Timeline for Convergence of a test statistic

8 events

when toggle format	what		by	license	comment
Oct 29, 2015 at 3:48	comment	added	Iosif Pinelis		I have corrected the calculation of the limit displacement.
Oct 29, 2015 at 3:47	comment	added	Iosif Pinelis		On the other hand, if e.g. $h_n\sqrt n\to\infty$ (and still $k^*/n\to\kappa\in(0,1)$), then the alternative limit distribution does not exist -- the entire probability mass is being swept away out of the containing space $D[0,1]$, that is, the probability distribution weakly converges to the zero measure on $D[0,1]$. So, in this case the hypotheses $H_0$ and $H_1$ are asymptotically mutually singular and thus highly distinguishable.
Oct 29, 2015 at 3:47	comment	added	Iosif Pinelis		The full alternative is too broad for one to be able to say something definite about the asymptotics of the distribution. However, if e.g. $h_n\sqrt n\to 0$, then the "alternative" limit distribution is the same as the "null" one (namely, that of the Brownian bridge). So, in this case the hypotheses $H_0$ and $H_1$ are asymptotically indistinguishable.
Oct 29, 2015 at 3:46	history	edited	Iosif Pinelis	CC BY-SA 3.0	corrected the calculation of the limit displacement
Oct 28, 2015 at 14:53	comment	added	bursttim		So it diverges, in general?
Oct 28, 2015 at 13:11	vote	accept	bursttim
Oct 28, 2015 at 13:08	history	edited	Iosif Pinelis	CC BY-SA 3.0	corrected the scaling
Oct 28, 2015 at 12:51	history	answered	Iosif Pinelis	CC BY-SA 3.0