Two major generalizations of the hyperbolic secant distribution have been proposed in the statistical literature which both introduce an additional parameter that governs the kurtosis of the generalized distribution. The generalized hyperbolic secant (GHS) distribution was introduced by Harkness and Harkness (1968) who considered the p-th convolution of a hyperbolic secant distribution. Another generalization, the so-called generalized secant hyperbolic (GSH) distribution was recently suggested by Vaughan 2002). In contrast to the GHS distribution, the cumulative and inverse cumulative distribution function of the GSH distribution are available in closedform expressions. We use this property to proof that the additional shape parameter of the GSH distribution is actually a kurtosis parameter in the sense of van Zwet (1964). --kurtosis ordering,hyperbolic secant distribution

Fischer, Matthias J.

Klein, Ingo

English

Research Papers in Economics

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Klein, Ingo; Fischer, Matthias J.
Working Paper
Kurtosis ordering of the generalized
secant hyperbolic distribution: a
technical note
Diskussionspapiere // Friedrich-Alexander-Universität Erlangen-Nürnberg, Lehrstuhl für
Statistik und Ökonometrie, No. 54/2003
Provided in cooperation with:
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
Suggested citation: Klein, Ingo; Fischer, Matthias J. (2003) : Kurtosis ordering of the generalized
secant hyperbolic distribution: a technical note, Diskussionspapiere // Friedrich-Alexander-
Universität Erlangen-Nürnberg, Lehrstuhl für Statistik und Ökonometrie, No. 54/2003, http://
hdl.handle.net/10419/29581Friedrich-Alexander-Universit¨ at
Erlangen-N¨ urnberg
Wirtschafts-und Sozialwissenschaftliche Fakult¨ at
Diskussionspapier
54 / 2003
Kurtosis ordering of the generalized secant hyperbolic
distribution — A technical note
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 ordering of the generalized secant hyperbolic
distribution – A technical note
Ingo Klein and Matthias Fischer
Department of Statistics and Econometrics,
University of Erlangen-Nuremberg
Abstract: Two major generalizations of the hyperbolic secant distribution
have been proposed in the statistical literature which both introduce an addi-
tional parameter that governs the kurtosis of the generalized distribution. The
generalized hyperbolic secant (GHS) distribution was introduced by Hark-
ness and Harkness (1968) who considered the p-th convolution of a hyper-
bolic secant distribution. Another generalization, the so-called generalized
secant hyperbolic (GSH) distribution was recently suggested by Vaughan
(2002). In contrast to the GHS distribution, the cumulative and inverse cu-
mulative distribution function of the GSH distribution are available in closed-
form expressions. We use this property to proof that the additional shape pa-
rameter of the GSH distribution is actually a kurtosis parameter in the sense
of van Zwet (1964).
Keywords: kurtosis ordering; hyperbolic secant distribution.
1 Introduction
The hyperbolic secant distribution — which was studied by Baten (1934) and Talacko
(1956) — has not received sufﬁcient attention in the literature, although it has a lot of
nice properties: All moments and the moment-generating function exist, it is reproduc-
tive (i.e. the class is preserved under convolution) and both the cumulative and the inverse
cumulative distribution function admit a closed form. In addition, the hyperbolic secant
distribution exhibits more leptokurtosis than the normal and even more than the logistic
distribution. Nevertheless, generalizations have been proposed that introduce an addi-
tional parameter which increase the ”kurtosis” of the hyperbolic secant distribution. At
ﬁrst, Baten (1934) discussed the sum of n independent random variables, each having the
same hyperbolic secant distribution. More general, Harkness and Harkness (1968) inves-
tigated the p-th convolution of hyperbolic secant variables for arbitrary positive p > 0
which, unfortunately, doesn’t admit a closed-form representation for the cumulative and
the inverse cumulative distribution function. Recently, Vaughan (2002) suggested a fam-
ily of symmetric distributions, the so-called generalized secant hyperbolic (GSH) dis-
tribution. This family includes both the hyperbolic secant and the logistic distribution.
Moreover, it closely approximates the Student t-distribution with corresponding kurtosis.
In addition, the moment-generating function and all moments exist, and the cumulative
and inverse cumulative distribution are again available in closed form. The range of ”kur-
tosis” — measured by the fourth standardized moments — varies from 1.8 to inﬁnity.
Within this note we proof that the additional parameter of the GSH distribution is indeed
a kurtosis parameter which preserves the kurtosis ordering of van Zwet (1964).2 The generalized secant hyperbolic (GSH) distribution
The above-mentioned standard 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 (1)
with
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 density from (1) is chosen so that X ∼ fGSH(x) has zero mean and unit variance,
the range of the ”kurtosis parameter” t is ∈ (−π,∞). The GSH distribution includes the
logistic distribution (t = 0) and the hyperbolic secant distribution (t = −π/2) as special
cases and the uniform distribution on (−
√
3,
√
3) as limiting case for t → ∞. Vaughan
(2002) derived, amongst other properties, the cumulative distribution function, depending
on the parameter t, given by
FGSH(x;t) =

