## 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 ...

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