tokens: We calculated a simple reduced-chi-square statistic for each DM model, finding for the YMW16 model and ∼10.2 for the NE2001 model. (The χ 2 statistic accounted only for uncertainties in the parallax distances, not in the electron-density models.) The similarity in these χ 2 values suggests the two models are comparable in their ability to predict distances to pulsars based on DM values, at least in the distance regime probed by millisecond pulsar timing parallax measurements (within ∼2.5 kpc of the Sun). Figure 4. Distances from timing parallax measurements (67% confidence) vs. distances from DM models for pulsars with significant timing parallax detections. The blue points show the NE2001 DM model distances, while the red points show the YMW16 model distances. The black dashed line shows a one-to-one relation. 4.2. Parallax Upper Limits The remaining 25 pulsars had timing parallax measurements with significances of less than 3 σ . Of these, 22 had positive parallaxes and only 3 had negative parallaxes in the formal timing fits. This is strong evidence that most of the non-detections have a true low-level parallax signal; otherwise, there would be comparable numbers of negative and positive parallax measurements. None of the negative parallaxes had large significance. We used 95%-confidence upper limits on parallax to compute 95%-confidence lower limits on distance for these pulsars. The details of this calculation are outlined in M16 . We have not attempted to correct for Lutz–Kelker bias (Verbiest et al. 2010 ), as we do not believe we have enough prior information on the spatial and luminosity distribution of the millisecond pulsar population to accurately correct for this bias (see M16 ). For the lower limits on distance, our 95% limit remains a conservative approach, in the sense that compensation for Lutz–Kelker bias would only push these limits higher. Among the pulsars with parallax upper limits, a few have significant discrepancies with previous measurements (PSRs J0636+5128, J1455−3330, J1640+1224, B1937+21). As with the pulsars discussed in Section 4.1 , the source of these discrepancies are not clear, but may result from differences in DM variation models or red-noise models. 4.3. Distance Constraints from Rotation and Orbital Period Derivatives Observed pulsar rotation period derivatives are a combination of intrinsic pulsar spin-down and kinematic terms due to the acceleration and transverse motion of the pulsar relative to the Sun (Shklovskii 1970 ; Nice Taylor 1995 ). By assuming that the pulsar is losing rotational energy, so that it has a positive rotation period derivative, an upper limit can be placed on the pulsar distance. See M16 Section 4.3 for details. Table 5 lists such “ distance” constraints, which we calculated following the procedures of M16 . In the table, we omitted distance limits greater than 10 kpc as physically uninteresting, and we did not calculate a constraint for PSR J1024−0719, as its observed rotation
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