Suppose we have the classic AFT (Accelerated Failure Time) model:
$$\log T = -X'\beta + \epsilon$$
I have seen the Likelihood corresponding to this model written in different ways.
For example, I have seen it formulated (i.e. written) relative to distribution of $T$:
$$ L(\beta, \sigma; T, D, X) = \prod_{i=1}^{n} [f(t_i; \mu_i, \sigma)]^{d_i} [S(t_i; \mu_i, \sigma)]^{1-d_i} $$
$$ \ln L(\beta, \sigma; T, D, X) = \sum_{i=1}^{n} d_i \ln f(t_i; \mu_i, \sigma) + (1-d_i) \ln S(t_i; \mu_i, \sigma) $$
And I have also seen it formulated relative to the distribution of $\epsilon$:
$$ L(\beta; T, D, X) = \prod_{i=1}^{n} [f(\epsilon_i)]^{d_i} [S(\epsilon_i)]^{1-d_i} $$
$$ \ln L(\beta; T, D, X) = \sum_{i=1}^{n} d_i \ln f(\epsilon_i) + (1-d_i) \ln S(\epsilon_i) $$
Is there an advantage to using either one of these formulations? Will they both give you the same parameter estimates?
My guess is that since AFT models usually involve Extreme Value Distributions, it might be easier to use one of these formulations compared to the other?
