Failure Time Distribution

The PHREG Procedure

Failure Time Distribution

Let T be a nonnegative random variable representing the failure time of an individual from a homogeneous population. The survival distribution function (also known as the survivor function) of T is written as

$S(t)={\rm Pr}(T \geq t)$

A mathematically equivalent way of specifying the distribution of T is through its hazard function. The hazard function h(t) specifies the instantaneous failure rate at t. If T is a continuous random variable, h(t) is expressed as

$h(t)= \lim_{\Delta t arrow 0^{+} } \frac{ {\rm Pr}(t \leq T \lt t + \Delta t|T \geq t) }{ \Delta t } = \frac{f(t)}{S(t)}$

where f(t) is the probability density function of T. If T is discrete with masses at x₁ < x₂ < ... , then h(t) is given by

$h(t)=\sum_{j} h_{j} \delta (t- x_{j})$

where

$\delta(u)=\{ 0 & {\rm if}u \lt 0 \ 1 & {\rm otherwise} .$

$h_{j}={\rm Pr}(T=x_{j}|T \geq x_{j})=\frac{ {\rm Pr}(T=x_{j}) }{S(x_{j}) }$

for j = 1, 2, ...

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