Online Appendix for “The Kumaraswamy
Distribution: Median-Dispersion ReParameterizations for Regression Modeling and
Simulation-Based Estimation”
Pablo A. Mitnik and Sunyoung Baek
1. Alternative dispersion orders weaker than the dispersive order
In the main text we claim that neither the right-spread order nor the dilation order are relevant
for examining whether the second parameter of re-parameterized Kumaraswamy distributions
with the same median and support establishes a dispersion order among them. In the case of
the right-spread order (also called “excess-wealth order”) this is simply because nonidentical
distributions with the same finite support are not related in terms of this order (Sordo
2009:Corollary 7). This is not the case, however, for the dilation order – that is, it is not the
case that any two distributions with the same finite support can only be related in terms of the
dilation order if they are identical (Muñoz-Perez and Sanchez-Gomez 1990: Theorem 3).
Therefore, showing why this order is not relevant for the problem at hand requires a more
detailed analysis.
The definition of the dilation order involves the notion of “convex order.” Using the
same notation as in the main text, Z is greater than 𝑊 in the convex order if 𝐸[𝜙(𝑍)] ≥
𝐸[𝜙(𝑊)] for all convex functions 𝜙: ℝ → ℝ, provided these expectations exist. In turn, the
random variable 𝑋 is greater than the random variable 𝑌 in the dilation order if X−𝐸(𝑋) is
greater than 𝑌 − 𝐸(𝑌) in the convex order (Shaked and Shanthikumar 2007:Chapter 3). A
necessary and sufficient condition for two distributions to be ordered by dilation, which we
use below, is the following (Shaked and Shanthikumar 2007:Theorem 3.A.8). If two random
1
variables X and Y have finite expectations then X is larger than Y in the dilation order if and
only if
1
1
1
−1 (𝑢)
−1 (𝑢)
𝑀(𝑣) =
∫ 𝐺
−𝐹
𝑑𝑢 − ∫ 𝐺 −1 (𝑢) − 𝐹 −1 (𝑢) 𝑑𝑢 ≤ 0,
1−𝑣 𝑣
0
for all 𝑣 ∈ [0,1).
Although it is not the case that any two distributions with the same finite support can
only be related in terms of the dilation order if they are identical, this does not imply that
Kumaraswamy distributions – and in particular those with the same median – are related in
term of this order. Determining whether this is the case analytically is very difficult, due to
the lack of a tractable-enough expression for the expectation of Kumaraswamy-distributed
variables.
In contexts like this, numerical evidence may be helpful. Hence, we conducted a
numerical analysis in which we evaluated 𝑀(𝑣) at 𝑣1 = 0.00000000001 and 𝑣2 =
0.99999999999 using pairs of Kumaraswamy-distributed variables Y and X with the same
median (𝜔 = 0.1, 0.3, 0.5, 0.7 or 0.9) and the following values of the second parameter (for
the first re-parameterization):
𝑑𝑝𝑦 = {0.3, 0.8, … , 7.8}
𝑑𝑝𝑥 = {𝑑𝑝𝑦 + 0.5, 𝑑𝑝𝑦 + 1, … , 8.3},
where 𝑑𝑝𝑦 and 𝑑𝑝𝑥 denote the parameter 𝑑𝑝 for variables Y and X, respectively. Hence, in
total, we evaluated 𝑀(𝑣1 ) and 𝑀(𝑣2 ) for 680 pairs of Kumaraswamy-distributed variables
with a common median.
