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July, 1978 Nonparametric Inference for a Family of Counting Processes
Odd Aalen
Ann. Statist. 6(4): 701-726 (July, 1978). DOI: 10.1214/aos/1176344247


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.


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Odd Aalen. "Nonparametric Inference for a Family of Counting Processes." Ann. Statist. 6 (4) 701 - 726, July, 1978.


Published: July, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0389.62025
MathSciNet: MR491547
Digital Object Identifier: 10.1214/aos/1176344247

Primary: 62G05
Secondary: 60G45 , 60H05 , 62G10 , 62M05 , 62M99 , 62N05

Keywords: counting process , empirical process , Inference for stochastic processes , intensity process , Martingales , nonparametric theory , point process , stochastic integrals , Survival analysis

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • July, 1978
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