Table 1.

Estimated test sizes (Eq. 11) for the first-digit statistic $V_{\left\{1\right\}}^{\left(t\right)}$, using the asymptotic quantile ${\chi }_{\mathrm{8,0.99}}^2$, and for the TS version of the procedure of Barabesi et al. (6), based on $T^{†}=\mathrm{85,500}$ Monte Carlo replicates for each configuration $\left(m_t,n_t\right)$, with $m_t\le n_t$

		$m_t$
No. of transactions	Test	1	5	10	20	40	80	100	200	500
$n_t=50$	$V_{\left\{1\right\}}^{\left(t\right)}$	0.053	0.027	0.018	0.014	0.011	—	—	—	—
	TS	0.024	0.003	0.001	0.000	0.000	—	—	—	—
$n_t=100$	$V_{\left\{1\right\}}^{\left(t\right)}$	0.071	0.045	0.027	0.016	0.012	0.011	0.011	—	—
	TS	0.049	0.013	0.004	0.001	0.000	0.000	0.000	—	—
$n_t=200$	$V_{\left\{1\right\}}^{\left(t\right)}$	0.094	0.069	0.047	0.026	0.016	0.012	0.011	0.010	—
	TS	0.070	0.035	0.013	0.003	0.001	0.000	0.000	0.000	—
$n_t=500$	$V_{\left\{1\right\}}^{\left(t\right)}$	0.132	0.126	0.097	0.062	0.031	0.017	0.016	0.012	0.010
	TS	0.103	0.084	0.049	0.017	0.003	0.000	0.000	0.000	0.000

Model 7 holds with ${\tau }_t=0$ for each trader. The nominal test size is $\alpha =0.01$.