Table A1 includes a small but pattern-revealing sample of our results, showing only
the signs of 𝑀(𝑣1 ) and 𝑀(𝑣2 ). Table A2 includes the complete set of results, with actual
figures instead of just signs. This table also shows the values of 𝑑𝑞𝑦 and 𝑑𝑞𝑥 corresponding
2
to the values of 𝑑𝑝𝑦 and 𝑑𝑝𝑥 employed in our analysis. Whenever 𝑀(𝑣1 ) and 𝑀(𝑣2 )
have different signs, the pairs of distributions in question are not ordered by dilation. Using
this criterion we found evidence that Kumaraswamy-distributed variables with a common
median are not ordered by dilation if 𝑑𝑝𝑦 > 1, regardless of the values of 𝜔 and 𝑑𝑝𝑥 ; or if
𝜔 > 0.5, regardless of the values of 𝑑𝑝𝑦 and 𝑑𝑝𝑥 . In addition, we found out that, in a
substantial number of cases in which 𝜔 ≤ 0.5 and 𝑑𝑝𝑦 < 1, Kumaraswamy-distributed
variables are not ordered by dilation either. For the remaining cases – only 48 of our 680
pairs of distributions – evaluating 𝑀(𝑣) at 𝑣1 and 𝑣2 did not provide evidence that the
distributions in each pair are not ordered by dilation. We cannot determine whether this
happened because these distributions are actually ordered by dilation or rather because pairs
of values of 𝑣 even closer to 0 and 1, respectively, would be needed to show that they are not
dilation-ordered. Nevertheless, the evidence from the numerical analysis indicates that the
dilation order should not be employed, either, to assess whether the second parameter in the
re-parameterizations advanced in the main text is a dispersion parameter.
[Tables A1 and A2 about here]
2. Proofs of propositions 4.1 to 4.3
Proposition 4.1. When 𝜔 → 𝑐 (𝜔 → 𝑏), the re-parameterized Kumaraswamy distributions
tend to the degenerate distribution with parameter 𝜌 = 𝑐 (𝜌 = 𝑏).
Proposition 4.2. When 𝑑𝑟 → 0, the Kumaraswamy distribution tends to the degenerate
distribution with parameter 𝜌 = 𝜔, for 𝑟 = 𝑝, 𝑞.
As the proofs of these propositions are rather simple and very similar for both reparameterizations, we only include here the proof for the first re-parameterization. Let X ~ Kp
(𝜔, 𝑑𝑝 , 𝑐, 𝑏), with quantile function 𝐻𝑝 . The quantile-spread of X is then:
3
𝑑𝑝
𝑄𝑆𝐻𝑝 (𝑢) = (𝑏 − 𝑐) {[1 − 𝑢 𝑔(𝑑𝑝 ,𝜔,𝑐,𝑏) ]
𝑑𝑝
− [1 − (1 − 𝑢) 𝑔(𝑑𝑝 ,𝜔,𝑐,𝑏) ] },
1
𝜔−𝑐
𝑑
where 𝑔(𝑑𝑝 , 𝜔, 𝑐, 𝑏) = ln [1 − ( 𝑏 − 𝑐 ) 𝑝 ] (ln 0.5)−1 . Taking limit for 𝜔 → 𝑐, 𝜔 → 𝑏,
and 𝑑𝑝 → 0 we get:
lim 𝑄𝑆𝐻𝑝 (𝑢) = (𝑏 − 𝑐){[1 − 𝑢0 ]𝑑𝑝 − [1 − (1 − 𝑢)0 ]𝑑𝑝 } = (𝑏 − 𝑐)(0 − 0) = 0
𝜔→ 𝑐
lim 𝑄𝑆𝐻𝑝 (𝑢) = (𝑏 − 𝑐){[1 − 𝑢∞ ]𝑑𝑝 − [1 − (1 − 𝑢)∞ ]𝑑𝑝 } = (𝑏 − 𝑐)(1 − 1) = 0
𝜔→ 𝑏
lim 𝑄𝑆𝐻𝑝 (𝑢) = (𝑏 − 𝑐){[1 − 𝑢∞ ]0 − [1 − (1 − 𝑢)∞ ]0 } = (𝑏 − 𝑐)(1 − 1) = 0.
𝑑𝑝 → 0
1/2
It is easy to see that 𝛿2 (𝑋) can be written as 𝛿2 (𝑋) = ∫0
𝑄𝑆𝐻𝑝 (𝑢) 𝑑𝑢. Hence,
given that 𝑄𝑆𝐻𝑝 (𝑢) is bounded and that in the three cases above 𝑄𝑆𝐻𝑝 (𝑢) → 0 for any 0 <
𝑢 < 0.5, it follows, using Lebesgue’s dominated convergence theorem, that 𝛿2 (𝑋) = 0
when
𝜔→ 𝑐 , 𝜔→ 𝑏
or 𝑑𝑝 → 0. As the degenerate distribution is the only
distribution with 𝛿2 (𝑋) = 0, in all three cases the Kumaraswamy distribution tends to the
degenerate distribution with location parameter equal to its median, as stated in Propositions
4.1 and 4.2.
