(T_{0}\). Each observed failure contributes its density \(f(x_{(i)})\). Each of the \(n - n_{u}\) items still working at \(T_{0}\) contributes only the information that its lifetime exceeds \(T_{0}\), with probability \(1 - F(T_{0})\), the survival function.
Option A uses all n densities as if every lifetime were observed. Option B ignores the censored items, which biases the estimate towards short lifetimes. Option C divides by the density at \(T_{0}\), which is not a probability of surviving past \(T_{0}\).
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