Nonparametric Estimation of Conditional Medians for Linear and Related Processes
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(2) Discussion Paper #2005-04. Nonparametric Estimation of Conditional Medians for Linear and Related Processes (The former title is Least Absolute Deviation Regression for Long-range Dependent Processes). Toshio HONDA. September 2005 October 2007(revised).
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(5) TOSHIO H ONDA.
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(227) Table 1: Parameters of tables Table 2. 3. 4. 5. 6. 7. 8. 9. γx. 2.5. 0.5. 2.5. 0.5. 2.5. 0.5. 2.5. 0.5. u(x). x2+x4. x2+x4. cos(x). cos(x). x2+x4. x2+x4. cos(x). cos(x). dist.. N(0,1). N(0,1). N(0,1). N(0,1). t3. t3. t3. t3.
(228) Table 2:γx=2.5, N(0,1), u(x)=x2+x4 0.5. γz h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 0.016. 0.026. 0.039. 0.013. 0.022. 0.033. 0.003. 0.017. 0.031. bias. 0.016. 0.026. 0.039. 0.013. 0.022. 0.033. 0.003. 0.017. 0.031. var. 0.271. 0.258. 0.248. 0.088. 0.070. 0.062. 0.074. 0.052. 0.043. mse. 0.271. 0.258. 0.249. 0.088. 0.071. 0.063. 0.074. 0.053. 0.044. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.347. 0.370. 0.403. 0.332. 0.361. 0.394. 0.325. 0.347. 0.383. bias. 0.035. 0.058. 0.091. 0.020. 0.049. 0.082. 0.013. 0.035. 0.071. var. 0.282. 0.263. 0.255. 0.103. 0.081. 0.069. 0.080. 0.057. 0.047. mse. 0.284. 0.266. 0.263. 0.103. 0.083. 0.076. 0.080. 0.058. 0.052. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 2.048. 2.127. 2.220. 2.044. 2.119. 2.208. 2.051. 2.124. 2.209. bias. 0.048. 0.127. 0.220. 0.044. 0.119. 0.208. 0.051. 0.124. 0.209. var. 0.321. 0.283. 0.272. 0.148. 0.109. 0.095. 0.121. 0.085. 0.068. mse. 0.323. 0.299. 0.320. 0.150. 0.123. 0.138. 0.124. 0.100. 0.112. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0.
(229) Table 3:γx=0.5, N(0,1), u(x)=x2+x4 0.5. γz h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 0.022. 0.028. 0.041. 0.014. 0.020. 0.031. 0.000. 0.009. 0.026. bias. 0.022. 0.028. 0.041. 0.014. 0.020. 0.031. 0.000. 0.009. 0.026. var. 0.295. 0.274. 0.266. 0.106. 0.085. 0.073. 0.077. 0.056. 0.047. mse. 0.295. 0.275. 0.267. 0.106. 0.085. 0.074. 0.077. 0.056. 0.048. NA4. 2. 1. 0. 0. 0. 0. 1. 0. 0. NA. 1. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.341. 0.364. 0.395. 0.332. 0.358. 0.389. 0.325. 0.348. 0.380. bias. 0.029. 0.052. 0.083. 0.020. 0.046. 0.077. 0.013. 0.036. 0.068. var. 0.298. 0.284. 0.279. 0.131. 0.103. 0.088. 0.099. 0.072. 0.055. mse. 0.299. 0.286. 0.285. 0.132. 0.105. 0.094. 0.099. 0.073. 0.060. NA4. 4. 1. 0. 9. 2. 1. 4. 1. 0. NA. 1. 0. 0. 0. 0. 0. 0. 0. 0. mean. 2.058. 2.114. 2.199. 2.041. 2.111. 2.196. 2.036. 2.103. 2.198. bias. 0.058. 0.114. 0.199. 0.041. 0.111. 0.196. 0.036. 0.103. 0.198. var. 0.415. 0.367. 0.356. 0.245. 0.198. 0.172. 0.213. 0.163. 0.171. mse. 0.418. 0.380. 0.396. 0.247. 0.210. 0.210. 0.215. 0.173. 0.210. NA4. 94. 41. 17. 87. 37. 18. 89. 43. 23. NA. 8. 3. 3. 10. 5. 2. 10. 3. 0.
(230) Table 4:γx=2.5, N(0,1), u(x)=cos(x) 0.5. γz h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 1.005. 0.999. 0.991. 1.002. 0.995. 0.986. 0.992. 0.990. 0.984. bias. 0.005. -0.001. -0.009. 0.002. -0.005. -0.014. -0.008. -0.010. -0.016. var. 0.272. 0.258. 0.248. 0.089. 0.071. 0.063. 0.074. 0.052. 0.043. mse. 0.272. 0.258. 0.248. 0.089. 0.071. 0.063. 0.074. 0.052. 0.043. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.889. 0.883. 0.876. 0.874. 0.873. 0.867. 0.867. 0.861. 0.855. bias. 0.011. 0.005. -0.002. -0.004. -0.005. -0.011. -0.011. -0.017. -0.023. var. 0.283. 0.263. 0.253. 0.103. 0.080. 0.068. 0.080. 0.056. 0.046. mse. 0.283. 0.263. 0.253. 0.103. 0.080. 0.068. 0.080. 0.057. 0.047. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.533. 0.541. 0.544. 0.528. 0.533. 0.527. 0.535. 0.540. 0.534. bias. -0.007. 0.001. 0.004. -0.012. -0.007. -0.013. -0.005. 0.000. -0.006. var. 0.320. 0.281. 0.271. 0.146. 0.109. 0.091. 0.122. 0.084. 0.064. mse. 0.320. 0.281. 0.271. 0.146. 0.109. 0.091. 0.122. 0.084. 0.065. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0.
