DBJ Discussion Paper Series, No.1401

Family of the Generalized Gamma Kernels:
A Generator of Asymmetric Kernels
for Nonnegative Data

Masayuki Hirukawa(Setsunan University)
Mari Sakudo(Development Bank of Japan)

June 2014

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in orderto protect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Family of the Generalized Gamma Kernels:
A Generator of Asymmetric Kernels
for Nonnegative Data
Masayuki Hirukaway
Setsunan University

Mari Sakudoz
Development Bank of JapanFirst Draft: May 2013This Version: June 2014
Abstract
Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversi…ed approaches. This paper proposes a family of theGeneralized Gamma kernels that is built on the probability density functionof the Generalized Gamma distribution (Stacy, E. W. (1962), ‘A Generalizationof the Gamma Distribution’, Annals of Mathematical Statistics, 33, 1187-1192)
and a few common conditions. The family can generate asymmetric kernels
that share appealing properties with the Modi…ed Gamma kernel (Chen, S. X.
(2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of the Institute of Statistical Mathematics, 52, 471-480). Asymptotics on
the kernels generated from the family can be delivered by manipulating theconditions directly, as with symmetric kernels.
Keywords: Asymmetric kernel; bias reduction; boundary correction; density estimation; kernel smoothing; Generalized Gamma kernels.
MSC 2010 Codes: 62G07; 62G20.

We would like to thank the editor Irène Gijbels and two anonymous referees for their commentsthat have substantially improved this paper. We are also grateful to Jin Seo Cho, Benedikt Funke,
Bruce Hansen, Susumu Imai, Yoshihide Kakizawa, Yoshihiko Nishiyama, Katsumi Shimotsu, andparticipants at Summer Workshop on Economic Theory 2012, New Zealand Econometric Study
Group Meeting 2013, and seminars at the Development Bank of Japan, Kyoto University, the University of Sydney and the University of Tokyo for constructive comments and suggestions. The …rst
author gratefully acknowledges …nancial support from Japan Society of the Promotion of Science(grant number 23530259). The views expressed herein and those of the authors do not necessarilyre‡ect the views of the Development Bank of Japan.
yCorresponding Author. Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi,
Neyagawa, Osaka, 572-8508, Japan; e-mail: hirukawa@econ.setsunan.ac.jp; phone: (+81)72-8398095; fax: (+81)72-839-8138.
z
Research Institute of Capital Formation, Development Bank of Japan, 9-7, Otemachi 1-chome,
Chiyoda-ku, Tokyo 100-8178, Japan; e-mail: marisakudo@gmail.com; phone (+81)3-3244-1241; fax(+81)3-3270-7084.

1

Introduction

Researchers and policy makers are often interested in the distributions of nonnegative economic and …nancial variables including incomes, wages, short-term interest
rates, and insurance claims. The distributions are empirically characterized by two
stylized facts, namely, (i) a natural boundary at the origin and (ii) concentrationof observations near the boundary and a long tail with sparse data.

When theirparametric speci…cations are found to be inappropriate, we resort to nonparametricdensity estimation.

However, in estimating the distributions with symmetric ker-nels, we must make two distinct modi…cations for the standard smoothing techniquesimultaneously. For (i), boundary correction methods should be employed; see, forinstance, Section 3 of Karunamuni and Alberts (2005) for a concise review of themethods.

For (ii), adaptive smoothing such as variable bandwidth methods (e.g.

