approximations are very good and the behavior of the exact densities, especially in the critical right hand tale of this distribution, are very accurate.

\[fig.dens\]

![Density plots of the estimated exact standardized distribution of $t_{\mathcal U}$ (red line), the standard Gaussian distibution (black line) and three bootstrap approximations (blue lines). Left panel, $T$=50 ($h=0.2$), right panel $T$=500 ($h=0.04$).](n50h02N.pdf "fig:"){height="6.0cm" width="7.0cm"} ![Density plots of the estimated exact standardized distribution of $t_{\mathcal U}$ (red line), the standard Gaussian distibution (black line) and three bootstrap approximations (blue lines). Left panel, $T$=50 ($h=0.2$), right panel $T$=500 ($h=0.04$).](n500h004.pdf "fig:"){height="6.0cm" width="7.0cm"}

We next investigate the finite sample size and power behavior of the bootstrap studentized test under the aforementioned variety of process parameters and three different sample sizes, $T=50$, $T=100$ and $T=200$. The results obtained are shown in Table \[tab:empsize-power\]. As it is evident from this table, the bootstrap studentized test has a very good empirical size and power behaviour even in the case of $T=50$ observations. The empirical sizes are close to the nominal ones and the empirical power of the test increases to one as the deviations from the null become larger (i.e., larger values of $a_2$) and/or the sample size increases.

  ----- ------- 