Nonparametric Inference for a Family of Counting Processes

Nonparametric Inference for a Family of Counting Processes

Let B = (N1, ⋯, Nk) be a multivariate counting process and let Ftbe the collection of all events observed on the time interval [ 0, t]. 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 appl...

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Veröffentlicht in:The Annals of Statistics. - Institute of Mathematical Statistics. - 6(1978), 4, Seite 701-726
1. Verfasser: Aalen, Odd (VerfasserIn)
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 1978
Zugriff auf das übergeordnete Werk:The Annals of Statistics
Schlagworte:Point process counting process intensity process inference for stochastic processes nonparametric theory empirical process survival analysis martingales stochastic integrals Mathematics mehr... Social sciences Behavioral sciences
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520 |a Let B = (N1, ⋯, Nk) be a multivariate counting process and let Ftbe the collection of all events observed on the time interval [ 0, t]. 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 N via Λ. A statistical model is defined by letting Λi(t) = αi(t)Yi(t), i = 1, ⋯, k, where α = (α1, ⋯, αk) is an unknown nonnegative function while Y = (Y1, ⋯, Yk), together with 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 α 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 βi(t) = ∫t 0αi(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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