Summary
LetX t be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equationdX t=Ytα(t, Zt) dt+dMt, whereY andZ are predictable covariate processes,M is a martingale and α is an unknown, nonrandom function. We study inference for α by introducing an estimator for\(A(t,z) = \int\limits_0^z {\int\limits_0^t {\alpha (s,x)dsdx} } \) and deriving a functional central limit theorem for the estimator. The asymptotic distribution turns out to be given by a Gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter Wiener process. This result is used to develop a test for independence ofX from the covariateZ, a test for time-homogeneity of α, and a goodness-of-fit test for the proportional hazards model α(t,z)=α1(t)a 2(z) used in survival analysis.
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Research supported by the Army Research Office under Grant DAAL03-86-K-0094
Research supported by the Air Force Office of Scientific Research under Contract F49620-85-C-0007
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McKeague, I.W., Utikal, K.J. Identifying nonlinear covariate effects in semimartingale regression models. Probab. Th. Rel. Fields 87, 1–25 (1990). https://doi.org/10.1007/BF01217745
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DOI: https://doi.org/10.1007/BF01217745