Abramson, 1982) would be a remedy.
Recently, asymmetric kernels with support on [0; 1) (e.g. Chen, 2000; Jin and
Kawczak, 2003; Scaillet, 2004) have emerged as a viable alternative that can accommodate the stylized facts. Although the kernels are relatively new in the literature,
several papers report favorable evidence from applying them to empirical models
in economics and …nance; see, for instance, Section 1 of Gospodinov and Hirukawa(2012) for a non-exhaustive list of the papers. The reason why asymmetric kernels
tend to work well for the distributions with two stylized facts is their property as a
combination of a boundary correction device and adaptive smoothing that has e¤ectsimilar to the variable bandwidth methods.
A cumbersome aspect of asymmetric kernels, however, is that so far exploring their
asymptotic properties has relied on kernel-speci…c and thus diversi…ed approaches.
In contrast, symmetric kernels are built on a set of common conditions, and their
1asymptotic properties can be implied straightforwardly by the conditions. Then, thispaper proposes a new class of asymmetric kernels that consists of a speci…c functional
form and a set of common conditions. Our aim is that as long as an asymmetric kernel
is generated from this class, we can deliver its asymptotic properties by manipulating
the conditions directly, rather than employing the techniques peculiar to the kernel.
A key issue of establishing the new class of asymmetric kernels is to choose theshape or functional form. While an apparent shape restriction (i.e. symmetry about
the origin) is imposed on symmetric kernels, there is no guidance on shapes of asymmetric kernels; actually, any shapes are admissible as long as the kernels have support
on [0; 1).

Recognizing that most of the papers on asymmetric kernels report su-

perior …nite sample performance of the Modi…ed Gamma kernel by Chen (2000), we
consider the kernel a reasonable benchmark of the functional form.

Accordingly,

among ‘close-cousins’of the Gamma probability density function (“pdf”), the pdf ofthe Generalized Gamma distribution (Stacy, 1962) is chosen as the functional form.
Combining the pdf with a set of common conditions, we …nally de…ne a family of theGeneralized Gamma (“GG”) kernels. As special cases, the GG kernels nest not only
the Modi…ed Gamma kernel but also the newly proposed Weibull and Nakagami-mkernels, which are built on the Weibull and Nakagami-m (Nakagami, 1943, 1960)
pdfs, respectively.
The GG kernels are designed to preserve all appealing properties that the Modi…ed
Gamma kernel possesses. First, by construction, the GG kernels are free of boundary bias and always generate nonnegative density estimates everywhere.

Second,when best implemented, each GG density estimator attains Stone’s (1980) optimal
convergence rate in the mean integrated squared error within the class of nonnegative
kernel density estimators. Third, the leading bias over the interior region for each GG

2density estimator contains only the second-order derivative of the true density overthe interior region, whereas the one for density estimators using some other asymmetric kernels (e.g. the Gamma kernel by Chen, 2000) consists of terms involving the…rst- and second-order density derivatives. Fourth, the variance of the GG estimatortends to decrease as the design point moves away from the boundary. This property
is particularly advantageous to estimating the distributions that have a long tail withsparse data. It is worth emphasizing that these properties can be demonstrated bymanipulating the common conditions directly.
This paper also demonstrates three additional properties of the GG density estimator. While the …rst property, applicability of two multiplicative bias correction
techniques studied in Hirukawa (2010) and Hirukawa and Sakudo (2014), is based on
random sampling, the remaining two properties allow for dependent sampling. Afterthe GG density estimator using weakly dependent data is shown to admit the same…rst-order bias and variance approximations as those using independent observations,
it is proven that the density estimator is consistent even when the true density becomes unbounded at the boundary. The Gamma and Modi…ed Gamma kernels are
known to share all three properties, and thus it is demonstrated that all in all, theGG kernels are endowed with the same attractive properties.
The remainder of this paper is organized as follows. Section 2 provides the de…nition of a family of the GG kernels and introduces three special cases. Convergenceproperties of the GG density estimator are also provided. Section 3 investigates threeappealing properties of the GG density estimator. Section 4 conducts Monte Carlosimulations to examine …nite sample properties of the GG density estimator. Section
5 summarizes the main results of the paper and suggests some research extensions.
Proofs of Theorems 1 and 2 are given in the Appendix. Besides, proofs of Theorems

34-6 and comprehensive simulation results are available on the …rst author’s webpage.
R1
This paper adopts the following notational conventions. (a) = 0 y a 1 exp ( y) dy(a > 0) is the gamma function; 1 f g signi…es an indicator function; and b c denotesdthe integer part. The expression ‘X = Y ’ reads “A random variable X obeys the
distribution Y .” The expression ‘Xn

Yn ’is used whenever Xn =Yn ! 1 as n ! 1.