Proposition 4.3. When 𝑑𝑟 → ∞, the Kumaraswamy distribution tends to the discrete
uniform distribution with possible values c and b, for 𝑟 = 𝑝, 𝑞.
Corollary: When 𝑐 = 0,
𝑏 = 1 and 𝑑𝑟 → ∞, the Kumaraswamy distribution tends to the
Bernoulli distribution with parameter 𝑠 = 0.5, for 𝑟 = 𝑝, 𝑞.
The proof of this proposition uses the following lemma.
Lemma. If Y ~ Kp (𝜔, 𝑑𝑝 , 0, 1), then
lim 𝜇𝑟′ (𝑌) = lim
𝑑𝑝 →∞
𝑑𝑝 →∞ ln(1
ln 0.5
−1
− ω𝑑𝑝 )
𝐵 (1 + 𝑟𝑑𝑝 ,
4
ln 0.5
−1
ln(1 − ω𝑑𝑝 )
) = 0.5,
where 𝜇𝑟′ (𝑌) is the rth moment around zero of Kp (𝜔, 𝑑𝑝 , 0, 1). The proof of this lemma
follows.
Substituting (6) in the expression for the rth moment around zero of K(p, q, 0, 1) used
in the introduction of the main text, the rth moment around zero of Kp (𝜔, 𝑑𝑝 , 0, 1) is
𝜇𝑟′ (𝑌) = 𝐿1 (𝑑𝑝 ) 𝐵 (𝐿2 (𝑑𝑝 ), 𝐿1 (𝑑𝑝 )),
where
−1
1
𝑑𝑝
𝐿1 (𝑑𝑝 ) = ln 0.5 [ln (1 − 𝜔 )]
𝐿2 (𝑑𝑝 ) = [1 + 𝑟 𝑑𝑝 ].
Γ (𝑣+𝑥)
Now, from the asymptotic expansion of a ratio of Gamma functions,
𝑣 𝑥−𝑦 [1 +
(𝑥−𝑦)(𝑥+𝑦−1)
2𝑣
Γ (𝑣+𝑦)
∝
1
+ 𝒪 (𝑣2 )] with |𝑣| → ∞ (Tricomi and Erdélyi 1951:133), where
𝑓 = 𝒪(𝑡) indicates that |𝑓| < 𝐴 𝑡 for some constant A, we can derive the following
asymptotic expansion for the Beta function by using 𝐵 (𝛼, 𝛽) =
𝐵(𝛼, 𝛽) ∝ Γ(𝛽) {1 −
𝛽(𝛽−1)
2𝛼
𝛤(𝛼)𝛤(𝛽)
𝛤(𝛼+𝛽)
:
1
+ 𝒪 (𝛼2 )} 𝛼 −𝛽 , with |𝛼| → ∞.
This expansion can be employed to represent the limit of 𝜇𝑟′ (𝑌) for 𝑑𝑝 → ∞ as:
lim 𝜇𝑟′ (𝑌) ∝
𝑑𝑝 →∞
lim Γ (1 + 𝐿1 (𝑑𝑝 )) {1 −
𝐿1 (𝑑𝑝 )[𝐿1 (𝑑𝑝 ) − 1]
𝑑𝑝 →∞
2 𝐿2 (𝑑𝑝 )
+𝒪(
1
𝐿2 (𝑑𝑝 )
2 )} 𝐿2 (𝑑𝑝 )
where we have used the property 𝑣 Γ(𝑣) = Γ(𝑣 + 1). As lim 𝐿1 (𝑑𝑝 ) = 0
𝑑𝑝 →∞
and lim 𝐿2 (𝑑𝑝 ) = ∞, we then have
𝑑𝑝 →∞
0
1
lim 𝜇𝑟′ (𝑌) ∝ Γ(1) {1 − ∞ + 𝒪 (∞)} lim 𝐿2 (𝑑𝑝 )
𝑑𝑝 →∞
𝑑𝑝 →∞
Using Γ(1) = 1, this entails
5
−𝐿1 (𝑑𝑝 )
.
