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Approximation of the basic martingale. (English) Zbl 0814.60043
Consider probability inequalities governing the approximation of the basic martingale, the rescaled difference between a counting process $$N_ n = \{N_ n(t)\}_{t \in [0, \infty)}$$ adapted to some filtration $$\{A_{nt}\}_{t \in [0, \infty)}$$ of $$\sigma$$-algebras contained in the probability space $$(\Omega,A,P)$$ and continuous compensator $$A_ n$$ of $$n^{-1} N_ n$$ with respect to $$\{A_{nt}\}_{t \in [0, \infty)}$$. A time-transformed Wiener process $$W_ n$$, which strongly approximates $$M_ n = n^{1/2} \{n^{-1} N_ n - A_ n\}$$ is constructed. A main result is exponential inequality for the distance (in the supremum metric) between $$M_ n$$ and the approximating process $$W_ n$$.
Application to the random censoring model is given. In the random censoring model it is more or less customary to refer to $$M_ n$$ as the basic martingale which converges to a standard time-transformed Wiener process $$\widetilde W_ n$$ as the sample size $$n$$ increases. The main result is used to study the behavior of statistics of the form $$T(M_ n)$$, which occur in the field of goodness-of-fit tests, in particular when the null hypothesis is composite. Some examples of interesting test statistics which belong to a larger class investigated by means of the empirical process approach are proposed. The statistic $$T(M_ n)$$ is analyzed for the special case of the counting process $$N_ n$$ corresponding to the first $$n$$ order statistics of a random sample of size $$(n+1)$$ from an exponential distribution with unknown mean $$\theta$$. The inequalities are compared to inequalities derived earlier for the random censoring model by using an empirical process approach.
Reviewer: N.Semejko (Kiev)

##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F15 Strong limit theorems 62G10 Nonparametric hypothesis testing
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