Lastly, in order to describe di¤erent asymptotic properties of an asymmetric kernel estimator across positions of the design point x > 0, we denote by “interior x”
and “boundary x” a design point x that satis…es x=b ! 1 and x=b !

for some0<< 1 as n ! 1, respectively.

2

A Family of the Generalized Gamma Kernels

2.1

De…nition

Let Y be distributed by the GG distribution GG ( ; ; ). Then, Y has the pdf
p (y; ; ; ) =y1

exp f (y= ) g
1 fy
( = )

0g :

(1)

It is known that the m-th uncentered moment of Y is given by
E (Y m ) =f( + m) = g
:
( = )

m

(2)

We now provide the de…nition of a family of the GG kernels that consists of several
common conditions. Before proceeding, it could be bene…cial to relate the de…nition
to probability density estimation. Below we present a set of regularity conditions forasymmetric kernel density estimation.
Assumption 1. The random sample fXi gni=1 is drawn from a univariate distribution
with a probability density function f having support on [0; 1).
Assumption 2. f is twice continuously di¤erentiable.
4

Assumption 3. The smoothing parameter b (= bn > 0) satis…es b + (nb) 1 ! 0 as
n ! 1.
To generate a kernel from the pdf (1), we allow ( ; ; ) to be a function of
the design point x > 0 and the smoothing parameter b, as in Chen (2000), Jin
and Kawczak (2003) and Scaillet (2004).

To put it another way, whenever werefer to the triplet ( ; ; ), it should be interpreted as a short-handed notation of( ; ; ) = ( b (x) ;b (x) ;

b (x)) unless otherwise noted.

Let the crude GG kernel

Given the random sample fXi gni=1 , we have thePdensity estimator f^GG0 (x) = (1=n) ni=1 KGG0 (Xi ; x; b).
be KGG0 (u; x; b) := p (u; ; ; ).

d

Let #x = GG ( ; ; ). Under Assumptions 2-3, a second-order Taylor expansion
n
o
n
o
^
^
of E fGG0 (x) around #x = x yields E fGG0 (x) = f (x) + E (#x x) f 0 (x) +(1=2) E (#xx)2 f 00 (x) + o E (#xx)2 . It follows that unless E (#x ) = x exactly(at least for interior x), the leading bias of f^GG0 would contain the term involving f 0 ,which is less desirable.

Although there are numerous choices of ( ; ; ) that canachieve E (#x ) = x, we adopt the simplest resolution that we set
x and employ the pdf of GG ( ;= x for interior( = ) = f( + 1) = g ; ) (which can be obtainedby the change of variable Z := [ ( = ) = f( + 1) = g] Y in (1)) as the kernel. In
the end, we reach the following de…nition of a family of the GG kernels.

De…nition 1. Let ( ; ; ) = ( b (x) ;b (x) ;

3b (x)) 2 R+ be a continuous functionof the design point x and the smoothing parameter b. For such ( ; ; ), consider
the pdf of GG ( ;( = ) = f( + 1) = g ; ), i.e.

u 1 exp
KGG (u; x; b) = nu( )= ( +1 )
o

+1

=

5

1 fu

0g :

(3)

This pdf is said to be a family of the GG kernels if it satis…es each of the following
conditions:
Condition 1.

=x
for x C1 b
, where 0 < C1 < 1 is some constant,'b (x) for x 2 [0; C1 b)

the function 'b (x) satis…es C2 b

C3 b for some constants 0 < C2'b (x)

C3 < 1, and the connection between x and 'b (x) at x = C1 b is smooth.
Condition 2.

