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Asymptotics 1 Example Xi X V arX ¢ ¡ iid ¹; ¾2 ; i = 1; 2; ::; n 1X = Xi n i 1X 1X 1X ) EX = E Xi = EXi = ¹=¹ n i n i n i " # 1X 1 X 2 ¾2 1 X : = V ar Xi = 2 V arXi = 2 ¾ = n i n i n i n » V arX X 2 ! 0 as n ! 1 ! ¹ as n ! 1: Concepts Probability Limits: Let Sn be a statistic whose properties depend on n. Then (weak consistency) plimSn = µ i¤ lim Pr [jSn ¡ µj < "] = 1 8" > 0: n!1 We say that Sn is a weakly consistent estimator of µ. Example (continued): plimX n = ¹? ¯ ¤ £ ¤ £¯ Pr ¯X n ¡ ¹¯ < " = Pr ¡" < X n ¡ ¹ < " " p ¢ p ¡ p # n Xn ¡ ¹ ¡ n" n" < < = Pr ¾ ¾ ¾ · p p ¸ ¡ n" n" = Pr <Z < ¾ ¾ p where Z » N (0; 1). As n ! 1, ¾n" ! 1 for all …xed " ) · p p ¸ ¡ n" n" <Z < = 1 lim Pr n!1 ¾ ¾ ) plimX n = ¹: 1 Alternatively, we say that, if i h Pr lim jSn ¡ µj < " = 1; n!1 then Sn is a strongly consistent estimator of µ. Example (continued): ¯ ¯ ¯ ¯ ¯ ¾ ¡ ¹ ¯¯ ¯X n ¡ ¹¯ = p¾ ¯ X n p = p jZj ¯ ¯ n ¾= n n where Z » N (0; 1) ) ¾ lim p jZj n!1 n = 0 i h ¯ ¯ ) Pr lim ¯X n ¡ ¹¯ < " = 1 8" > 0: n!1 Thus, X n is a strongly consistent estimator of ¹. New example: ½ X n with probability 1 ¡ Sn = n with probability n1 Note that Pr [jSn ¡ ¹j < "] = ( h Pr jZj < 0 p i n" ¾ 1 n : with probability 1 ¡ with probability n1 1 n and that, as n ! 1, n1 ! 0 and the second term disappears. ) Sn is a weakly consistent estimator of ¹. However, lim jSn ¡ ¹j n!1 does not exist. So Sn is not a strongly consistent estimator of ¹. Explain why this example is relevant. 3 Properties plim c = c plim cXn = cplim Xn plim (Xn + Yn ) = plimXn + plimYn plim (Xn Yn ) = (plimXn ) (plimYn ) plim g (X1n ; X2n ; ::; Xmn ) = g (plimX1n ; plimX2n ; ::; plimXmn ) Compare consistency and unbiasedness: If EXn ! ¹ and V arXn ! 0, then plim Xn = ¹. The converse is not true. Examples: 2 1. ¢ ¡ » iid ¹; ¾2 m 1 X = Xi m i=1 Xi Sn for …xed m. Then ESn V arSn = ¹; ¾2 = 9 0 as n ! 1 m ) plim Sn does not exist. 2. Xi Sn ¢ ¡ » iid ¹; ¾2 " m #¡1 1X = Xi : n i=1 Then plimSn But £ ¤¡1 plim X n " m 1X = plim Xi n i=1 = #¡1 1 : ¹ ¢¡1 ¡ ¢¡1 ¡ ESn = E X n 6= EX n and, in fact, for many cases does not exist. 4 Central Limit Theorem Let Then ¢ ¡ Xi » iid ¹; ¾ 2 ; i = 1; 2; ::; n: ¢ p ¡ n X ¡ ¹ =¾ » N (0; 1) for a large class of distributions for Xi . The CLT generalizes in many ways, the most important being to allow for heterogeneity in Xi . 3 One more example: let U » Â2m , V » Â2n with U; V independent. Then Z= U=m » Fm;n : V=n What happens as n ! 1? De…ne V = n X Wi i=1 where Wi » iidÂ21 ) EWi = 1; V arWi = 2 V =1 ) plim n U U=m ! ) Z= as n ! 1 V=n m ) mZ ! U » Â2m : 4