Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 38 / 2

Use the paper's notation $F$ for the survivor function, not for a cumulative distribution function. For a nonnegative continuous event time,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.

The probability density function is the negative derivative of the survivor function:Indeed the probability in a short interval is $F(t)-F(t+\epsilon )$, and division by its width gives the derivative relation.

By the definition of conditional probability,ThereforeIn particular $d\log F/dt=-h$ and $F(t)=e^{-H(t)}$ when $F(0)=1$.

Continuity of a nonnegative event-time distribution excludes an atom at zero, soThe 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.

On $\{T>c\}$, $X=c$ and the integrated hazard equals the constant $H(c)$. ThusFor the following finite calculations take $F(c)>0$; endpoint cases follow by a limit.

The event contribution is an ordinary density integral. Since $f=-F^{\prime }$ and $H^{\prime }=h$, integration by parts givesConsequently

Adding the event and censored contributions cancels the boundary terms:This is the mean accumulated hazard before fixed censoring identity. If $F(c)=0$, interpret the boundary term by its limiting value: $FH=-F\log F\to 0$. The identity then gives one.

Let $D={\sum }_{j=1}^n{1}_{\{T_j\le c_j\}}$ be the observed event count. For each subject, the fixed-censoring identity gives $\mathbb{E}H(X_j)=\mathbb{P}(T_j\le c_j)$. Linearity of expectation therefore yieldsIndependence between subjects is not needed for this expectation identity. The common $H$ presumes a common marginal hazard; with heterogeneous subject-specific hazards, replace it by $H_j$ for subject $j$. Random censoring admits the analogous conditional argument when its independence assumptions hold.