1, and for x 2 [0; C1 b),;

constant 1

satis…es 1

C4 < 1. Moreover, connections ofandC4 for some
at x = C1 b, if any,

are smooth.
1 + (C5 =x) b + o (b) for x C1 b
, for
O (1)
for x 2 [0; C1 b)( ) ( +2 )2 =f ( +1 )gsome constant 0 < jC5 j < 1.

Condition 3. Mb (x) :=Condition 4. Hb (x) := 21=( ) (2 )=( +1 ) ( 2 1 )( +1 )

1

1 + o (1) for interior x.
O (1)for boundary x
f (

1)+1

g

1

VI ( ) (xb) 2
Condition 5. Ab; (x) :=
(
1)+1
2
1
VB ( ) b1
f ( )g2 R+ , where constants 0 < VI ( ) ; VB ( ) < 1 depend only on .

for interior x ,for boundary x

Conditions 1 and 2 form a legitimate kernel from the GG pdf. It follows from'b (x) = O (b) uniformly on [0; C1 b) in Condition 1 that the kernel is well-de…ned in
the vicinity of the origin. Whilekernel near the origin,

1 in Condition 2 also ensures boundedness of the

1 controls the tail-behavior of the kernel and establishes

an exponential rate of the tail decay. The condition also allows for each of

and

to

be a piecewise function of (x; b) like , where the connection is made at x = C1 b; thecommon connection point simpli…es asymptotic analyses substantially. In addition,smooth connection requirements in Condition 1 and 2 are inspired by the constructionof the Modi…ed Gamma kernel; see p.473 of Chen (2000) for details.

6

Moreover,Condition 3 and Conditions 4 and 5 are the requirements for valid approximations tothe bias and variance of the GG density estimatorn
1X
f^GG (x) =KGG (Xi ; x; b) ;n i=1

respectively. Although all the common conditions appear to be high-level ones, it is
not hard to …nd a few special cases that satisfy them. Examples of the GG kernels
are provided shortly.

2.2

Convergence Properties of Probability Density Estimators Using the GG Kernels

2.2.1

Local Property

Bias-Variance Tradeo¤. Before providing examples of the GG kernels, readers
may wonder whether the kernels are a truly legitimate one, i.e. they can yield aconsistent density estimator.

To answer this question, we begin this section withpresenting the theorem on approximations to the bias and variance of f^GG (x).
Theorem 1. Under Assumptions 1-3, the bias of f^GG (x) can be approximated byon
B1 (x; f ) b, where
Bias f^GG (x)B1 (x; f ) =andb (x) = f'b (x)(C5 =2) xf 00 (x) for x C1 b;
0
for x 2 [0; C1 b)b (x) f (x)xg =b = O (1). On the other hand, the variance of f^GG (x)can be approximated byno^
V ar fGG (x)

(

p
1
nb1=2
VI (2) f (x) = x for interior x(nb) 1 VB (2) f (x)for boundary x:

The theorem states that Conditions 1-5 do lead to familiar properties of asymmetric kernel density estimators.1

By construction, f^GG (x) is free of boundary

1

Conditions 1-5 even establish approximations to the bias and variance of nonparametric regression estimators (e.g. local constant and local linear estimators) using the GG kernels.

7

bias and nonnegative everywhere. The bias of f^GG (x) is O (b), and its variance ison1
for interior x and O (nb) 1 for boundary x.
O nb1=2

Furthermore, as pointed out by Chen (2000) and Scaillet (2004), a unique featureof asymmetric kernel density estimators is that the variance coe¢ cient decreases as xpincreases. This is shared with f^GG (x) in that VI (2) f (x) = x decreases as the design
point x moves away from the boundary. The shrinking variance coe¢ cient re‡ects
that more data points can be pooled to smooth in areas with fewer observations.
This property is particularly advantageous to estimating the distributions that havea long tail with sparse data, such as those of the economic and …nancial variables
mentioned at the beginning of Section 1.
Mean Squared Error (“MSE”). For interior x, the MSE of f^GG (x) can be approximated byno^
M SE fGG (x)

b

2

C52
4

2

x2 ff 00 (x)g +VI (2) f (x)
p :
nb1=2
x

(4)