   
   
1 + 1
t arccot

−
exp(c2(t)x)+cos(t)
sin(t)

for t ∈ (−π,0),
exp(πx/
√
3)
1+exp(πx/
√
3) for t = 0,
1 − 1
tarccoth

exp(c2(t)x)+cosh(t)
sinh(t)

for t > 0.
the inverse distribution function, given 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.
However, it was not proved that the ”kurtosis parameter” t is actually a kurtosis parameter
in the sense of van Zwet(1964). This will be done in the next section.
3 GSH distribution and kurtosis ordering
Van Zwet (1964) introduced a partial ordering of kurtosis S on the set of symmetric
distribution functions Fs. Let F,G ∈ Fs and µF denote the location of symmetry of F,
then S is deﬁned by
(A) F S G : ⇐⇒ G
−1(F(x)) is convex for x > µF
and means that G has higher kurtosis than F. Balanda and MacGillivray (1990) gener-
alized 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.In the sense of Balanda and MacGillivray (1990), an arbitrary continuous, monotone in-
creasing distribution function F has less kurtosis than an equally distribution function G
if
(B) F S G : ⇐⇒ SG(S
−1
F (x)) is convex for x > F
−1(0.5).
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. This means that the spread function essentially coincides with the quantile function.
It can be shown that (A) and (B) coincide in this case. The aim of this note is to prove
that the parameter t from (1) is a kurtosis parameter in the sense of van Zwet (1964).
Proposition 3.1 (Kurtosis ordering) Assume that X1 (X2) follows a generalized secant
hyperbolic distribution with parameter t1 (t2) and corresponding cumulative distribution
functions F1 (F2). If t1 < t2, then F2 S F1, i.e. F1 has higher kurtosis than F2 (in the
sense of van Zwet).
Proof: To prove this result, we distinguish between 4 cases:
Case 1: −π ≤ t1 < t2 < 0,
Case 2: −π ≤ t1, t2 = 0,
Case 3: t1 = 0, t2 > 0,
Case 4: 0 < t1 < t2
and refer to transitivity which was shown by Oja (1981, Theorem 5.1). According to
equation (A), we have to show that
F
−1
1 (F2(u)) is convex for 1/2 ≤ u < 1
or, equivalently,
A(u) ≡
∂F
−1
1 (u)/∂u
∂F
−1
2 (u)/∂u
is strictly monotone increasing for 1/2 ≤ u < 1.
Case 1: Assume −π ≤ t1 < t2 < 0.
Preliminary remarks: If −π ≤ t1 < t2 < 0 and 1/2 ≤ u < 1, then t1(u − 1/2) ∈
[−π/2,0] and t2(u − 1/2) ∈ [−π/2,0] . Moreover, both sin(x) and cos(x) are strictly
monotone increasing on [−π/2,0].
Part 1: Paraphrasing A(u). Firstly,
∂F
−1
i (u)
∂u
= ti/c2(ti)
h
cot(tiu) + cot(ti(1 − u))
i
for 0 < u < 1,i = 1,2.
Consequently,
A(u) =
t1/c2(t1)
t2/c2(t2)
·
cot(t1u) + cot(t1(1 − u))
cot(t2u) + cot(t2(1 − u))
.Using cot(α) + cot(β) =
sin(α+β)
sin(α)sin(β) (see Bronstein and Semendjajew, [2.5.2.1.1]),
A(u) =
t1/c2(t1)
t2/c2(t2)
·
sin(t1)
sin(t1u)sin(t1(1−u))
sin(t2)
sin(t2u)sin(t2(1−u))
=
t1/c2(t1)sin(t1)
t2/c2(t2)sin(t2)
·
sin(t2u)sin(t2(1 − u))
sin(t1u)sin(t1(1 − u))
.
Now, because π ≤ t1 < t2 < 0, sin(t1) < 0 and sin(t2) < 0. Hence,
K(t1,t2) ≡
t1/c2(t1)sin(t1)
t2/c2(t2)sin(t2)
> 0.
Finally, using sin(α)sin(β) = 1/2(cos(α−β)−cos(α+β)) (see Bronstein and Semend-
jajew, [2.5.2.1.1]),
A(u) = K(t1,t2) ·
cos(2t2(u − 1/2)) − cos(t2)
cos(2t1(u − 1/2)) − cos(t1)
> 0,
because cos(x) is strictly monotone increasing for x ∈ [−π,0).
Part 2: Proof of the convexity. We have to show that A(u) is strictly monotone increasing
on [1/2,1). This is true, if A0(u) > 0 for [1/2,1):
A
0(u) =
K(t1,t2)
N
n
− sin(2t2(u − 1/2))2t2

