Leptokurtic distributions can be generated by applying certain non-linear transformations to a standard normal random variable. Within this work we derive general conditions for these transformations which guarantee that the generated distributions are ordered with respect to the partial ordering of van Zwet for symmetric distributions and the partial ordering of MacGillivray for arbitrary distributions. In addition, we propose a general power transformation which nests the H-, J- and K-transformations which have already been proposed in the literature. Within this class of power transformations the above mentioned condition can be easily verified and the power can be interpreted as parameter of leptokurtosis. --Power kurtosis transformation,leptokurtosis,kurtosis ordering

Fischer, Matthias J.

Klein, Ingo

English

Research Papers in Economics

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Klein, Ingo; Fischer, Matthias J.
Working Paper
Kurtosis transformation and kurtosis
ordering
Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für
Statistik und Ökonometrie, No. 53/2003
Provided in cooperation with:
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Suggested citation: Klein, Ingo; Fischer, Matthias J. (2003) : Kurtosis transformation and
kurtosis ordering, Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg,
Lehrstuhl für Statistik und Ökonometrie, No. 53/2003, http://hdl.handle.net/10419/29609Friedrich-Alexander-Universit¨ at
Erlangen-N¨ urnberg
Wirtschafts-und Sozialwissenschaftliche Fakult¨ at
Diskussionspapier
53 / 2003
Kurtosis transformation and kurtosis ordering
Ingo Klein and Matthias Fischer
Lehrstuhl f¨ ur Statistik und ¨ Okonometrie
Lehrstuhl f¨ ur Statistik und empirische Wirtschaftsforschung
Lange Gasse 20 · D-90403 N¨ urnbergKurtosis transformation and kurtosis ordering
Ingo Klein and Matthias Fischer
Department of Statistics and Econometrics,
University of Erlangen-Nuremberg
Abstract: Leptokurtic distributions can be generated by applying certain
non-linear transformations to a standard normal random variable. Within
this work we derive general conditions for these transformations which guar-
antee that the generated distributions are ordered with respect to the partial
ordering of van Zwet for symmetric distributions and the partial ordering
of MacGillivray for arbitrary distributions. In addition, we propose a gen-
eral power transformation which nests the H-, J- and K-transformations
which have already been proposed in the literature. Within this class of power
transformations the above mentioned condition can be easily veriﬁed and the
power can be interpreted as parameter of leptokurtosis.
Keywords: Power kurtosis transformation; leptokurtosis; kurtosis ordering.
1 Introduction
Flexible distribution families which accommodate both skewness and kurtosis can be gen-
erated if we transform a standard Gaussian distribution with certain non-linear transfor-
mations. One or both tails of the distribution can be shortened or increased by means of
separate skewness and kurtosis transformations. If these transformations are parameter-
ized the corresponding parameter can be interpreted as skewness and/or kurtosis parame-
ter. However, this term only makes sense if these parameters actually measure skewness
and/or kurtosis of distributions. To verify this we have to prove that general accepted con-
ditions for a skewness and/or kurtosis measures are satisﬁed. One of these requirements is
that a partial skewness and kurtosis ordering is preserved. Whereas van Zwet (1964) pro-
posed kurtosis orderings for symmetrical distributions, Balanda and MacGillivray (1990)
introduced kurtosis orderings for arbitrary distributions. For certain transformations it
was already shown that these orderings are preserved. Take, for example, the exponen-
tial transformation – the so-called H-transformation – which was proposed by Tukey
(1977) and extensively discussed by MacGillivray (1992). For other transformations,