The smoothing parameter value that minimizes the right-hand side of (4) isVI (2) f (x)bGG =C52 ff 00 (x)g22=5

x 1 n 2=5 :(5)Observe that the MSE-optimal smoothing parameter is O n 2=5 = O (h 2 ), where
h is the MSE-optimal bandwidth for density estimators using nonnegative symmetrickernels. Therefore, when best implemented, the approximation to the MSE becomesnof^GG (x)5 2 1=5
2=5C
fVI (2)g4=5 ff 00 (x)g ff (x)g4=5 n 4=5 :4 5noObserve that M SE f^GG (x) depends only on f (x) and not on x itself.
M SE

(6)
The

optimal MSE of f^GG (x) for interior x becomes O n 4=5 , which is also the optimal
convergence rate in the MSE of nonnegative symmetric kernel density estimators.
8no
On the other hand, for boundary x, M SE f^GG (x) = O (b2 + n 1 b 1 ), which yields
the MSE-optimal smoothing parameter byGG = O n 1=3 and the optimal MSE of
O n 2=3 .
2.2.2

Global Property

The inferior rate in the optimal MSE of f^GG (x) for boundary x does not a¤ect its
R1
R1
pglobal property. If 0 fxf 00 (x)g2 dx and 0 ff (x) = xg dx are both …nite, thenapplying the trimming argument in Chen (2000, p.476) approximates the mean integrated squared error (“MISE”) of f^GG (x) asno
M ISE f^GG (x)

b

C52
4

2

Z 1
0

VI (2)
fxf (x)g dx + 1=2
nb
00

2

Z 1
0

f (x)
p dx:
x

(7)

The smoothing parameter value that minimizes the right-hand side of (7) is
#2=5
R1
p
VI (2) 0 ff (x) = xg dx
n 2=5 :
bGG =
R
2
2 1
00
C5 0 fxf (x)g dx"(8)Therefore, when best implemented, the approximation to the MISE becomes
n
o
^
M ISE
fGG (x)

5
C52
4

Z 1
0

1=5
2

00

fxf (x)g dx

VI (2)

Z 1
0

f (x)
p dx
x

4=5

n 4=5 :

Note that O n 4=5 is the optimal convergence rate of the MISE within the class ofnonnegative kernel estimators in Stone’s (1980) sense.
2.2.3

A Note on Implementation

Choosing the smoothing parameter b is an important practical issue.

Below weconsider a very simple, Silverman’s (1986) rule-of-thumb type choice rule.

Al-though in principle the rule is built on the MISE (7) as the criterion, we makea couple of modi…cations.
ence density.

First, the unknown f is replaced by a known refer-

For simplicity, we choose the pdf of G ( ; !) ( ; ! > 0), i.e. g (x) =

9

1

x

exp ( x=!) 1 fx

0g = f!

( )g, as the reference.

Second, the criterion ismodi…ed to the asymptotic weighted mean integrated squared error (“AWMISE”)no
AW M ISE f^GG (x) := b2

C52
4

Z 1
0

where the weighting function w (x)VI (2)
fxg (x)g w (x) dx+ 1=2
nb
2

00

Z 1
0

g (x)
p w (x) dx;x

0 must be chosen to ensure …niteness of two

integrals. Given the speci…cation of g (x), it turns out that w (x) = x3 ful…lls thisrequirement. Then, the AWMISE is simpli…ed tono
AW M ISE f^GG (x) = b2

C52 C (2 )
4 2( )

VI (2) ! 5=2 ( + 5=2)
( )

1
nb1=2

+

;

where
C

=

1
(
2)2 (
1)2 (
4
1
+ (3
4) (
1) ( )
2
1
1+( + 1)
+ ( )
4
2

2) (
1
2
3
+
2

1)2 ( )+(

1) ( )+

1
2( + 1):As a consequence, the AWMISE-optimal smoothing parameter is given by
bzz
GG =