·

cos(2t1(u − 1/2)) − cos(t1)

−

− sin(2t1(u − 1/2))2t1

·

cos(2t2(u − 1/2)) − cos(t2)
o
with N ≡ [cos(2t1(u − 1/2)) − cos(t1)]2 > 0. Using the monotony of the cosinus on
[−π,0),
−2t2(cos(2t1(u − 1/2)) − cos(t1)) > 0 and − 2t1(cos(2t2(u − 1/2)) − cos(t2)) > 0
for 1/2 ≤ u < 1 and −π ≤ t1 < t2 < 0. Deﬁning
K
∗(t1,t2,u) ≡ min
n
−2t2(cos(2t1(u−0.5))−cos(t1));−2t1(cos(2t2(u−0.5))−cos(t2))
o
,
we get
A
0(u) ≥
K(t1,t2)K∗(t1,t2,u)
N

sin(2t2(u − 1/2)) − sin(2t1(u − 1/2))

=
K(t1,t2)K∗(t1,t2,u)
N

2sin(t2(u − 1/2))cos(t2(u − 1/2))
− 2sin(t1(u − 1/2))cos(t1(u − 1/2))

,
where we used sin(2α) = 2sin(α)cos(α) (see Gradshteyn and Ryzhik, [1.333.1]). Ac-
cording to the preliminary remarks,
sin(t2(u − 1/2))cos(t2(u − 1/2)) − sin(t1(u − 1/2))cos(t1(u − 1/2)) > 0for −π ≤ t1 < t2 < 0 and 1/2 ≤ u < 1 implying that A0(u) > 0 for 1/2 ≤ u < 1.
Case 2: Assume t1 ∈ [−π,0) and t2 = 0.
The inverse distribution function of a GSH variable with t2 = 0 is given by
F
−1
2 (u) =
√
3
π
ln

u
1 − u

, 0 < u < 1.
Consequently, for 0 < u < 1,
∂F
−1
2 (u)
∂u
=
√
3
π

1
u
+
1
1 − u

=
√
3
π

1
u(1 − u)