however, like the K-transformation (Haynes et al.,1997) or the J-transformation (Fischer
and Klein, 2003) it has not yet been veriﬁed.
Within this work these transformations are embedded in a general class of power transfor-
mation. Moreover it is shown that the exponent of this class of power transformation can
be understood as kurtosis parameter in the sense of preserving the kurtosis ordering of van
Zwet for symmetrical distributions. More general, it is also demonstrated that the kurtosis
ordering of van Zwet is preserved for arbitrary (not necessary parameterized) twice dif-
ferentiable transforms whose ratio of derivatives satisﬁes certain conditions concerning
the monotony.2 Kurtosis orderings
Van Zwet (1964) introduced a kurtosis ordering (more precise, a partial ordering) on the
set of all symmetric, continuous and strictly monotone increasing distributions. In this
concept, a symmetric distribution F has less kurtosis than a symmetric distribution G
(F S G), if G−1(F(x)) is convex for x > F −1(0.5), where F −1 and G−1 denote the
inverse distribution function (or quantile function) of F and G, respectively. Obviously,
F −1(0.5) is the median of the distribution of F.
Balanda and MacGillivray (1990) generalized this partial ordering of van Zwet by using
so-called spread functions deﬁned as symmetric differences of quantiles:
SF(u) = F
−1(u) − F
−1(1 − u), u ≥ 0.5.
SF is monotone increasing on [1/2,1). If F is symmetric, F −1(u) = −F −1(1 − u) for
u > 0.5, so that
SF(u) = 2F
−1(u) u ≥ 0.5
and
S
−1
F (x) = F(x/2) for x > F −1(0.5).
This means that (for symmetric distributions) the spread function essentially coincide
with the quantile function. In the sense of Balanda and MacGillivray (1990), an arbitrary
continuous, monotone increasing distribution function F has less kurtosis than an equally
distribution function G (F S G) if SG(S
−1
F (x)) is convex for x > F −1(0.5). If F and G
are symmetric distributions,
SG(S
−1
F (x)) = 2G
−1(2F(x)) = G
−1(F(x)) x ≥ F
−1(0.5),
implying that in this special case the orderings of van Zwet and MacGillivray coincide.
This justiﬁes the identical notation S.
3 Verifying the property of ordering by means of the sec-
ond derivation
It is well-known that a twice differentiable function is convex if its second derivative is
positive. Consequently, we have to investigate the second derivative of
SG(S
−1
F (x)) for x ≥ F −1(0.5),
provided that it exists. Setting u = F(x), the ﬁrst derivative is given by
a(u) ≡
∂SG(u)
∂u
·
1
S0
F(u)
=
S0
G(u)
S0
F(u)
for u ≥ 0.5,
implying that the second derivative is given by
a
0(u) =
S00
G(u)S0
F(u) − S0
G(u)S00
F(u)
S0
F(u)2 for u ≥ 0.5.a0(u) is positive, if
(A1) S
00
G(u)S
0
F(u) ≥ S
0
G(u)S
00
F(u) for u ≥ 0.5.
In terms of density function f and distribution function F,
S
0
F(u) =
1
f(F −1(u))
+
1
f(F −1(1 − u))
for u ≥ 0.5
and
S
00
F(u) =
f0(F −1(1 − u))
f(F −1(1 − u))3 −
f0(F −1(u))
f(F −1(u))3 for u ≥ 0.5.
Note that a necessary condition for the existence of S00
F(u) for u ≥ 0.5 is that the density
f has to be differentiable and strictly positive on R.
Condition (A1) can be simpliﬁed for symmetric distributions F and G. Using the so-
called score function
ψF(x) ≡ −
f0(x)
f(x)
, for x ∈ R
we have
(A2) ψG(G
−1(u))
f(F −1(u))
g(G−1(u))
− ψF(F
−1(u)) ≥ 0 for u ≥ 0.5.
If ψG(x) > 0 for x > 0, equation (A2) is equivalent to
(A3)
ψF(F −1(u))
ψG(G−1(u))
−
f(F −1(u))
g(G−1(u))
≤ 0 for u > 0.5.
This means that the ratio of the score functions has to dominate the ratio of the density
functions uniformly for u > 0.5.
Example 3.1 (GSH distribution) The generalized secant hyperbolic (GSH) distribution
– which is able to model both thin and fat tails – was introduced by Vaughan (2002) and
has density
fGSH(x;t) = c1(t) ·
exp(c2(t)x)
exp(2c2(t)x) + 2a(t)exp(c2(t)x) + 1
, x ∈ R,t > −π (1)
with normalizing constants
a(t) = cos(t), c2(t) =
q
π2−t2
3 , c1(t) =
sin(t)
t · c2(t), for − π < t ≤ 0,
a(t) = cosh(t), c2(t) =
q
π2+t2
3 , c1(t) =
sinh(t)
t · c2(t), for t > 0
.
The inverse distribution function is given in closed form by
F
−1
GSH(u;t) =

  
  
