Advanced Econometrics Takeshi Amemiya
Asymptotic Normality of the Median
Let {Yt), t = 1, 2,. . . , T, be a sequence of i. i.d. random variables with common distribution function Fand density function f. The population median M is defined by
F(Af) = ^. (4.6.1)
We assume F to be such that M is uniquely determined by (4.6.1), which follows from assuming/(у) > 0 in the neighborhood of у = M. We also assume that f'(y) exists for у > M in a neighborhood of M. Define the binary random variable W'/a) by
Wia) = 1 if y, isa
= 0 if У,<а
for eveiy real number a. Using (4.6.2), we define the sample median ттЪу
(4.6.3)
The median as defined above is clearly unique.11
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The asymptotic normality of mT can be proved in the following manner: Using (4.6.3), we have for any у
Define
P,= l-P(Yt<M+ T~l/2y).
Then, because by a Taylor expansion
P, = ~ T~ipf(M)y - СЦТ-'), (4.6.5)
we have
^(І = Р&т + (KT-W) %f(M)y, (4.6.6)
where Wf = Wt(M + T~ 1/2y) and ZT = Г_1/222_,(1У* — P,). We now derive the limit distribution of ZTusing the characteristic function (Definition 3.3.1). We have
T
E exp (/AZr) = П Eexp [ikT~i/2(W* — Pt)] (4.6.7)
r-1
= П (p‘ exp [ІЇТ-'КЦ - P')]
t-l
+ (1 - P,)exp {-iXT-wp,))
-* exp (—A2/8),
where the third equality above is based on (4.6.5) and the expansion of the exponent: ex= 1 + x + 2~lx2 + . . . . Therefore ZT-*N(0, 4_I), which implies by Theorem 3.2.7
ZT+O(T-l'2)^N(0,4-'). (4.6.8)
Finally, from (4.6.4), (4.6.6), and (4.6.8), we have proved
yff(mT~M) -* iV[0, 4->/(МГ2]. (4.6.9)
The consistency of mT follows from statement (4.6.9). However, it also can be proved by a direct application of Theorem 4.1.1. Let mT be the set of the в points that minimize12
ST='2lYl-e-'2iYl-M. (4.6.10)
f-I f-1
Then, clearly, mTE. mT. We have
Q ж plim T~lST= 6 + 2 jMАДА) ей - 20 /(А) ей, (4.6.11)
where the convergence can be shown to be uniform in 0. The derivation of
(4.6.11) and the uniform convergence will be shown for a regression model, for which the present i. i.d. model is a special case, in the next subsection. Because
---- 1+2 F(6) (4.6.12)
and
^ = 2Ш, (4.6.13)
we conclude that Q is uniquely minimized at в = M and hence mris consistent by Theorem 4.1.1.
Next, we shall consider two complications that prevent us from proving the asymptotic normality of the median by using Theorem 4.1.3: One is that dST/dd = 0 may have no roots and the other is that d*ST/de2 = 0 except for a finite number of points. These statements are illustrated in Figure 4.2, which depicts two typical shapes of ST.
Despite these complications, assumption C of Theorem 4.1.3 is still valid if we interpret the derivative to mean the left derivative. That is to say, define for Д>0
(і) ^ = 0 has roots
0 9
.... dSj
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( и) =0 has no roots
dST ST(6) - ST(d-A)
------- л-------
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Then, from (4.6.10), we obtain
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Because (W£M)) are i. i.d. with mean і and variance |, we have by Theorem 3.3.4 (Lindeberg-Levy CLT)
Assumption В of Theorem 4.1.3 does not hold because (PST/de2 = 0 for almost every 6. But, if we substitute [<PQ/de2]M for plim Г-1[0257-/302]л/іп assumption В of Theorem 4.1.3, the conclusion of the theorem yields exactly the right result (4.6.9) because of (4.6.13) and (4.6.16).
