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ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S U p ‐STATISTIC
Author(s) -
Dahlhaus Rainer
Publication year - 1985
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1985.tb00411.x
Subject(s) - mathematics , kolmogorov–smirnov test , asymptotic distribution , empirical distribution function , central limit theorem , infimum and supremum , goodness of fit , test statistic , brownian bridge , statistic , statistics , combinatorics , statistical hypothesis testing , brownian motion , estimator
. In this paper the asymptotic behaviour of Bartlett's U p ‐statistic for a goodness‐of‐fit test for stationary processes, is considered. The asymptotic distribution of the test process is given under the assumption that a central limit theorem for the empirical spectral distribution function holds. It is shown that the U p ‐statistic tends to the supremum of a tied down Brownian motion. By a counterexample we refute the conjecture that this distribution is in general of the Kolmogorov‐Smirnov type. The validity of the central limit theorem for the spectral distribution function is then discussed. Finally a goodness‐of‐fit test for ARMA‐processes based on the estimated innovation sequence is given, and it is shown that this test statistic is asymptotically Kolmogorov‐Smirnov distributed.