1
c2(t) ln

sin(tu)
sin(t(1−u))

f¨ ur t ∈ (−π,0),
√
3
π ln
 
u
1−u

f¨ ur t = 0,
1
c2(t) ln

sinh(tu)
sinh(t(1−u))

f¨ ur t > 0.
Klein and Fischer (2003) showed that t is indeed a kurtosis parameter in the sense of van
Zwet (1964). For t1 = −π/2 and t2 = 0.5 the curve of the ratio of the score functions
and of the densities from condition (A3) is exemplarily shown in ﬁgure 1, below.Figure 1: Ratio of score- and density functions
4 A general symmetric kurtosis transformation
Let Z be a random variable which is symmetric around the median 0 and which has
continuous distribution function. Deﬁne
Y ≡ Z · W(Z)
where W is a suitable kurtosis transformation. Hoaglin (1983) postulated some plausible
requirements to a suitable transformation T of kurtosis. Firstly, W should preserving
symmetry, i.e. W(z) = W(−z) for z ∈ R and we therefore have to discuss W only
on the positive axis. Secondly, the initial distribution Z should hardly be transformed
in the centre, i.e. W(z) ≈ z for z ≈ 0. Finally, in order to increase the tails of the
distribution, we have to assure that W is accelerated strictly monotone increasing for
positive z > 0, i.e. W 0(z) > 0 and W 00(z) > 0 for z > 0. Consequently, W is strictly
monotone increasing and convex for z > 0. Conversely, a shortening of the tails takes
place, either if W is strictly monotone increasing with negative second derivation or if
W is not monotone but concave for z > 0. Differentiability and monotony imply that
W 0(0) = 0.
Examples which satisfy the aforementioned conditions are:
1. H(z) = exp(1/2hz2) = exp(1/2z2)h for h ∈ R (Tukey, 1977),
2. J(z) = cosh(z)j for j ∈ R (Fischer and Klein, 2003),
3. K(z) = (1 + z2)k for k ∈ R (Tukey, 1964).
The following example shows that the transformations H,J and K are special cases of
the general power transformation
A(z) ≡
 
∞ X
i=0
aiz
2i
!r
for r ∈ R, (2)
where the weights ai, i = 0,1,2,... have to be chosen that the limes of the power series P∞
i=0 aiz2i exists for all z ∈ R.Example:
1. Tukey’s H-transformation can be written as
H(z) =

e
1/2z2h
=
 
∞ X
i=0
1
2ii!
z
2i
!h
.
It is obvious that ai = 1/(2ii!) in equation (2).
2. From
cosh(z) = 1/2e
z + 1/2e
−z = 1/2
∞ X
i=0

zi
i!
+
(−z)i
i!

=
∞ X
i=0
1
2i!
z
2i.
we see that Fischer and Klein’s J-transformation has the power series representation
J(z) =
 
∞ X
i=0
1
2i!
z
2i
!j
,
that means ai = 0 for odd i and ai = 1/(2i!) for even i in equation (2).
3. Setting a0 = 1, a1 = 1 and ai = 0, i > 1 in equation (2) leads to Tukey’s K-
transformation.
The ﬁrst and second derivatives of A are
A
0(z) = r
 