4

1

VI (2) ! 5=2 ( ) ( + 5=2)
C52 C (2 )2=5

n 2=5 :

In practice, parameters ( ; !) need to be replaced by their method-of-moments estimates (^ ; !
^ ).2
It would be possible to choose b in a more sophisticated manner. For instance,
replacing two integrals in (7) with their nonparametric estimators using the GG kernels, we could derive an analog to the solve-the-equation plug-in method by Sheather
and Jones (1991). Alternatively, we may rely on the cross-validation method (e.g.
Bouezmarni and Rombouts, 2010, p.250). Investigating these methods is left for our
future research.
2

Simulation results of GG density estimators with the rule-of-thumb smoothing parameters
plugged in are available on the …rst author’s webpage.

10

2.3

Examples of the GG Kernels

This section introduces three special cases of the GG kernels. The sole reason why thethree kernels are listed as examples is that for each of these kernels, approximations
to the gamma functions that appear in Mb (x), Hb (x) and Ab; (x) in Conditions 3-5
are readily available; see the proof of Theorem 2 in Appendix for details.
It is worth emphasizing that there could be many other examples belonging tothis family.

As Stacy (1962, p.1187) states, for instance, functions of a standard

normal variate (e.g. its positive even powers, its modulus, and all positive powers of
its modulus) will generate the GG kernels. Moreover, for a given pdf, choices of thefunctional forms of ( ; ; ) may not be unique, as long as they satisfy Conditions
1-5; in other words, we could even generate two or more kernels from the same pdf
by making changes in ( ; ; ).
2.3.1

ExamplesModi…ed Gamma Kernel. The Modi…ed Gamma (“MG”) kernel in Chen (2000)
turns out to be an immediate example of the GG kernels. Put( x;x
for x 2bb2( ; )=
2
1 x+ 1; x4b + b for x 2 [0; 2b)4 band= 1 in (3). Then, it collapses to
KGG (u; x; b) =u

1

exp f u= ( = )g
1 fu
( = ) ( )

(9)

0g ;

which is the pdf of the Gamma distribution G ( ; = ). Observe that=b (x) in

Chen (2000, p.473) and = = b. It follows that the above kernel …nally reduces tothe MG kernel
KM G( b (x);b) (u) =u b (x) 1 exp ( u=b)
1 fu
b b (x) f b (x)g
11

0g :

Weibull Kernel. To obtain the Weibull (“W”) kernel, let
8 q
2x
<
;x
for x 2bb( ; )=: 1 x + 1; x2 + b for x 2 [0; 2b)
2 b
4b
and

=

in (3). Then, it becomes1u
KGG (u; x; b) =

exp

h n

u
= (1+1= )f = (1 + 1= )g

o i

1 fu

0g :

Because the right-hand side is the pdf of the Weibull distribution W ( ; = (1 + 1= )),the W kernel can be de…ned as3
KW ( ; = (1+1= )) (u)== (1 + 1= )1u
= (1 + 1= )u
= (1 + 1= )

exp

1 fu

0g :

Nakagami-m Kernel. Use exactly the same ( ; ) as for the MG kernel but put= 2 in (3). Then, it reduces to
2u

1

exp

KGG (u; x; b) =

h

u=
2

2
+1
2

=

2

which is the pdf of the Nakagami-m distribution N M
due to Nakagami (1943, 1960).4

2

+1
2

=

i

1 fu

=2; ( =2) [0g ;( =2) = f( + 1) =2g]2This distribution is frequently applied in tele-communications engineering as the distribution that can describe signal intensity of3This de…nition di¤ers from the one given in Kuruwita, Kulasekera and Padgett (2010, Table 1),
who construct their Weibull kernel from a di¤erent motivation. Apparently, their de…nition
KW (u; x; b) =

1
bx

u
x

1=b 1

exp

u
x

1=b

1 fu

0g

tends to be unbounded near the origin.
4
The pdf of N M ( ; !) (1=2; ! > 0) is
p (y; ; !) =

2
!

