Avd.

Matematisk statistik

Sf 2955: Computer intensive methods :

SCALE PARAMETER/ Timo Koski

The notation
F (x; θ)
denotes a distribution function that depends on a parameter θ. For example,

1 − e−x/θ if x ≥ 0
F (x; θ) =
0 if x < 0,

is the distribution function of the exponential distribution Exp (θ).

1 Definition of a Scale Parameter

We define a scale parameter.

Definition 1.1 Assume θ > 0 in F (x; θ). Then θ is a scale parameter, if it

holds for all x that x
F (x; θ) = H , (1.1)
θ
where H(x) is a distribution function.

To take an example, θ is scale parameter in the exponential distribution

Exp (θ), as we have x
F (x; θ) = H ,
θ
where 
1 − e−x if x ≥ 0
H(x) =
0 if x < 0.

1
2 Properties, Pivotal Variables
We observe two lemmas.

Lemma 2.1 Assume X ∈ F (x; θ). θ is a scale parameter if and only if the
distribution of
X
θ
does not depend on θ.
X
Proof: ⇒: Assume θ is a scale parameter. Then the distribution of θ
is given
by  
X
P ≤ z = P (X ≤ θz)
θ
since θ > 0 by definition. Then, as we assume that θ is a scale parameter,
 
θz
P (X ≤ θz) = F (θz; θ) = H = H (z) ,
θ
or  
X
P ≤z = H(z)
θ
which says that Xθ has distribution which does not depend on θ.
⇐: We assume that Xθ has distribution, say H, which does not depend on
θ > 0. Then  
X x
F (x; θ) = P (X ≤ x; θ) = P ≤ ;θ
θ θ
since θ > 0. But by assumption
 
X x x
P ≤ ;θ = H
θ θ θ

where the distribution function H (z) does not depend on θ. But then we
have for any x shown that
x
F (x; θ) = H
θ
which in view of (1.1) gives the claim as asserted.
This lemma says that Xθ is a pivotal variable.

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Remark 2.1 Note that a pivotal variable need not be a statistic − the va-
riable can depend on parameters of the model, but its distribution must not.
If it is a statistic, then it is known as an ancillary statistic. Pivotal quantities
are used to the construction of test statistics, e.g., Student’s t−statistic is
pivotal for a normal distribution with unknown variance (and mean). Pivotal
variables provide in addition a method of constructing confidence intervals,
and the use of pivotal quantities improves performance of the bootstrap, as
defined later.

d
Lemma 2.2 Assume F (x; θ) has a density dx F (x; θ) = f (x; θ) for all x.
Then θ is a scale parameter if and only if
1 x
f (x; θ) = g , (2.2)
θ θ
where g is a probability density.
Proof: ⇒: If F (x; θ) has an integrable derivative and θ is a scale parameter,
then it holds from (1.1 ) that
d d x 1 x
f (x; θ) = F (x; θ) = H = g
dx dx θ θ θ
d
where we have set g(x) = dx
H (x), which is a density function, as H is a
distribution function.
⇐: We assume that
1 x
f (x; θ) = g .
θ θ
Then Z x
1 x u
Z
F (x; θ) = f (u; θ)du = g du.
−∞ θ −∞ θ
Here we make a change of variable t = uθ , du = θdt and thus
Z x
1 x u
Z x
θ
g du = g (t) dt = G .
θ −∞ θ −∞ θ
d
where dx
G (x) = g(x). In other words we have obtained for every x that
x
F (x; θ) = G ,
θ
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and since G (x) is a distribution function, the desired assertion follows by
(1.1).

3 Examples and the Scale-Free property

We give first some further examples.

1. X ∈ N (µσ, σ 2 ). We guess that σ is a scale parameter. The pertinent

density is
1 (x−µσ)2
f (x; σ) = √ e− 2σ2
2πσ
and some elementary algebra gives
x −µ)2
1 (σ 1 x
f (x; σ) = √ e− 2 = g ,
2πσ σ σ

where g is the density of N(µ, 1) or

1 (z−µ)2
g (z) = √ e− 2
2π
and the desired conclusion follows by (2.2).

2. X ∈ R (0, θ). Then we take x ∈ [0, 1] and get

 
X θx
P ≤ x = P (X ≤ θx) = =x
θ θ

as the distribution function of R(0, θ) is F (z) = θz . The desired conclu-

sion follows now from the first lemma. We have shown that Xθ ∈ R(0, 1),
which is, however, immediate from the definitions.

3. Finally, we take the Weibull distribution, so that

 k
1 − e−(x/λ) if x ≥ 0
F (x; λ) =
0 if x < 0.

For k = 1 gives the exponential distribution, and k = 2 gives the

Rayleigh distribution. It is obvious that λ > 0 is scale parameter.

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 Suppose that X ∈ Exp (θ) is a lifetime measured in seconds. Then θ =
E [X] is also measured in seconds. We might convert seconds to minutes.
But the probability x
P (X ≤ x) = H ,
θ
is the same whether we measure in minutes or seconds, i.e., it is invariant
with respect to scale or the units of measurement, as is quite reasonable.