∞ X
i=0
aiz
2i
!r−1 ∞ X
i=1
2iaiz
2i−1 = rA(z)C1(z) with C1(z) =
∞ P
i=1
2iaiz2i−1
∞ P
i=0
aiz2i
and
A
00(z) = rA(z)((r − 1)C2(z) + C3(z))
with
C2(z) =
P∞
i=1 2iaiz2i−1
P∞
i=0 aiz2i
2
= C1(z)
2 ≥ 0 for all z ∈ R (3)
and
C3(z) =
P∞
i=1 2i(2i − 1)aiz2i−2
P∞
i=0 aiz2i (4)
for z ≥ 0. Symmetry of A is given by construction. A(0) = 1 is satisﬁed if a0 = 1.
A0(0) = 0 is satisﬁed because of C1(0) = 0, if A(z) ≥ 0 f¨ ur z ≥ 0. The monotony
condition A0(z) > 0 holds, if r > 0 and
∞ X
i=1
2iaiz
2i−1 ≥ 0for z > 0. A is tail increasing, if A00(z) > 0 which in turn is satisﬁed, if r > 0 and
r > 1 − C3(z)/C2(z)
for all z > 0. A transformation A with all this properties will be called a general power
kurtosis transformation in the sequel. The untransformed distribution (here: standard
Gaussian) is obtained for r = 0. What remains is to proof that the exponent r can be
interpreted as kurtosis parameter in the sense of van Zwet (1964).
5 Kurtosis ordering of general kurtosis transformations
Let Z be a standard Gaussian random variable. Deﬁne two random variables
Yi = Z · Wi(Z), i = 1,2,
where Wi, i = 1,2 are kurtosis transformations which are twice differentiable for z ∈ R
and symmetric around 0. It will be demonstrated in the next theorem that Y1 has less
kurtosis than Y2 if the condition
(B)
W
(p)
2 (z)
W
(p)
1 (z)
≥
W
(p−1)
2 (z)
W
(p−1)
1 (z)
for z > 0 and p = 1,2
holds, provided that Wi(z) > 0, W 0
i(z) > 0 and W 00
i (z) > 0 for z > 0 and i = 1,2. This
means that the ratios of the p-derivatives of Wi are monoton increasing in p.
Theorem: Suppose that Z is a standard normal random variable. Deﬁne Yi = Z ·
Wi(Z) with distribution function Fi and assume that Wi is twice differentiable on R with
Wi(z) > 0, W 0
i(z) > 0, W 00
i (z) > 0 for z > 0 and symmetric around 0 for i = 1,2. If
condition (B) is satisﬁed for p = 1,2, then F1 S F2.
Proof: Because of the symmetry of Fi,
SFi(u) = 2F
−1
i (u) = 2zWi(z) f¨ ur z = Φ−1(u), u ≥ 0.5.
for i = 1,2. Consequently,
S
0
Fi(Φ(z)) = 2(Wi(z) + zW
0
i(z)) for z ≥ 0
and
S
00
Fi(Φ(z)) = 2(2W
0
i(z) + zW
00
i (z)) for z ≥ 0,
i = 1,2. Hence,
1/4
h
S
00
F2(Φ(z))S
0
F1(Φ(z)) − S
00
F1(Φ(z))S
0
F2(Φ(z))
i
= (2W
0
2(z) + zW
00
2 (z))(W1(z) + zW
0
1(z)) − (2W
0
1(z) + zW
00
1 (z))(W2(z) + zW
0
2(z))
= 2W
0
2(z)W1(z) + 2W
0
2(z)zW
0
1(z) + zW
00
2 (z)W1(z) + z
2W
0
1(z)W
00
2 (z)
−2W
0
1(z)W2(z) − 2W
0
1(z)zW
0
2(z) − zW
00
1 (z)W2(z) − z
2W
0
2(z)W
00
1 (z)
= 2(W
0
2(z)W1(z) − W
0
1(z)W2(z)) + z(W
00
2 (z)W1(z) − W
00
1 (z)W2(z))
+z
2(W
00
2 (z)W
0
1(z) − W
00
1 (z)W
0
2(z)),so that condition (B) is satisﬁed, if this term is positive. This is true for W1(z) > 0,
W 0
1(z) > 0, W 00
1 (z) > 0 ,
W 0
2(z)
W 0
1(z)
≥
W2(z)
W1(z)
und
W 00
2 (z)
W 00
1 (z)
≥
W 0
2(z)
W 0
1(z)
for z > 0. 
In the special case of a general power kurtosis transformation different exponents deﬁne
different transformations
Yi = Z · Ai(Z) =
 
∞ X
l=1
alz
2l
!ri
, ri ∈ R, i = 1,2.
If r is actually a kurtosis parameter, we should be able to show that Y1 with correspond-
ing parameter r1 has less kurtosis than Y2 with corresponding parameter r2 > r1. For
this purpose, condition (B) has to be veriﬁed for the ratios of the derivatives of A2(z) and
A1(z).
Lemma: Deﬁne
Ai(z) ≡
 
∞ X
l=1
alz
2l
!ri
, i = 1,2, r ∈ R
and suppose that the power series in brackets converges. Furthermore, assume A0
i(z) ≥ 0
and A00
i(z) ≥ 0 for z > 0 and i = 1,2. If r2 > r1 > 0,
A
(p)
2 (z)
A
(p)
1 (z)
≥
A
(p−1)
2 (z)
A
(p−1)
1 (z)
for z > 0 and p = 1,2.
Proof: It has already been shown that
A
0
i(z) = riAi(z)C1(z) with C1(z) =
P∞
l=1 2lalz2l−1
P∞
l=0 alz2l for i = 1,2
and
A
00
i(z) = riAi(z)((ri − 1)C2(z) + C3(z)) for i = 1,2
with C2(z) ≥ 0 and C3(z) as deﬁned in equations (3) and (4). Consequently, for r1 > 0,
A0
2(z)
A0
1(z)
−
A2(z)
A1(z)
=
A2(z)
A1(z)

r2
r1
− 1

≥ 0,
if, and only if r2 ≥ r1. Furthermore, for r1 > 0 we have
A00
2(z)
A00
1(z)
−
A0
2(z)
A0
1(z)
=
r2A2(z)
r1A1(z)

