Inferential Statistics using the Coefficient of Variation ...

Academic Forum 26 2008-09
Here are graphs of some probability distribution functions for the standard deviation and a
formula for its mean and variance:
Graph of S pdf for n=2,3,4,5 using σ=1
E[
S]
=
⎛ n ⎞
2Γ⎜
⎟
⎝ 2 ⎠
−
σ → σ
⎛ n −1⎞
n −1Γ⎜
⎟
⎝ 2 ⎠
2
⎡ ⎛ n ⎞
⎢ 2Γ⎜
⎟
⎢ ⎝ 2
Var[
S]
= 1 −
⎠
⎢
⎛ n −1⎞
⎢ ( n −1)
Γ⎜
⎟
⎢⎣
⎝ 2 ⎠
2
as n → ∞
⎤
⎥
⎥ 2
σ
⎥
⎥
⎥⎦
Mean and Variance of S
→ 0 as n → ∞
To find a formula for the distribution of the CV, consider the joint distribution of ( X , S)
. By
independence, the joint probability density function is
f with support +
R × R . The
probability density function for S was derived earlier, and the probability mass function for is
2
n ⎛ n(
x − µ ) ⎞
easily computed to be f X
( x)
= exp
, ∈ R
2
2
⎜−
2
⎟ x . Therefore, the cumulative
πσ ⎝ σ ⎠
distribution function for CV is
⎡ S ⎤
FCV
( c)
= P⎢
≤ c⎥
= P[ ( S ≤ cX ) ∩ ( X > 0)
] + P[
X < 0]
⎣ X ⎦
(If n is sufficiently large, then
∞ cx
= f ( x)
f ( s)
dsdx + P[
X < 0], c > 0
P [ CV < 0] ≈ 0.)
∫ ∫
0 0
X
S
f X S
Differentiate the
cumulative
distribution
function to obtain
the probability
density function:
=
∞
∫
0
⎛ n ⎞
= ⎜ ⎟
⎝ π ⎠
c > 0
f
CV
n ⎡ n(
x − µ )
exp⎢−
2
2πσ
⎣ 2σ
1
2
c
n−2
( n −1)
⎛ n −1⎞
n
Γ⎜
⎟σ
2
⎝ 2 ⎠
∞
'
( c)
= FCV
( c)
= ∫ f
n−1
2
n−2
2
2
⎤
⎥
⎦
0
( cx)
X
n−2
⎛ n −1⎞
Γ⎜
⎟2
⎝ 2 ⎠
2
⎡ nµ
⎤
exp⎢
−
2 ⎥
⎣ 2σ
⎦
( x)
f
∞
∫
0
x
n−3
2
x
S
n−1
d(
cx)
( cx)
dx + 0
dc
⎡n
−1⎤
⎢ 2
⎣ σ ⎥
⎦
n−1
2
⎡ ( n −1)(
cx)
exp⎢−
2
⎣ 2σ
2
⎡ ( n −1)
c + n
exp⎢−
x
2
⎣ 2σ
2
2
⎤
⎥dx
⎦
nµ
x ⎤
+ ,
2 ⎥dx
σ ⎦
21