−𝐿1 (𝑑𝑝 )
,
lim 𝜇𝑟′ (𝑌) = lim 𝐿2 (𝑑𝑝 )
𝑑𝑝 →∞
−𝐿1 (𝑑𝑝 )
𝑑𝑝 →∞
lim 𝜇𝑟′ (𝑌) = lim exp[−𝐿1 (𝑑𝑝 ) ln 𝐿2 (𝑑𝑝 )].
𝑑𝑝 →∞
𝑑𝑝 →∞
Replacing 𝐿1 (𝑑𝑝 ) and 𝐿2 (𝑑𝑝 ) by their definitions, we obtain
lim 𝜇𝑟′ (𝑌) = exp [(−1) ln 0.5 lim
𝑑𝑝 →∞
𝑑𝑝 →∞
1
ln (1+𝑟 𝑑𝑝 )
1
].
(A1)
ln(1− 𝜔𝑑𝑝 )
𝑥−1 2𝑘−1
Using the series representation ln 𝑥 = 2 ∑∞
𝑘=1 2𝑘−1 (𝑥+1)
with |𝑥| > 0 (Gradshteyn and
Ryzhik 2007:53), the limit in the hand right side of (A1) can be expressed as
1
𝑑𝑝
lim ln (1 + 𝑟 𝑑𝑝 ) [ln (1 − 𝜔 )]
𝑑𝑝 →∞
2𝑘−1
∞
∑
−1
=
1
𝑑𝑝
∞
𝑟 𝑑𝑝
1
[ lim
]
𝑘=1 2𝑘 − 1 𝑑𝑝 →∞ 2 + 𝑟 𝑑𝑝
∑
{
1
−𝜔
[ lim
1]
𝑘=1 2𝑘 − 1 𝑑𝑝 →∞
2 − 𝜔 𝑑𝑝
−1
∞
1
1
2𝑘−1
2𝑘−1
(1)
(−1)
=∑
[∑
] = −1.
𝑘=1 2𝑘 − 1
𝑘=1 2𝑘 − 1
2𝑘−1
−1
}
∞
(A2)
Substituting (A2) in (A1), we get
lim 𝜇𝑟′ (𝑌) = exp[(−1)ln 0.5 (−1)] = 0.5
𝑑𝑝 →∞
which concludes the proof of the lemma.
We can now proceed with the proof of proposition 4.3.
′
(𝑋) = 𝑞 𝐵 (1 +
First re-parameterization. Substituting 𝜇𝑟−𝑗
𝑟−𝑗
𝑝
, 𝑞) in (4), and then
𝑟
using 𝐵(1, 𝛽) = 𝛽 −1 and ( ) = 1, we obtain:
𝑟
𝜇𝑟′ (𝑋) = (𝑏 − 𝑐)𝑟 ∑
𝑟
𝑗=0
𝑐 𝑗
𝑟−𝑗
𝑟
(𝑗 ) (
) 𝑞 𝐵 (1 +
, 𝑞)
𝑏−𝑐
𝑝
6
= (𝑏 − 𝑐)𝑟 ∑
𝑟−1
𝑗=0
𝑐 𝑗
𝑟−𝑗
𝑐 𝑟
𝑟
𝑟
(𝑗) (
) 𝑞 𝐵 (1 +
, 𝑞) + (𝑏 − 𝑐)𝑟 ( ) (
) 𝑞 𝐵(1, 𝑞)
𝑟 𝑏−𝑐
𝑏−𝑐
𝑝
= (𝑏 − 𝑐)𝑟 ∑
𝑟−1
𝑗=0
𝑐 𝑗
𝑟−𝑗
𝑟
(𝑗 ) (
) 𝑞 𝐵 (1 +
, 𝑞) + 𝑐 𝑟 .