(r2 − 1)C2(z) + C3(z)
(r1 − 1)C2(z) + C3(z)
− 1

≥ 0
if, and only if r2 ≥ r1. 
In that sense the parameter h, j und k of the kurtosis transformations H, J und K can be
seen as kurtosis parameters.6 Summary
Within this work we derived conditions – based on the derivatives of density functions,
distribution functions, spread functions and general kurtosis and power kurtosis transfor-
mation – under which the kurtosis orderings of van Zwet and MacGillivray, respectively,
are preserved. In particular it was shown that the class of general power kurtosis trans-
formation is well suited to generate leptokurtic distributions which are characterized by a
well-deﬁned kurtosis parameter.
References
[1] Balanda, K. P. and H. L. MacGillivray: Kurtosis and spread. The Canadian Journal
of Statistics, 18(1):17-30, 1990.
[2] Fischer, M. and I. Klein: Kurtosis modelling by means of the j-transformation.
Diskussion paper 52, University of Erlangen-N¨ urnberg, 2003.
[3] Klein, I. and M. Fischer: Kurtosis ordering of the generalized secant hyperbolic dis-
tribution – A technical note. Diskussion paper 54, University of Erlangen-N¨ urnberg,
2003.
[4] Fischer, M., Horn, A. and I. Klein: Tukey-type distributions: The interplay between
skewness and kurtosis parameter in the context of ﬁnancial return data. Diskussion
paper 53, University of Erlangen-N¨ urnberg, 2003.
[5] Hoaglin, D. C.: Summarizing shape numerically: The g − and − h distributions, in:
Hoaglin, D. C. and F. Mosteller and J. W. Tukey (eds.): Data analysis for tables, trends,
and shapes, Wiley, New York, 461-513, 1983.
[6] MacGillivray, H. L.: Shape properties of the g-and-h Johnson families. Communica-
tion in Statistics (Theory and Methods), 21(5):1233-1250, 1992.
[7] MacGillivray, H. L. and K. P. Belanda: The relationship between skewness and kur-
tosis. Australian Journal of Statistics, 30(3):319-337, 1988.
[8] Rayner, G.D.andH.L.MacGillivray: Weightedquantile-basedestimationforaclass
of transformation distributions. Computational Statistics & Data Analysis, 39:401-
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[9] Tukey, J. W.: Exploratory data analysis. Addison-Wesley, Reading, M.A., 1977.
[10] Van Zwet, W. R.: Convex Transformations of Random Variables. Mathematical Cen-
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[11] Vaughan, D. C.: The Generalized Hyperbolic Secant distribution and its Applica-
tion. Communications in Statistics – Theory and Methods, 31(2):219-238, 2002.Adresses of the authors:
Prof. Dr. Ingo Klein
Lehrstuhl f¨ ur Statistik und ¨ Okonometrie
Universit¨ at Erlangen-N¨ urnberg
Lange Gasse 20
D-90403 N¨ urnberg
Tel. +60 911 5320271
Fax +60 911 5320277
Elec. Mail: Ingo.Klein@wiso.uni-erlangen.de
http://www.statistik.wiso.uni-erlangen.de
Dr. Matthias Fischer
Lehrstuhl f¨ ur Statistik und ¨ Okonometrie
Universit¨ at Erlangen-N¨ urnberg
Lange Gasse 20
D-90403 N¨ urnberg
Tel. +60 911 5320271
Fax +60 911 5320277
Elec. Mail: Matthias.Fischer@wiso.uni-erlangen.de
http://www.statistik.wiso.uni-erlangen.de

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Exploratory data analysis.

Kurtosis and spread.

Kurtosis modelling by means of the j-transformation. Diskussion paper 52, University of Erlangen-N¨ urnberg,

Kurtosis ordering of the generalized secant hyperbolic distribution – A technical note. Diskussion paper 54, University of Erlangen-N¨ urnberg,

Shape properties of the g-and-h Johnson families.

Summarizing shape numerically: The g −

The Generalized Hyperbolic Secant distribution and its Application. Communications in Statistics – Theory and Methods, 31(2):219-238, 2002.Adresses of the authors:

Tukey-type distributions: The interplay between skewness and kurtosis parameter in the context of ﬁnancial return data. Diskussion paper 53, University of Erlangen-N¨ urnberg,

Weightedquantile-basedestimationforaclass of transformation distributions.

Kurtosis transformation and kurtosis ordering

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Kurtosis transformation and kurtosis ordering

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