(231) Table 5:γx=0.5, N(0,1), u(x)=cos(x) 0.5. γz h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 1.011. 1.001. 0.991. 1.002. 0.993. 0.982. 0.988. 0.982. 0.979. bias. 0.011. 0.001. -0.009. 0.002. -0.007. -0.018. -0.012. -0.018. -0.021. var. 0.295. 0.275. 0.265. 0.106. 0.085. 0.072. 0.078. 0.056. 0.047. mse. 0.295. 0.275. 0.265. 0.106. 0.085. 0.073. 0.078. 0.056. 0.047. NA4. 2. 1. 0. 0. 0. 0. 1. 0. 0. NA. 1. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.884. 0.878. 0.872. 0.876. 0.871. 0.864. 0.868. 0.862. 0.857. bias. 0.006. 0.000. -0.006. -0.002. -0.007. -0.014. -0.010. -0.016. -0.021. var. 0.297. 0.282. 0.279. 0.131. 0.103. 0.089. 0.099. 0.072. 0.053. mse. 0.297. 0.282. 0.279. 0.131. 0.104. 0.089. 0.099. 0.072. 0.054. NA4. 4. 1. 0. 9. 2. 1. 4. 1. 0. NA. 1. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.551. 0.545. 0.538. 0.532. 0.539. 0.531. 0.530. 0.534. 0.536. bias. 0.011. 0.005. -0.002. -0.008. -0.001. -0.009. -0.010. -0.006. -0.004. var. 0.410. 0.358. 0.333. 0.238. 0.179. 0.146. 0.197. 0.143. 0.150. mse. 0.410. 0.358. 0.333. 0.238. 0.179. 0.146. 0.197. 0.143. 0.150. NA4. 94. 41. 17. 87. 37. 18. 89. 43. 23. NA. 8. 3. 3. 10. 5. 2. 10. 3. 0.
(232) Table 6:γx=2.5, t3, u(x)= x2+x4. γz. 0.5. h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. -0.002. 0.005. 0.019. 0.000. 0.012. 0.026. 0.004. 0.016. 0.029. bias. -0.002. 0.005. 0.019. 0.000. 0.012. 0.026. 0.004. 0.016. 0.029. var. 0.224. 0.213. 0.208. 0.049. 0.041. 0.037. 0.027. 0.020. 0.017. mse. 0.224. 0.213. 0.209. 0.049. 0.041. 0.038. 0.027. 0.020. 0.018. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.321. 0.345. 0.377. 0.334. 0.357. 0.387. 0.334. 0.358. 0.388. bias. 0.009. 0.033. 0.065. 0.022. 0.045. 0.075. 0.022. 0.046. 0.076. var. 0.241. 0.230. 0.223. 0.060. 0.048. 0.042. 0.038. 0.027. 0.023. mse. 0.242. 0.231. 0.228. 0.060. 0.050. 0.048. 0.038. 0.029. 0.028. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 2.033. 2.098. 2.179. 2.057. 2.112. 2.189. 2.042. 2.105. 2.186. bias. 0.033. 0.098. 0.179. 0.057. 0.112. 0.189. 0.042. 0.105. 0.186. var. 0.313. 0.270. 0.261. 0.139. 0.091. 0.081. 0.117. 0.070. 0.057. mse. 0.314. 0.279. 0.293. 0.143. 0.104. 0.116. 0.118. 0.081. 0.092. NA4. 2. 0. 0. 1. 0. 0. 2. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0.
