Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 2, pp. 202--220

TESTING FOR INDEPENDENCE BETWEEN HITTING TIMES OF BIVARIATE PQD PROCESSES

By

 T. RAMALINGAM and NADER EBRAHIMI, Northern Illinois University

SUMMARY.  Dependence among the components of a multivariate process induces a host of probabilistic structure among the hitting times of the process. Ebrahimi (1987) has initiated a novel, direct approach to studying the dependence structures of hitting times of bivariate processes. This paper studies the problem of testing for independence between the hitting times of two stochastic processes whose hitting times are known to be positive quadrant dependent (PQD). To this end, we first show that such processes have independent hitting times iff the corresponding record processes are uncorrelated. In the context of a stationary bivariate Gaussian processes, this characterisation is in turn shown to be equivalent to the PQD hitting times of the component Gaussian processes being independent iff the component processes are independent.These results are then used in developing formal nonparametric tests for the above problem in the cases where the two processes are either Gaussian or have increasing sample paths. The models treated herein have applicability in such areas as reliability and time series analysis. Our results can be extended to the case of multivariate processes. 

AMS (1980) subject classification.  62G20, 60G60.

Key words and phrases. PQD(NQD) random variables, characterising PQD(NQD) property, shock models, record processes, random fields, bivariate stationary Gaussian processes, weak convergence, lilkelihood ratio tests.

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This article in mathematical reviews.