zbMATH — the first resource for mathematics

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)

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