Figure 1

The cumulative distribution function, $\mathcal{N}(\delta )$. Each set of three lines corresponds to, from left to right, $x=0.00$, 0.04, and 0.08. The leftmost set of three lines is for $L=10$, and the rightmost set is for $L=40$. The three-parameter fit, Eq. (5) is plotted in black for each case and is indistinguishable from the ensemble data (blue).Reuse & Permissions

Figure 2

The parameters $a$, $b$, and $c$ of Eq. (5). On the left panels, the parameters are plotted vs $x$ for fixed values of $L$. On the right panels, we have plots of the parameters vs $L$ for fixed $x$. From data like this, we extracted $a_L(x),b_L(x)$, and $c_L(x)$.Reuse & Permissions

Figure 3

The effect of depletion on $p(\delta )$ is shown here, where we have plotted the ensemble average of $p(\delta )$. The set of three lines on the left is for $\delta ={\delta }_3^i(10)$, while $\delta ={\delta }_3^i(40)$ is on the right. Each set of three lines corresponds to $x=0.00$, 0.04, and 0.08. The parameterized $p(\delta ,x)$, Eq. (7), is plotted in black and is barely distinguishable from the ensemble average in blue.Reuse & Permissions

Figure 4

The results of the MLM calculation for 10 randomly chosen depleted members of the GOE with $x=0.00$, 0.04, and 0.08. The left-hand side has $L=20$, and the right-hand side has $L=40$.Reuse & Permissions

Figure 5

The results of the MLM calculation for randomly chosen spectra with $x=0.04$ and 0.12.Reuse & Permissions

Figure 6

The results of the MLM calculation for $x=0.02$, 0.04, $\dots$, 0.18 and 0.20, for all $L$. ${\sigma }_x$ was used for error bars. We used this graph as a guide for using ${\sigma }_x=0.006$ for $20⩽L⩽40$ for all $x$.Reuse & Permissions

Figure 7

The mean value of $\sigma$ averaged over $L$, top panel, and $x$, bottom panel. The dashed line in the upper panel is the result for the analysis based on Eq. (4).Reuse & Permissions

Figure 8

A comparison between ${\Delta }_3(L)$ vs $L$ for the Bohigas expression and the ensemble averages.Reuse & Permissions

Figure 9

The test results for using the Bohigas expression to extract $x$.Reuse & Permissions

Figure 10

The results of the MLM calculation for neutron resonance data. In panel (a) we have ${}^{158}\text{Gd}$. The results for the full set (solid line) are between those for the subset containing just the lowest 60 levels (lower dashed line) and the subset containing the top 32 levels (upper line). In panel (b), we have ${}^{58}\text{Ni}$. Nothing can be concluded from this about $x$. In panel (c), we see the full spectrum of ${}^{152}\text{Sm}$ (solid line) above the lowest 70 levels (lower dashed line. In panel (d), we have the full ${}^{234}\text{U}$ data set (solid line), looking more incomplete that the first 75 levels alone (middle dashed line), while the levels 76 to 118 (lower dashed line) look like a complete subset. In panel (e), we have the full ${}^{236}\text{U}$ set of 81 levels looking more incomplete than the lowest 69 levels. Finally, in panel (f), we see the ${}^{235}\text{U}$ data. The $j=3$ subset is the solid line, and the dashed line is the $j=4$ subset. Note the range on the vertical axis.Reuse & Permissions

Figure 11

The results of the MLM calculation for acoustic data for sets 1, 2, and 4. The other sets had $x>0.3$. The mean values for the interval $20⩽L⩽40$ are 0.20, 0.22, and 0.23 respectively.Reuse & Permissions

Figure 12

Lower panel $p(\delta )$ for spectra with $x=0.08$ depletion (left peak) and $y=0.08$ intruders. They are quite close together, and it is hard to distinguish the effect of missed levels from intruders. The ensemble average data and the fits are superimposed and indistinguishable. The upper panel shows the parameters $a,\hspace{0.5em}b,$ and $c$ for in the fit Eq. (7). The solid lines are for intruders and the dashed lines are for missed levels.Reuse & Permissions

Figure 13

The ${\Delta }_3(L)$ and $\mathcal{N}(\delta )$ for shell model spectra with $J^{+}T=2^{+}1$ (dashed line) and $J^{+}T=2^{+}0$ (dotted line) compared to the GOE result (solid line) for pure GOE spectra.Reuse & Permissions