Bayes and EB benchmarkig for SAE. Dissertation 2012.pdf

Table 4 1 contains the direct hierarchical bayes and

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Table 4-1 contains the direct, hierarchical Bayes, and benchmarked Bayes esti- mates and associated posterior RMSEs (PRMSE) for each of the small domains. It also contains the percent increase in the PRMSE in the benchmarked Bayes estimates com- pared to the HB estimates. The direction of adjustment depends on the sign of p ¯ ˆ θ B w , and the amount of adjustment depends on the relative magnitudes of η i s i (1 + ζ i s i ) 1 . For the given dataset, the adjustments are always positive since p > ¯ ˆ θ B w . Also, with the present choice of weights, the percent increase in PRMSE over the HB estimators is quite small, which somewhat justifies the choice of the given HB model. Moreover, the amount of adjustment is typically more for domains with smaller sample sizes as compared to those with larger samples as we would like to see. Consider, for example, domain 12 with a sample size of 11 and se(HB) equal to 0.056. Since the constraint weight ζ i is inversely proportional to the posterior variance, we expect to see a larger adjustment in this domain, which is precisely what occurs. We also note that this domain also shows the largest percent increase in RPMSE, which is 1.616. Similar expected behavior occurs for domain 79 with sample size 122, which ends up having the smallest overall adjustment. 51

