The Annals of Statistics

Nonparametric Inference for a Family of Counting Processes

Odd Aalen

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Let $\mathbf{B} = (N_1, \cdots, N_k)$ be a multivariate counting process and let $\mathscr{F}_t$ be the collection of all events observed on the time interval $\lbrack 0, t\rbrack.$ The intensity process is given by $\Lambda_i(t) = \lim_{h \downarrow 0} \frac{1}{h}E(N_i(t + h) - N_i(t) \mid \mathscr{F}_t)\quad i = 1, \cdots, k.$ We give an application of the recently developed martingale-based approach to the study of $\mathbf{N}$ via $\mathbf{\Lambda}.$ A statistical model is defined by letting $\Lambda_i(t) = \alpha_i(t)Y_i(t), i = 1, \cdots, k,$ where $\mathbf{\alpha} = (\alpha_1, \cdots, \alpha_k)$ is an unknown nonnegative function while $\mathbf{Y} = (Y_1, \cdots, Y_k),$ together with $\mathbf{N},$ is a process observable over a certain time interval. Special cases are time-continuous Markov chains on finite state spaces, birth and death processes and models for survival analysis with censored data. The model is termed nonparametric when $\mathbf{\alpha}$ is allowed to vary arbitrarily except for regularity conditions. The existence of complete and sufficient statistics for this model is studied. An empirical process estimating $\beta_i(t) = \int^t_0 \alpha_i(s) ds$ is given and studied by means of the theory of stochastic integrals. This empirical process is intended for plotting purposes and it generalizes the empirical cumulative hazard rate from survival analysis and is related to the product limit estimator. Consistency and weak convergence results are given. Tests for comparison of two counting processes, generalizing the two sample rank tests, are defined and studied. Finally, an application to a set of biological data is given.

Article information

Ann. Statist. Volume 6, Number 4 (1978), 701-726.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G10: Hypothesis testing 62M99: None of the above, but in this section 62N05: Reliability and life testing [See also 90B25] 60G45 60H05: Stochastic integrals 62M05: Markov processes: estimation

Point process counting process intensity process inference for stochastic processes nonparametric theory empirical process survival analysis martingales stochastic integrals


Aalen, Odd. Nonparametric Inference for a Family of Counting Processes. Ann. Statist. 6 (1978), no. 4, 701--726. doi:10.1214/aos/1176344247.

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