Counting Process

Definition

A counting process $N(t)$ is a right-continuous process that records the cumulative number of events of a point process in $[0,t]$ (nondecreasing, integer-valued, unit jumps). A multivariate counting process $N=(N_1,\ldots,N_k)$ collects several such processes. Its intensity process is $\lambda_i(t)=\lim_{h\downarrow0}\tfrac1h\,\mathbb E[N_i(t+h)-N_i(t)\mid\mathcal F_t]$, the instantaneous event rate given the observed history $\mathcal F_t$. Counting-process theory (Aalen 1978) makes this the unifying language for survival, event-history, and duration data, with censoring handled naturally through the filtration.

Key Ideas

• Multiplicative intensity model. Aalen's model sets $\lambda_i(t)=\alpha_i(t)\,Y_i(t)$: an unknown nonnegative rate function $\alpha_i$ (the hazard) times an observable predictable at-risk process $Y_i$ (e.g., the number of individuals still at risk). It is nonparametric when $\alpha_i$ is unrestricted apart from regularity, and it nests censored survival data, finite-state continuous-time Markov chains, and birth-and-death processes.
• Compensator and martingale. The compensator is $\Lambda_i(t)=\int_0^t\lambda_i(s)\,ds$, and $M_i(t)=N_i(t)-\Lambda_i(t)$ is a square-integrable martingale with predictable variation $\langle M_i\rangle(t)=\int_0^t\lambda_i(s)\,ds$. "Observed minus expected, given the past" is mean-zero — the property that drives all the inference.
• Nelson-Aalen estimator. The cumulative hazard $B_i(t)=\int_0^t\alpha_i(s)\,ds$ is estimated by $\hat B_i(t)=\int_0^t \tfrac{J_i(s)}{Y_i(s)}\,dN_i(s)$ — add $1/Y_i$ at each observed event while anyone is at risk. It generalizes the empirical cumulative hazard and is the additive-hazard companion of the Kaplan-Meier product-limit (survival) estimator.
• Inference by stochastic integration. Estimators and test statistics are written as stochastic integrals of predictable processes against the martingale $M_i$; their variances are optional/predictable-variation estimators, and the martingale central limit theorem yields consistency and weak convergence to Gaussian processes — with no i.i.d. or fixed-censoring assumption.
• Two-sample and rank tests. Weighted differences of estimated increments give two-sample tests that generalize the log-rank / Mantel-Haenszel and other rank tests to arbitrary censoring and time-dependent risk sets.

How It Works

1. Model the event counts $N_i$ and specify the predictable at-risk process $Y_i$ from the observation/censoring scheme.
2. Assume the multiplicative intensity $\lambda_i=\alpha_i Y_i$; form the martingale $M_i=N_i-\int\alpha_i Y_i$.
3. Estimate the cumulative hazard by the Nelson-Aalen stochastic integral $\hat B_i(t)=\int_0^t (J_i/Y_i)\,dN_i$.
4. Estimate its variance from the predictable variation and invoke the martingale CLT for confidence bands.
5. For group comparisons, integrate a predictable weight against the difference of increment estimates to obtain a rank-type test statistic with a martingale-based variance.

Why It Matters

• Unifies survival analysis. Kaplan-Meier estimation, the log-rank test, and Cox's proportional-hazards partial likelihood all become special cases or close relatives of the counting-process construction — one theory instead of many ad hoc derivations.
• Handles arbitrary censoring. Because censoring enters only through the predictable $Y_i$ and the filtration, right-censoring, staggered entry, and time-dependent risk sets are accommodated without special-casing.
• Rigorous, general asymptotics. The martingale CLT replaces i.i.d.-based limit theory, giving valid inference for dependent, time-evolving event data — the probabilistic backbone of duration and event-history modeling.

Open Questions

• Mathematical overhead. The square-integrable-martingale / stochastic-integral machinery is heavy relative to the applied payoff, and downstream methods usually hide it.
• Beyond multiplicative intensity. Additive-hazard, frailty (random-effects), and interval-censored extensions require enlarging the model, and identifiability can become delicate.
• Small-sample behavior. The asymptotics are large-sample; with few events the Gaussian approximation to Nelson-Aalen bands and rank tests can be poor.

Related

• Parametric Survival Model
• Interval Censoring
• Cure Rate Model
• Sequential Ordinal Model
• Odd Aalen
• Aalen (1978) Nonparametric Inference for a Family of Counting Processes
• Kalbfleisch (1978) Non-parametric Bayesian Analysis of Survival Time Data
• John D. Kalbfleisch
• Spatial Point Process
• Frailty Model