# 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  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 ( и) =0 has no roots dST ST(6) – ST(d-A)

——- л——-  Then, from (4.6.10), we obtain   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).