zbMATH — the first resource for mathematics

Projection approach to distribution-free testing for point processes. Regular models. (English) Zbl 1458.62189
Summary: We create the notion of equivalence between different martingale models for point processes. This allows to map one model into another model in the same equivalence class. Therefore the distribution of test statistics for goodness of fit testing needs to be calculated in only once, for “standard” model, in each equivalence class. The equivalence classes are surprisingly broad, and thus the economy on computational work is considerable. Namely, any such class includes a non-time homogeneous Poisson model. Therefore it is sufficient to know the distribution of test statistics only for Poisson models.
The situation, therefore, becomes comparable to testing simple hypothesis about a continuous distribution function for a sample of i.i.d. random variables with continuous distribution $$F$$, when it is sufficient to consider $$F$$, uniform on $$[0,1]$$. However, for point processes we consider here parametric cases, and the nature of equivalence is entirely different.
Reviewer: Reviewer (Berlin)
MSC:
 62M09 Non-Markovian processes: estimation 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G44 Martingales with continuous parameter 62G10 Nonparametric hypothesis testing
Full Text:
References:
 [1] P. K. Andersen, O. Borgan, R. D. Gill, N. Keiding,Statistical Models Based on Counting Processes. Springer Series in Statistics. Springer-Verlag, New York, 1993. · Zbl 0769.62061 [2] D. J. Daley, D. Vere-Jones,An Introduction to the Theory of Point Processes. vol. I. Elementary Theory and Methods. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2003;vol. II. General Theory and Structure. Second edition. Probability and its Applications (New York). Springer, New York, 2008. · Zbl 1026.60061 [3] A. F. Karr,Point Processes and their Statistical Inference. Second edition. Probability: Pure and Applied, 7. Marcel Dekker, Inc., New York, 1991. · Zbl 0733.62088 [4] E. Khmaladze, The use of Omega-square Tests for Testing Parametric Hypotheses.Theory Probab. Appl.24(1979), no. 2, 283-302. · Zbl 0447.62049 [5] E. Khmaladze, Martingale approach in the theory of goodness-of-fit tests.Theory of Probability and Its Applications 26(1982), no. 2, 240-257. · Zbl 0481.60055 [6] E. Khmaladze, Note on distribution free testing for discrete distributions.Ann. Statist.41(2013), no. 6, 2979-2993. · Zbl 1294.62095 [7] E. Khmaladze, Unitary transformations, empirical processes and distribution free testing.Bernoulli22(2016), no. 1, 563-588. · Zbl 1345.60094 [8] E. Khmaladze, Distribution free testing for conditional distributions given covariates.Statist. Probab. Lett.129 (2017), 348-354. · Zbl 1379.62036 [9] L. Roberts, Distribution free testing of goodness of fit in a one dimensional parameter space.Statist. Probab. Lett. 99(2015), 215-222. · Zbl 1396.62094 [10] G.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.