Use the paper's notation

F

for the survivor function, not for a cumulative distribution function. For a nonnegative continuous event time,

F (t) = P (T > t), h (t) = ε ↓ 0 lim \frac{P ( t < T \leq t + ε ∣ T > t )}{ε}, H (t) = \int_{0}^{t} h (s) d s .

(1)

The hazard function is defined where survival is positive, and the last expression is the integrated hazard. These definitions assume the usual absolutely continuous survival distribution so that the displayed density and hazard derivatives exist almost everywhere.

Solution

 0  0

The probability density function is the negative derivative of the survivor function:

f (t) = - F^{'} (t), F (t) = \int_{t}^{\infty} f (s) d s .

(1)

Indeed the probability in a short interval is

F (t) - F (t + ε)

, and division by its width gives the derivative relation.

Solution

 0  0

By the definition of conditional probability,

h (t) = lim_{ε ↓ 0} \frac{F ( t ) - F ( t + ε )}{εF ( t )} = \frac{f ( t )}{F ( t )} .

(1)

Therefore

f (t) = h (t) F (t) .

(2)

In particular

d lo g F / d t = - h

and

F (t) = e^{- H (t)}

when

F (0) = 1

The integrated hazard value follows directly from an integral over an empty interval. If a different model allowed an atom at zero, the first conclusion would require modification; that is not the continuous model used here.