(233) Table 7:γx=0.5, t3, u(x)= x2+x4 0.5. γz h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. -0.004. 0.006. 0.017. 0.016. 0.023. 0.038. 0.014. 0.020. 0.033. bias. -0.004. 0.006. 0.017. 0.016. 0.023. 0.038. 0.014. 0.020. 0.033. var. 0.245. 0.230. 0.262. 0.065. 0.053. 0.047. 0.037. 0.026. 0.022. mse. 0.245. 0.230. 0.263. 0.066. 0.053. 0.048. 0.037. 0.026. 0.023. NA4. 4. 1. 0. 2. 1. 0. 1. 0. 0. NA. 1. 1. 0. 0. 0. 0. 0. 0. 0. mean. 0.324. 0.349. 0.377. 0.325. 0.349. 0.381. 0.327. 0.352. 0.384. bias. 0.012. 0.037. 0.065. 0.013. 0.037. 0.069. 0.015. 0.040. 0.072. var. 0.293. 0.269. 0.254. 0.088. 0.066. 0.067. 0.075. 0.046. 0.038. mse. 0.293. 0.270. 0.258. 0.088. 0.068. 0.072. 0.075. 0.048. 0.043. NA4. 16. 8. 3. 20. 14. 5. 26. 11. 7. NA. 4. 2. 2. 0. 0. 0. 1. 0. 0. mean. 2.029. 2.090. 2.162. 2.040. 2.092. 2.174. 2.051. 2.100. 2.174. bias. 0.029. 0.090. 0.162. 0.040. 0.092. 0.174. 0.051. 0.100. 0.174. var. 0.418. 0.415. 0.381. 0.231. 0.195. 0.240. 0.176. 0.171. 0.120. mse. 0.418. 0.423. 0.408. 0.233. 0.203. 0.271. 0.179. 0.181. 0.151. NA4. 193. 89. 53. 185. 84. 52. 184. 87. 58. NA. 27. 16. 12. 30. 13. 5. 28. 16. 6.
(234) Table 8:γx=2.5, t3, u(x)=cos(x). γz. 0.5. h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 0.986. 0.979. 0.972. 0.988. 0.986. 0.979. 0.992. 0.990. 0.983. bias. -0.014. -0.021. -0.028. -0.012. -0.014. -0.021. -0.008. -0.010. -0.017. var. 0.224. 0.213. 0.208. 0.049. 0.041. 0.037. 0.027. 0.020. 0.017. mse. 0.224. 0.214. 0.209. 0.049. 0.041. 0.038. 0.027. 0.020. 0.017. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.863. 0.860. 0.853. 0.876. 0.871. 0.865. 0.877. 0.873. 0.866. bias. -0.015. -0.018. -0.025. -0.002. -0.007. -0.013. -0.001. -0.005. -0.012. var. 0.241. 0.229. 0.222. 0.060. 0.047. 0.042. 0.038. 0.027. 0.022. mse. 0.241. 0.229. 0.222. 0.060. 0.048. 0.042. 0.038. 0.027. 0.022. NA4. 0. 0. 0. 0. 0. 0. 0. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0. mean. 0.522. 0.520. 0.521. 0.543. 0.535. 0.531. 0.532. 0.530. 0.533. bias. -0.018. -0.020. -0.019. 0.003. -0.005. -0.009. -0.008. -0.010. -0.007. var. 0.313. 0.269. 0.256. 0.139. 0.086. 0.072. 0.115. 0.066. 0.050. mse. 0.313. 0.270. 0.257. 0.139. 0.086. 0.072. 0.115. 0.066. 0.050. NA4. 2. 0. 0. 1. 0. 0. 2. 0. 0. NA. 0. 0. 0. 0. 0. 0. 0. 0. 0.
(235) Table 9:γx=0.5, t3, u(x)=cos(x). γz. 0.5. h. 0.0. 0.5. 1.0. 1.5. 2.5. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. 0.2. 0.3. 0.4. mean. 0.985. 0.980. 0.969. 1.004. 0.997. 0.991. 1.002. 0.993. 0.986. bias. -0.015. -0.020. -0.031. 0.004. -0.003. -0.009. 0.002. -0.007. -0.014. var. 0.245. 0.231. 0.260. 0.065. 0.053. 0.047. 0.037. 0.026. 0.022. mse. 0.245. 0.231. 0.261. 0.065. 0.053. 0.047. 0.037. 0.026. 0.022. NA4. 4. 1. 0. 2. 1. 0. 1. 0. 0. NA. 1. 1. 0. 0. 0. 0. 0. 0. 0. mean. 0.867. 0.865. 0.856. 0.868. 0.867. 0.862. 0.870. 0.869. 0.863. bias. -0.011. -0.013. -0.022. -0.010. -0.011. -0.016. -0.008. -0.009. -0.015. var. 0.295. 0.269. 0.254. 0.087. 0.064. 0.065. 0.074. 0.045. 0.037. mse. 0.295. 0.269. 0.254. 0.087. 0.064. 0.065. 0.074. 0.045. 0.037. NA4. 16. 8. 3. 20. 14. 5. 26. 11. 7. NA. 4. 2. 2. 0. 0. 0. 1. 0. 0. mean. 0.536. 0.531. 0.525. 0.547. 0.536. 0.544. 0.562. 0.549. 0.552. bias. -0.004. -0.009. -0.015. 0.007. -0.004. 0.004. 0.022. 0.009. 0.012. var. 0.389. 0.379. 0.340. 0.225. 0.171. 0.201. 0.153. 0.141. 0.085. mse. 0.389. 0.379. 0.340. 0.225. 0.171. 0.201. 0.153. 0.141. 0.086. NA4. 193. 89. 53. 185. 84. 52. 184. 87. 58. NA. 27. 16. 12. 30. 13. 5. 28. 16. 6.
(236)
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