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I'm currently reading The Statistical Analysis of Failure Time Data by Kalbfleisch and Prentice and had trouble at arriving at the expression for the survivor function of a random variable $T$ having both discrete and continuous components. The setup is the following:

Let $T$ be a random variable on $[0,\infty)$ with survivor function $F(t)=P(T>t)$. Then

  • if $T$ is absolutely continuous with density $f$, then the hazard function $\lambda$ can be defined as $$ \lambda(t):=\lim_{h\to 0^+}\frac{P(t\leq T<t+h\mid T\geq t)}{h}=\frac{f(t)}{F(t)} $$ for $t\geq 0$, and hence we have $$ F(t)=\exp\left(-\int_0^t\lambda(s)\,\mathrm ds\right),\quad t\geq 0. $$

  • if $T$ is discrete taking on the values $0\leq a_1<a_2<\cdots$, then we define the hazard at $a_i$ as $$ \lambda_i=P(T=a_i\mid T\geq a_i),\quad i=1,2,\ldots. $$ Then we can show that $$ F(t)=\prod_{j\mid a_j\leq t}(1-\lambda_j),\quad t\geq 0. $$

These expressions for the survivor functions I am ok with. Now they write the following:

More generally, the distribution of $T$ may have both discrete and continuous components. In this case, the hazard function can be defined to have the continuous component $\lambda_c(t)$ and discrete components $\lambda_1,\lambda_2,\ldots$ at the discrete times $a_1<a_2<\cdots$. The overall survivor function can then be written $$ F(t)=\exp\left(-\int_0^t\lambda_c(s)\,\mathrm ds\right)\prod_{j\mid a_j\leq t}(1-\lambda_j).\tag{1} $$

That $T$ has both discrete and continuous components means that the distribution of $T$ is of the form $$ P_T(\mathrm dx)=f_c(x) \lambda(\mathrm dx)+\sum_{j=1}^\infty b_j \delta_{a_j}(\mathrm dx) $$ or equivalently $$ P(T\in A)=\int_A f_c(x)\,\mathrm dx+\sum_{j\mid a_j\in A} b_j $$ for some sequence $a_1<a_2<\cdots$ and $b_i\in (0,1)$ and some non-negative measurable function $f_c$ with $\int_0^\infty f_c\,\mathrm d\lambda+\sum_{i=1}^\infty b_j=1$. If we define $$ \lambda_c(t)=\frac{f_c(t)}{P(T\geq t)}=\frac{f_c(t)}{F(t)},\quad t\neq a_i, $$ and $$ \lambda_i=P(T=a_i\mid T\geq a_i), $$ then how do I show (and is it even true) that the survivor function of $T$ is given by $(1)$?

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    $\begingroup$ I don't see an issue here: the authors consider relations between survivor function and hazard function separately for the case when there is density, and for the case when the distribution is purely discrete. In $(1)$ they combine the results for the distribution that has a continuous part and an purely discrete one. Now, for $t\neq a_j$ they define $\lambda_c(t) = f(t)/F(t)$, whereas for $t = a_j$ they use the second formula. $\endgroup$ – Ilya Jun 25 '13 at 9:44
  • $\begingroup$ @Ilya: Thanks for the response. I guess my question is, what is the definition of a variable having both a continuous part and a purely discrete part? $\endgroup$ – Stefan Hansen Jun 25 '13 at 9:49
  • $\begingroup$ Well, I don't think that in a measure-theory oriented course you would find a formal definition of it, but most likely they mean that the distribution $\mu_T$ of $T$ is given by $$ \mu_T(\mathrm dx) = f_c(x)\lambda(\mathrm dx) + \sum_{j}b_j \delta_{a_j}(\mathrm dx) $$ where $f_c$ is some "sub-density" function, and $\lambda$ is the Lebesgue measure. $\endgroup$ – Ilya Jun 25 '13 at 9:51
  • $\begingroup$ @Ilya: That makes sense, but is it obvious that if $T$ has distribution given by $\mu_T$ above, and we define $\lambda_c$ according to $f_c$, and $\lambda_1,\lambda_2,\ldots$ according to $a_1,a_2,\ldots$, then its survivor function is given by $(1)$? $\endgroup$ – Stefan Hansen Jun 25 '13 at 10:50
  • $\begingroup$ To be honest, in the current shape it is even not very obvious how do they define $\lambda_c$ and $\lambda_j$ in such a case. $\endgroup$ – Ilya Jun 25 '13 at 10:51
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The function $F(t) = \mathsf P(T>t) = 1-\mathsf P(T\leq t)$ is clearly of RCLL class on $[0,\infty)$. As a result, the definitions of continuous part of the hazard function $\lambda_c$ and discrete parts allow you computing $F$ by integrating $\lambda_c$ in between of the jumps, and applying jump conditions at $t = a_j$. The latter have the following shape: $$ \lambda_j = \mathsf P(T = a_j\mid T\geq a_j) = \frac{F(a_j-) - F(a_j)}{F(a_j-)}\implies F(a_j) = F(a_j-)(1-\lambda_j) $$ where $F(t-):=\lim_{s\uparrow t}F(s)$.

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    $\begingroup$ Changed some $S$'s to $F$'s - hopefully it's what you intented to write. Thanks for your help. $\endgroup$ – Stefan Hansen Jun 25 '13 at 12:04
  • $\begingroup$ @StefanHansen: thanks $\endgroup$ – Ilya Jun 25 '13 at 12:32
  • $\begingroup$ Stefan what I don't understand is that in the book by Kalbfleisch and Prentice they define $F(t) = P(T \geq t)$ on page 6 instead of $F(t) = P(T > t)$ the way you define it. Hence in the denominator of the fraction in the post by @Ilya I fail to see how $P(T \geq a_j)$ can get turned to $\lim\limits_{t\to a_j^{-}}F(t)$ given the book's definition shouldn't it be just F(a_j). Which then raises the question, how did the authors get to the general survival function you quote in (1)? :S $\endgroup$ – user1200428 Dec 9 '14 at 14:18
  • $\begingroup$ Also their notation is horrible!! $\endgroup$ – user1200428 Dec 9 '14 at 14:27

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