𝑏−𝑐
𝑝
1
Substituting (6) and 𝑑𝑝 = 𝑝 in the last expression and taking limit for 𝑑𝑝 → ∞, we get
lim 𝜇𝑟′ (𝑋) = (𝑏 − 𝑐)𝑟 ∑
𝑑𝑝 →∞
𝑟−1
𝑗=0
𝑐 𝑗
𝑟
(𝑗 ) (
) lim 𝐿(𝑑𝑝 , 𝑟 − 𝑗) + 𝑐 𝑟
𝑏 − 𝑐 𝑑𝑝 →∞
where
𝐿(𝑑𝑝 , 𝑟 − 𝑗) =
with ω
̃=
𝜔−𝑐
𝑏−𝑐
ln 0.5
−1
ln(1 − ω
̃ 𝑑𝑝 )
𝐵 (1 + (𝑟 − 𝑗)𝑑𝑝 ,
ln 0.5
−1
ln(1 − ω
̃ 𝑑𝑝 )
),
.
As 𝐿(𝑑𝑝 , 𝑟 − 𝑗) is the (𝑟 − 𝑗)th moment around zero of a variable with distribution
Kp (𝜔, 𝑑𝑝 , 0, 1), the Lemma applies and lim 𝐿(𝑑𝑝 , 𝑟 − 𝑗) = 0.5. Hence,
𝑑𝑝 →∞
lim 𝜇𝑟′ (𝑋) = 0.5 (𝑏 − 𝑐)𝑟 ∑
𝑑𝑝 →∞
𝑟−1
𝑗=0
𝑐 𝑗
𝑟
(𝑗 ) (
) + 𝑐𝑟.
𝑏−𝑐
𝑟
𝑏 𝑟
𝑐 𝑟
𝑐 𝑗
Now, applying the binomial theorem, (𝑏−𝑐) = (1 + 𝑏−𝑐) = ∑𝑟𝑗=0 ( 𝑗 ) (𝑏−𝑐) . Therefore,
𝑟
𝑐 𝑗
𝑏 𝑟
𝑐 𝑟
∑𝑟−1
𝑗=0 ( 𝑗 ) (𝑏−𝑐 ) = (𝑏−𝑐 ) − (𝑏−𝑐 ) . We thus have
lim 𝜇𝑟′ (𝑋) = 0.5 (𝑏 − 𝑐)𝑟 [(
𝑑𝑝 →∞
𝑏 𝑟
𝑐 𝑟
) −(
) ] + 𝑐 𝑟 = 0.5 (𝑏 𝑟 + 𝑐 𝑟 ).
𝑏−𝑐
𝑏−𝑐
(A3)
The rth moment of a discrete uniform distribution with possible values 𝛼 and 𝛽 is
𝐸(𝑍 𝑟 ) = ∑𝑍=𝛼,𝛽 𝑍 𝑟 Pr(𝑍) =
𝛼𝑟 +𝛽 𝑟
2
.Therefore, from (A3) and the fact that both the
Kumaraswamy distribution and the discrete uniform distribution have bounded support, it
follows that when 𝑑𝑝 → ∞ the Kumaraswamy distribution converges to the discrete uniform
distribution with possible values c and b, as stated in Proposition 4.3.
7
Second re-parameterization. Given the one-to-one mapping between the dispersion
parameters of the first and second re-parameterization introduced in the proof of Proposition
3.1, it is the case, using (7), that
lim 𝑑𝑝 = lim 𝐷(𝑑𝑞 ; ω) = lim
𝑑𝑞 → ∞
𝑑𝑞 → ∞
̃
ln ω
𝑑𝑞 → ∞ ln(1−0.5𝑑𝑞 )
= ∞.
From this and the proof of Proposition 4.3 for the first re-parameterization, the corresponding
proof for the second re-parameterization follows immediately.
References
Gradshteyn, I. and I. Ryzhik (2007). Table of Integrals, Series and Products. New York,
Academic Press.
Muñoz-Perez, J. and A. Sanchez-Gomez (1990). "Dispersive ordering by Dilation." Journal
of Applied Probability 27(2): 440-444.
Shaked, M. and G. Shanthikumar (2007). Stochastic Orders. New York, Springer
Sordo, M. (2009). "On the Relationship of Location-Independent Riskier Order to the Usual
Stochastic Order." Statistics and Probability Letters 79: 155-157.
Tricomi, F. G. and A. Erdélyi (1951). "The Asymptotic Expansion of a Ratio of Gamma
Functions." Pacific Journal of Mathematics 1: 133-142.
8