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

Подпись: (4.6.2)= 0 if У,<а

for eveiy real number a. Using (4.6.2), we define the sample median ттЪу

Подпись: mT = infimage348(4.6.3)

The median as defined above is clearly unique.11

image349 Подпись: (4.6.4)

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

Подпись: Figure 4.2 Complications proving the asymptotic normality of the median

( и) =0 has no roots

Подпись: (4.6.14)dST ST(6) – ST(d-A)

——- л——-

image353 Подпись: (4.6.15)

Then, from (4.6.10), we obtain

image355 Подпись: N{0, 1). Подпись: (4.6.16)

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).

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