Minimization Property of the Median m & |µ − m| ≤ σ
Fritz Scholz
Fact 1: |E(Y )| ≤ E(|Y |) by generalization of the triangle inequality.
Also from: |Y | ≥ Y ⇒ E(|Y |) ≥ E(Y ) and |Y | ≥ −Y ⇒ E(|Y |) ≥ E(−Y ) = −E(Y ),
hence |E(Y )| ≤ E(|Y |).
Fact 2: E(Y 2 ) ≥ (E(|Y |))2 since var(|Y |) = E(Y 2 ) − (E(|Y |))2 ≥ 0.
Fact 3 (Minimization Property of the Median): If m denotes the median of Y then
E(|Y − m|) ≤ E(|Y − a|) for any constant a, in particular for a = µ = E(Y ).
Thus the median m minimizes E(|Y − a|) over a.
This important property of the median parallels that of a = µ minimizing E[(Y − a)2 ].
Proof: We use the following notation: IA (x) = 1 when x ∈ A and IA (x) = 0 otherwise. Without
loss of generality we assume a < m.
We have the following identity (verified by checking for x ≤ a, x ≥ m and a < x < m):
|x − a| − |x − m| = (a − m)I(−∞,a] (x) + (m − a)I[m,∞) (x) + (2x − a − m)I(a,m) (x) .
Simple manipulation turns that into
|x − a| − |x − m| =
=
=
=
2(x − a)I(a,m) (x) + (a − m)I(a,m) (x) + (a − m)I(−∞,a] (x) + (m − a)I[m,∞) (x)
2(x − a)I(a,m) (x) + (a − m)I(−∞,m) (x) + (m − a)I[m,∞) (x)
2(x − a)I(a,m) (x) + (a − m)I(−∞,m) (x) + (m − a)(1 − I(−∞,m) (x))
2(x − a)I(a,m) (x) + (m − a) + 2(a − m)I(∞,m) (x) .
Using the random variable Y for x and taking expectations on both sides we get
E(|Y − a|) − E(|Y − m|) = E 2(Y − a)I(a,m) (Y ) + (m − a) − 2(m − a)P (Y < m) .
By definition of the median we have P (Y < m) ≤ 1/2 and P (Y > m) ≤ 1/2 (or equivalently
P (Y ≥ m) ≥ 1/2 and P (Y ≤ m) ≥1/2) and thus (m − a) − 2(m − a)P (Y < m) ≥ 0.
Furthermore, E 2(Y − a)I(a,m) (Y ) ≥ 0 since the random variable within E( ) is nonnegative.
Thus E(|Y − a|) − E(|Y − m|) ≥ 0, q.e.d.
The proof for |µ − m| ≤ σ now becomes a 1-liner:
Fact1
Fact3
Fact2
|µ − m| = |E(Y ) − m| = |E(Y − m)| ≤ E(|Y − m|) ≤ E(|Y − µ|) ≤
q
E(|Y − µ|2 ) = σ
q.e.d.