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Table 4-1. Table of estimates using Theorem 1 Domain n i Direct HB se(HB) e i PRMSE % inc* 1 10 0.126 0.135 0.04889 0.141 0.04927 0.784 2 0 - - - - - - 3 24 0.063 0.088 0.03198 0.091 0.03213 0.467 4 28 0.146 0.150 0.04471 0.158 0.04538 1.514 5 20 0.138 0.125 0.04138 0.132 0.04193 1.320 6 17 0.112 0.130 0.04290 0.135 0.04318 0.662 7 78 0.097 0.108 0.02746 0.110 0.02757 0.396 8 66 0.274 0.214 0.04228 0.219 0.04265 0.880 9 5 0.173 0.127 0.04820 0.132 0.04846 0.542 10 6 0 0.143 0.05422 0.150 0.05471 0.908 11 7 0 0.140 0.05159 0.147 0.05208 0.941 12 11 0.335 0.164 0.05630 0.175 0.05721 1.616 13 7 0.133 0.119 0.04613 0.126 0.04657 0.969 14 2 0 0.097 0.04217 0.100 0.04229 0.288 15 27 0 0.075 0.02837 0.077 0.02846 0.315 16 29 0.113 0.129 0.03963 0.134 0.03992 0.723 17 27 0.12 0.125 0.03925 0.130 0.03955 0.757 18 14 0 0.093 0.03571 0.097 0.03589 0.492 19 77 0.131 0.125 0.02931 0.128 0.02943 0.426 20 75 0.223 0.188 0.03762 0.192 0.03788 0.697 21 3 0 0.100 0.04185 0.104 0.04204 0.449 22 6 0 0.115 0.04532 0.119 0.04554 0.489 23 8 0 0.153 0.05529 0.160 0.05579 0.902 24 9 0 0.119 0.04514 0.124 0.04542 0.625 25 10 0 0.094 0.03777 0.098 0.03799 0.576 26 6 0 0.095 0.04043 0.098 0.04056 0.329 27 32 0.098 0.114 0.03617 0.118 0.03636 0.547 28 23 0 0.083 0.03176 0.086 0.03193 0.531 29 25 0.187 0.122 0.04021 0.127 0.04050 0.711 30 23 0.226 0.145 0.04482 0.150 0.04509 0.602 31 71 0.118 0.118 0.02996 0.121 0.03004 0.282 32 50 0.109 0.109 0.03134 0.112 0.03152 0.567 33 2 0 0.115 0.04834 0.119 0.04856 0.439 34 2 0 0.119 0.04902 0.123 0.04911 0.189 35 8 0.108 0.117 0.04439 0.122 0.04471 0.720 36 7 0 0.125 0.04829 0.130 0.04851 0.456 37 9 0.062 0.098 0.03949 0.103 0.03987 0.966 38 17 0 0.073 0.02929 0.076 0.02937 0.279 39 24 0.117 0.136 0.04386 0.142 0.04423 0.836 40 20 0 0.096 0.03532 0.100 0.03560 0.816 41 50 0.163 0.156 0.03848 0.160 0.03875 0.703 * in PRMSE 52
Table 4-1. Continued Domain n i Direct HB se(HB) e i PRMSE % inc* 42 38 0.141 0.122 0.03592 0.126 0.03612 0.558 43 76 0.104 0.114 0.02822 0.116 0.02829 0.251 44 73 0.142 0.121 0.02961 0.124 0.02975 0.468 45 2 0 0.129 0.05255 0.135 0.05284 0.546 46 3 0 0.096 0.04157 0.099 0.04171 0.345 47 10 0 0.098 0.03858 0.101 0.03872 0.371 48 7 0 0.125 0.04711 0.130 0.04739 0.596 49 10 0.087 0.116 0.04245 0.121 0.04275 0.692 50 5 0 0.126 0.05070 0.131 0.05096 0.510 51 23 0.038 0.089 0.03253 0.092 0.03264 0.355 52 21 0.243 0.143 0.04584 0.149 0.04625 0.907 53 31 0.114 0.125 0.03789 0.129 0.03814 0.671 54 18 0.202 0.158 0.04941 0.164 0.04971 0.601 55 74 0.094 0.094 0.02554 0.096 0.02562 0.312 56 83 0.204 0.186 0.03699 0.191 0.03729 0.814 57 2 0 0.140 0.05667 0.145 0.05688 0.363 58 1 0 0.093 0.04170 0.096 0.04177 0.163 59 2 0 0.094 0.04115 0.097 0.04125 0.232 60 8 0.112 0.132 0.04798 0.138 0.04831 0.689 61 16 0.202 0.156 0.05206 0.163 0.05260 1.040 62 3 0.301 0.158 0.06362 0.165 0.06395 0.516 63 33 0.055 0.093 0.03118 0.096 0.03131 0.421 64 28 0.105 0.118 0.03743 0.122 0.03764 0.560 65 33 0.126 0.133 0.03899 0.138 0.03928 0.731 66 13 0.393 0.207 0.06393 0.216 0.06460 1.038 67 70 0.079 0.096 0.02635 0.098 0.02642 0.283 68 75 0.179 0.159 0.03425 0.163 0.03444 0.537 69 1 1 0.146 0.06075 0.149 0.06081 0.107 70 2 0.361 0.157 0.05946 0.162 0.05968 0.372 71 4 0 0.104 0.04291 0.108 0.04307 0.384 72 2 0 0.226 0.08082 0.235 0.08134 0.643 73 45 0.271 0.208 0.04578 0.215 0.04632 1.189 74 10 0 0.096 0.03854 0.100 0.03873 0.485 75 83 0.149 0.134 0.03002 0.137 0.03017 0.489 76 59 0.113 0.126 0.03247 0.129 0.03263 0.505 77 68 0.338 0.270 0.04638 0.276 0.04685 1.024 78 39 0.098 0.107 0.03331 0.111 0.03346 0.452 79 122 0.11 0.106 0.02348 0.108 0.02355 0.285 80 125 0.308 0.293 0.03805 0.298 0.0383 0.651 81 7 0 0.122 0.04681 0.126 0.04704 0.484 82 12 0 0.097 0.03752 0.101 0.03771 0.498 83 13 0.049 0.122 0.04428 0.128 0.04458 0.678 * in PRMSE 53

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Table 4-1. Continued Domain n i Direct HB se(HB) e i PRMSE % inc* 84 4 0 0.126 0.04915 0.131 0.04940 0.506 85 32 0.189 0.172 0.04641 0.178 0.04674 0.699 86 10 0.135 0.132 0.04844 0.137 0.04878 0.700 87 52 0.192 0.157 0.03765 0.162 0.03800 0.925 88 65 0.153 0.140 0.03346 0.143 0.03366 0.584 89 71 0.285 0.226 0.04132 0.231 0.04164 0.763 90 57 0.086 0.103 0.02965 0.106 0.02979 0.446 91 153 0.149 0.135 0.02387 0.136 0.02394 0.297 92 138 0.308 0.277 0.03542 0.281 0.03563 0.593 93 10 0 0.121 0.04513 0.126 0.04547 0.761 94 16 0.067 0.107 0.03885 0.110 0.03901 0.404 95 18 0.108 0.144 0.04668 0.150 0.04707 0.816 96 14 0.111 0.138 0.04731 0.143 0.04759 0.595 * in PRMSE 54
CHAPTER 5
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• Spring '16
• Yessi
• The Land, Estimation theory, Mean squared error, Bayes estimator, Empirical Bayes method

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