( )

y2

1

exp

!

y 2 1 fy0g :It is widely recognized that the Nakagami-m distribution was …rst proposed in Nakagami (1960).
In reality, however, the distribution originates as early as in Nakagami (1943).

12

short-wave fading. In the end, the Nakagami-m (“NM”) kernel is de…ned as
KN M ( =2;( =2)[
= h( =2)= f( +1)=2g]2 ) (u)

2
2

2

=

=2
2
+1
2

2

i =2

u2( 2 ) 1 exp
2

"

2
2

2

=

+1
2

2u

2

#

1 fu

FIGURE 1 ABOUT HERE
2.3.2

Asymptotic Results on Density Estimators Using the MG, W and
NM Kernels

Figure 1 plots the shapes of the MG, W and NM kernels at four di¤erent design
points (x = 0; 1; 2; 4) at which the smoothing is performed.

For reference, theGamma (“G”) kernel (Chen, 2000) is also drawn in each panel.5 It is worth notingthat for all plotted functions, the value of the smoothing parameter is …xed at b = 0:2.
When smoothing is made at the origin (Panel (a)), the NM kernel collapses to a halfnormal pdf, whereas all others reduce to an exponential pdf.

As the design point

moves away from the boundary (Panels (b)-(d)), the shape of each kernel varies andbecomes ‡atter; in other words, each kernel changes the amount of smoothing in a
locally adaptive manner.

It is worth emphasizing that unlike variable bandwidth

methods (e.g. Abramson, 1982), adaptive smoothing for these kernels is achieved bya single smoothing parameter, which makes them much more appealing in empiricalwork.

We can also observe that the more distant the design point is from theboundary, the closer the shapes become to a symmetric one; in particular, shapes of
the NM kernel become almost symmetric for a large x, as its functional form suggests.
Convergence properties of density estimators using the MG, W and NM kernels
are presented in Theorem 2 below.

Obviously, the functional form of ( ; ; ) for

5This kernel does not belong to the GG kernels, because it can be obtained by setting ( ; ) =(x=b + 1; x + b) in (9).

13

0g :

each of the three kernels satis…es Conditions 1-2. Hence, to demonstrate Theorem
2, it su¢ ces to check that Conditions 3-5 hold for each kernel.
Theorem 2. Let f^j (x) be the probability density estimator using the kernel j 2
fM G; W; N M g. Then, under Assumptions 1-3, the bias and variance of f^j (x) canno^be approximated by Bias fj (x)B1;j (x; f ) b andnoV ar f^j (x)

(

p
1
nb1=2
VI;j (2) f (x) = x for interior x(nb) 1 VB;j (2) f (x)for boundary x;

where VB;j (2) can be found in Appendix, and values of C5;j and explicit forms of
B1;j (x; f ) and VI;j (2) are as follows:

j
MG
W
NM
Moreover,B1;j (x; f )x 2bx 2 [0; 2b)000(1=2) xf (x)b (x) f (x)( 2 =24) xf 00 (x) b (x) f 0 (x)0(1=4) xf 00 (x)b (x) f (x)

C5;j
1
2
=12
1=2

b (x) = f(1=2) (x=b)

VI;j (2)p
1= (2 p )
1= p
2 2
1= 2

1g2 = O (1) in this case.

It follows from (5) and Theorem 2 that the MSE-optimal smoothing parametersof three kernel density estimators for interior x arebM G =bW =bN M =

"

1
f (x)
p
2
ff 00 (x)g2
2 3=2
23=2
p

12
2

2

2=5

x 1 n 2=5 ;f (x)ff 00 (x)g2f (x)ff 00 (x)g2#2=5

x 1 n 2=5 ;2=5x 1 n 2=5 :Recognize that bN M = 2bM G holds; even bN M = 2bM G is the case by (8), wherebN M and bM G are MISE-optimal smoothing parameters of the NM and MG kernels,
respectively. Then, by (6), the optimal-MSEs of f^M G (x) and f^N M (x) for interior x

