0
$\begingroup$

The survival function with respect to time is defined to be

$S(t) = 1 - F(T) = \int_{t}^{\infty}f(x)dx$

With a bit of algebraic manipulation, one arrives at the time dependent hazard rate function,

$\lambda(t) = \frac{f(t)}{S(t)}$

which says that

the probability density function $f(t)$ of the cumulative probability density function $F(t)$ is

$f(t) = \lambda(t) S(t)$

Now, suppose there exists $n$ samples to be observed for a duration $t_{i}$ where $i$ is the i-th sample.

If a death occurs at time $t_{i}$, then the likelihood function is described to be

$L_{i} = f(t_{i}) = S(t_{i})\lambda(t_{i})$

However, if a death does not occur, the likelihood function is described to be

$L_{i} = f(t_{i}) = S(t_{i})$ where $\lambda(t) = 1$.

What is the significant of the physical intuition behind the fact that $\lambda(t) = 1$? Mathematically, $\lambda(t) = 1$ implies that $f(t) = S(t)$ to suggest that the probability density function for an occurrence of death for sample $i$ intersects with the survivability function S(t) for sample $i$.

Any help to further illuminate my doubts is greatly appreciated.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.