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Intensity of counting processes

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The intensity of a counting process is a measure of the rate of change of its predictable part. If a stochastic process is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is

where is a martingale and is a predictable increasing process. is called the cumulative intensity of and it is related to by

.

Definition

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Given probability space and a counting process which is adapted to the filtration , the intensity of is the process defined by the following limit:

.

The right-continuity property of counting processes allows us to take this limit from the right.[1]


Estimation

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In statistical learning, the variation between and its estimator can be bounded with the use of oracle inequalities.

If a counting process is restricted to and i.i.d. copies are observed on that interval, , then the least squares functional for the intensity is

which involves an Ito integral. If the assumption is made that is piecewise constant on , i.e. it depends on a vector of constants and can be written

,

where the have a factor of so that they are orthonormal under the standard norm, then by choosing appropriate data-driven weights which depend on a parameter and introducing the weighted norm

,

the estimator for can be given:

.

Then, the estimator is just . With these preliminaries, an oracle inequality bounding the norm is as follows: for appropriate choice of ,

with probability greater than or equal to .[2]

References

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  1. ^ Aalen, O. (1978). Nonparametric inference for a family of counting processes. The Annals of Statistics, 6(4):701-726.
  2. ^ Alaya, E., S. Gaiffas, and A. Guilloux (2014) Learning the intensity of time events with change-points[permanent dead link]