Estimation and Inference of Discontinuity in Density
Estimation and Inference of Discontinuity in Density
Estimation and Inference of Discontinuity in Density
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Actual SizeSize Distortion0.35(a)0.06(b)0.30.050.250.040.20.030.150.10.05W G00 0.05 0.1 0.15 0.2 0.25Nom<strong>in</strong>al Sizel Gl45 o L<strong>in</strong>e0.020.0100.010 0.05 0.1 0.15 0.2 0.25Nom<strong>in</strong>al SizeFigure 1: (a) P-value plots <strong>and</strong> (b) P-value discrepancy plots (Davidson <strong>and</strong> MacK<strong>in</strong>non, 1998) forW G ; `G <strong>and</strong> ` tests when n = 2000:Table 4: <strong>Estimation</strong> <strong>and</strong> test<strong>in</strong>g <strong>of</strong> the discont<strong>in</strong>uity <strong>of</strong> the density <strong>of</strong> enrollments at multiples <strong>of</strong> 40(accord<strong>in</strong>g to Maimonides’s rule, Angrist <strong>and</strong> Lavy, 1999) for …fth graders. The b<strong>in</strong>n<strong>in</strong>g <strong>and</strong> locallikelihood methods are used with various smooth<strong>in</strong>g b<strong>and</strong>widths.B<strong>in</strong>n<strong>in</strong>g methodLocal likelihood methodc h b fl b fr b GWald W G EL `G b fl b fr b EL `40 15 :0056 :0080 :0024 2:666 0:444 :0039 :0114 :0075 20:6720 :0052 :0089 :0037 7:888 1:331 :0040 :0114 :0074 26:9425 :0050 :0094 :0044 13:57 2:321 :0040 :0114 :0074 33:2830 :0054 :0099 :0045 16:23 2:856 :0045 :0116 :0072 36:3680 15 :0115 :0095 :0019 1:141 1:014 :0081 :0140 :0059 8:30120 :0112 :0089 :0024 2:389 1:964 :0085 :0116 :0030 3:36625 :0108 :0087 :0021 2:395 1:534 :0087 :0107 :0021 2:09230 :0105 :0089 :0016 1:698 1:001 :0088 :0107 :0020 2:350120 15 :0092 :0050 :0043 7:920 3:906 :0064 :0078 :0014 0:57720 :0088 :0048 :0039 9:355 3:977 :0066 :0070 :0003 0:04525 :0075 :0047 :0029 7:005 2:403 :0060 :0063 :0003 0:06930 :0066 :0045 :0021 4:988 1:263 :0055 :0060 :0005 0:178160 15 :0027 :0010 :0017 4:670 3:253 :0017 :0013 :0003 0:14220 :0025 :0010 :0015 5:176 2:397 :0018 :0012 :0006 0:73025 :0023 :0010 :0012 4:550 1:710 :0017 :0013 :0005 0:58630 :0021 :0011 :0011 4:316 1:456 :0017 :0013 :0004 0:47326
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