Estimating Lifetime Distribution

Study types

• Case report
• Observational studies: prospective (before), retrospective (after), cross-sectional (single point in time)

For person \(i\), denote \(t_i\) as the lifetime. Then, for a cohort of \(n\) persons, the lifetimes can be modeled as \(T_1,T_2,\ldots,T_n\) which are iid with df \(F_T(t)\).

Nonparametric method

Empirical df

\[F_n(t)=\begin{cases} 0 & \text{if }t<T_{(1)} \\ \dfrac{i}{n} & \text{if }T_{(i)}\leq t<T_{(i+1)} \\ 1 & \text{if }t\geq T_{(n)} \end{cases} \] Alternatively, for \(t_{(k)}\leq t<t_{(k+1)}\), \[\hat{S}(t)=\prod_{j=1}^k\left(1-\dfrac{d_j}{n_j}\right)\]

Greenwood’s formula

\[\mathrm{Var}(\hat{S}(t))\approx\hat{S}^2(t)\sum_{j=1}^k\frac{d_j}{n_j(n_j-d_j)}\]

Parametric Method

Assume that the lifetime follows a distribution with parameter \(\theta\), then \(\hat\theta\) is obtained from solving \[s(\mathbf{x};\theta)=\frac{\partial}{\partial\theta}\ln L(\theta;\mathbf{x})=0\] where \(\mathbf{x}=(t_1,t_2,\ldots,t_n)\) is the observed data.

Uncensored data

\[L(\theta;\mathbf{x})=f(\mathbf{x};\theta)=\prod_{i=1}^nf(t_i;\theta)\] Under exponential distribution with mean \(1/\lambda\), \(\hat\lambda=\dfrac{n}{\sum_{i=1}^nt_i}\).

Censored data

\[L(\theta;\mathbf{x})=\prod_{i=1}^nf(t_i)^{\delta_i}S(t_i)^{1-\delta_i}\] where \(\delta_i=1\) if the observation is uncensored and \(0\) otherwise.

Under exponential distribution with mean \(1/\lambda\), \(\hat\lambda=\dfrac{\sum_{i=1}^n\delta_i}{\sum_{i=1}^nt_i}\).