.
Thus, for −π < t2 < 0 and 0 < u < 1,
A(u) =
∂F
−1
1 (u)/∂u
∂F
−1
2 (u)/∂u
=
t1/c2(t1)[cot(t1u) + cot(t1(1 − u))]
√
3/π · 1/(u(1 − u))
=
t1/c2(t1)
√
3/π
sin(t1)(u(1 − u))
sin(t1u)sin(t1(1 − u))
=
π sin(t1)
c2(t1)t1
√
3
·
(t1u)(t1(1 − u))
sin(t1u)sin(t1(1 − u))
.
The ﬁrst derivation is given by
A
0(u) = K(t1)

t1 sin(t1u) − (t1)2ucos(t1u)
sin(t1u)2 ·
t1(1 − u)
sin(t1(1 − u))
+
−t1 sin(t1(1 − u)) + (t1)2(1 − u)cos(t1(1 − u))
sin(t1(1 − u))2 ·
t1u
sin(t1u)

=
K(t1)t1t1ut1(1 − u)
sin(t1u)sin(t1(1 − u))
h 1
t1u
− cot(t1u)
i
−
h 1
t1(1 − u)
− cot(t1(1 − u))
i
for −π < t1 < 0 and 0 0.
Note that a series expansion to the cot(x) for 0 < |x| < π (see Gradshteyn and Ryzhik,
[1.441.7]) is given by
cot(x) =
1
x
−
∞ X
i=1
22i|B2i|
(2i)!
x
2i−1, (2)
where Bi, i = 1,2,..., denotes the numbers of Bernoulli. Applying (2) for x = t1u and
x = t1(1 − u),
A
0(u) = K(t1)t1
t1ut1(1 − u)
sin(t1u)sin(t1(1 − u))
 
∞ X
i=1
22i|B2i|
(2i)!
(t1u)
2i−1 −
∞ X
i=1
22i|B2i|
(2i)!
(t1(1 − u))
2i−1
!
.
For −π < t1 < 0,
K(t1)t1
t1ut1(1 − u)
sin(t1u)sin(t1(1 − u))
< 0.The term in brackets is negative because t1(1−u) > t1u and (t1(1−u))2i−1 > (t1u)2i−1,
i = 1,2,... for 1/2 0 for
−π < t1 < 0 and 1/2 t1.
As calculated above,
∂F
−1
2 (u)
∂u
=
t2
c2(t2)
·
h
coth(t2u) + coth(t2(1 − u))
i
, 0 0 for t2 > 0. It now follows that
A(u) =
∂F
−1
1 (u)/∂u
∂F
−1
2 (u)/∂u
=
√
3/π
t2/c2(t2)
·
1/(u(1 − u))
coth(t2u) + coth(t2(1 − u))
=
√
3t2/π
sinh(t2)/c2(t2)
·
sinh(t2u)sinh(t2(1 − u))
t2ut2(1 − u)
.
Deﬁning
K(t2) ≡
√
3t2/π
sinh(t2)/c2(t2)
> 0 for t2 > 0,
then – analogue to case 2, but now using hyperbolic functions – for t2 > 0,
A
0(u) = K(t2)t2 ·
sinh(t2u)sinh(t2(1 − u))
t2ut2(1 − u)
·
h
coth(t2u) − 1/(t2u)
i
−
h
coth(t2(1 − u)) − 1/(t2(1 − u))
i
Now deﬁne z(x) ≡ coth(x) − 1/x which is strictly monotone increasing for x > 0,
because by means of sinh(x) > x for x > 0,
z
0(x) = −
1
(sinh(x))2 +
1
x2 > 0.
Hence, from t2(1 − u) < t2u for 1/2 0 follows
h
coth(t2u) − 1/(t2u)
i
−
h
coth(t2(1 − u)) − 1/(t2(1 − u))
i
> 0,
This completes the proof of case 3.Case 4: Assume t1 > 0 and t2 > t1.
Similar to case 1 we have for 0 < u < 1
A(u) =
t1/c2(t1)
t2/c2(t2)
·
sinh(t1)
sinh(t1u)sin(t1(1−u))
sinh(t2)
sin(t2u)sinh(t2(1−u))
=
t1/c2(t1)sinh(t1)
t2/c2(t2)sinh(t2)
·
sinh(t2u)sinh(t2(1 − u))
sinh(t1u)sinh(t1(1 − u))
.
Deﬁning
K(t1,t2) ≡
t1/c2(t1)
t2/c2(t2)
sinh(t1)
sinh(t2)
> 0,
the ﬁrst derivative of A is given by
A
0(u) = K(t1,t2) ·
sinh(t2u)sinh(t2(1 − u))
sinh(t1u)sinh(t1(1 − u))
·
h
t2(coth(t2u) − coth(t2(1 − u))) − t1(coth(t1u) − coth(t1(1 − u)))
i
.
Now,
z(t) = coth(tu) − coth(t(1 − u)) is stictly monotone increasing for t > 0, (3)
if the ﬁrst derivative z0(t) is positive: For t > 0, 1/2 0 ⇐⇒
t(1 − u)
[sinh(t(1 − u)))]2 −
tu
[sinh(tu)]2 > 0 ⇐⇒
1 −
tu
t(1 − u)
[sinh(t(1 − u))]2
[sinh(tu)]2 > 0 ⇐⇒
[sinh(t(1−u))]2
t(1−u)
[sinh(tu)]2
tu
< 1.
To prove the last inequality, we show that f(x) =
[sinh(x)]2
x is monotone increasing for
x > 0. Using x ≥ tanh(x),
f
0(x) =
2sinh(x)cosh(x)x − [sinh(x)]2
x2 =
[cosh(x)]2
x2
 
