Integrals

• Lebesgue-Stieltjes Integral
  • Definition
    Suppose $G(·)$ is a right-continuous, non-decreasing step function having jumps at $x_1, x_2, \cdots$. Then for any function $f(·)$, we define the integral $$ \int^b_a f(x) ~ d G(x) \equiv \sum_{j ~: ~ a < x_j \leq b} f(x_j) \cdot \{G(x_j) - G(x_j-)\} = \sum_{j ~ : ~ a < x_j \leq b} f(x_j) \cdot \Delta G(x_j) $$ where $ \Delta G(x_j) = G(x_j) - G(x_j-) $. This is called a Lebesgue-Stieltjes integral. If $G(·)$ is continuous with derivative $g(·)$, then we define $\int^b_a f(x) ~ d G(x)$ to be the Lebesgue integral $\int^b_a f(x)g(x)~ dx $. Thus, we can define a Lebesgue-Stieltjes integral $\int^b_a f(x) ~ d G(x)$ for $G(·)$ either absolutely continuous or a step function.
  • Example
    1. Survival data; 1-sample problem
       $$ (U_i, \delta_i) ~~~ i = 1, 2, \cdots, n $$
       Define the stochastic process $G(·)$ by : 
       $$ \begin{align}
       G(t) 
       & \equiv \sum^n_{i = 1} 1(U_i \leq t, \delta_i = 1) \\
       & = \# \text{ of failures observed on or before } t.
       \end{align} $$
       As before, let $\tau_1 < \tau_2 < \cdots < \tau_K$ denote the distinct failure times, 
       $$ \begin{align}
       d_j & = \# \text{ failures at } \tau_j \\
       Y(\tau_j) & = \# \text{ at risk at } \tau_j 
       \end{align} $$
       Then, for example, if $f(t) = t$,
       $$ \begin{align}
       \int^\infty_0 f(t) ~ d G(t)  
       & = \sum^K_{j = 1} f(\tau_j) \cdot \Delta G(\tau_j) \\
       & = \sum^K_{j = 1} \tau_j \cdot d_j .
       \end{align} $$
       Or, if $f(t) = \sum^n_{i = 1} 1 (U_i \geq t) = \# \text{ at risk at }t$,
       $$ \begin{align}
       \int^\infty_0 f(t) ~ d G(t)  
       & = \sum^K_{j = 1} f(\tau_j) \cdot \Delta G(\tau_j) \\
       & = \sum^K_{j = 1} Y(\tau_j) \cdot d_j .
       \end{align} $$
    2. Survival data; 1-sample problem - 2
       $$ (U_i, \delta_i) ~~~ i = 1, 2, \cdots, n $$
       Consider $$ \hat \Lambda(t) = \int^t_0 \frac{dN(u)}{Y(u)} $$ where $$ N(t) = \sum^n_{i = 1} 1(U_i \leq t, \delta_i = 1) $$ and $$ Y(t) = \sum^n_{i = 1} 1 (U_i \geq t) .$$
       Then it is easily shown, using the notation used for the Kaplan-Meier estimator, that $$ \hat \Lambda(t) = \sum_{\tau_j \leq t} \frac{d_j}{Y(\tau_j)} = \text{ Nelson-Aalen estimator}.$$