2tanh(x)x − [tanh(x)]
2
≥
[cosh(x)]2
x2 tanh(x)x ≥ 0.
From equation (3) follows that
coth(t2u) − coth(t2(1 − u)) > coth(t1u) − coth(t1(1 − u))
and
t2
h
coth(t2u) − coth(t2(1 − u))
i
− t1
h
coth(t1u) − coth(t1(1 − u))
i
> 0.
Consequently, A0(u) > 0 for 1/2 < u < 1. References
[1] Balanda, K. P. and H. L. MacGillivray: Kurtosis and spread. The Canadian Journal
of Statistics, 18(1):17-30, 1990.
[2] Baten, W. D.: The Probability Law for the Sum of n Independent Variables, each
Subject to the Law (1/2h)sech(πx/2h). Bulletin of the American Mathematical Society,
40:284-290, 1934.
[3] Bronstein, I. N. and K. A. Semendjajew: Taschenbuch der Mathematik. Verlag Harri
Deutsch, Frankfurt(Main), 1980.
[4] Gradshteyn, I. S. and I. M. Ryzhik: Table of Integrals, Series, and Products. Aca-
demic Press, New York, 2000.
[5] Harkness, W. L. and M. L. Harkness: Generalized Hyperbolic Secant Distributions.
Journal of the American Statistical Association, 63:329-337, 1968.
[6] Oja, H.: On location, scale, skewness and kurtosis of univariate distributions. Scan-
dinavian Journal of Statistics 8, 154–168.
[7] Van Zwet, W. R.: Convex Transformations of Random Variables. Mathematical Cen-
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[8] Vaughan, D. C.: The Generalized Hyperbolic Secant distribution and its Application.
Communications in Statistics – Theory and Methods, 31(2):219-238, 2002.
Adress 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

Convex Transformations of Random Variables.

Generalized Hyperbolic Secant Distributions.

Kurtosis and spread.

On location, scale, skewness and kurtosis of univariate distributions.

Ryzhik: Table of Integrals, Series, and Products.

Semendjajew: Taschenbuch der

The Generalized Hyperbolic Secant distribution and its Application. Communications in Statistics – Theory and Methods,

The Probability Law for the Sum of n Independent Variables, each Subject to the Law (1/2h)sech(πx/2h).

Kurtosis ordering of the generalized secant hyperbolic distribution: a technical note

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Kurtosis ordering of the generalized secant hyperbolic distribution: a technical note

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