14

have the relation
M SE

n
o
f^M G (x)

M SE
5

n
o
f^N M (x)2=51

ff (x)g4=5 ff 00 (x)g

47=5

2=5

n 4=5 :

(10)

The right-hand side is also the optimal-MSE of the density estimator using the Gaussian kernel. In other words, when best implemented, density estimators using thesekernels become …rst-order asymptotically equivalent, and both kernels on [0; 1) arein a sense equivalent to the Gaussian kernel on ( 1; 1). In contrast, when best
implemented, the MSE of f^W (x) for interior x can be approximated byon
M SE f^W (x)

5

2

47=5

24

2=5
2=5

ff (x)g4=5 ff 00 (x)gn 4=5 :

Comparing the factors of (10) and (11) reveals that 5=47=5 (1= )2=5
and 5=47=5 ( 2 =24)

2=5

(11)
0:454178 : : :0:503178 : : : Therefore, we can see that f^W (x) is slightly

ine¢ cient than f^M G (x) and f^N M (x) under the best-case scenario.

3

Properties of the GG Density Estimator

This section explores three properties of the GG density estimator other than thosedescribed in Section 2.2.

After demonstrating applicability of multiplicative bias

correction techniques for independent observations, we show the remaining two properties for weakly dependent observations, namely, validity of the …rst-order bias andvariance approximations in GG density estimation that are stated in Theorem 1, andweak consistency of GG density estimators when the true density is unbounded at theorigin. It is known that the G and MG kernels possess all these properties. Therefore, this section establishes that these appealing properties essentially inhere in the
GG kernels.

15

3.1

Nonparametric Multiplicative Bias Correction for GG
Density Estimation

We start with examining whether GG density estimators in general admit multiplicative bias correction (“MBC”) techniques. While Hagmann and Scaillet (2007) and
Gustafsson et al. (2009) propose semiparametric MBC procedures using asymmetrickernels, we concentrate on two classes of nonparametric MBC methods studied in
Hirukawa (2010) and Hirukawa and Sakudo (2014). The …rst class of MBC estimators, originally proposed by Terrell and Scott (1980), is de…ned asno 11 c n
o 1cc
f~T S;GG (x) = f^GG;b (x)f^GG;b=c (x);where, f^GG;b (x) and f^GG;b=c (x) signify GG density estimators using smoothing parameters b and b=c, and c 2 (0; 1) is some predetermined constant that does not dependon the design point x.

The second class of MBC estimators due to Jones, Lintonnoand Nielsen (1995) is implied by the identity f (x) = f^GG;b (x) f (x) =f^GG;b (x) andde…ned as

f~JLN;GG (x) = f^GG;b (x)(1 X KGG (Xi ; x; b)n i=1 f^GG;b (Xi )n):Both f~T S;GG (x) and f~JLN;GG (x) always generate nonnegative density estimates everywhere by construction.
To develop convergence properties of MBC estimators, we modify Assumptions
2-3 as follows. Discussions on these assumptions can be found in Hirukawa (2010)
and Hirukawa and Sakudo (2014).
Assumption 2a. f has four continuous and bounded derivatives, and f (x) > 0 for
a given design point x > 0.
Assumption 3a. The smoothing parameter b satis…es b + (nb3 )16

1! 0 as n ! 1.

The next theorem refers to acceleration in bias convergence via the bias correction
methods. The proof is similar to the ones for Theorems 1-2 of Hirukawa and Sakudo(2014), and this it is omitted.
noTheorem 3. If Assumptions 1, 2a and 3a hold, and E f^GG (x) admits the expanno^sion E fGG (x) = f (x) + B1 (x; f ) b + B2 (x; f ) b2 + o (b2 ), where B1 (x; f ) (which

can be found in Theorem 1) and B2 (x; f ) are kernel-speci…c functions depending on
x and derivatives of f , then the bias and variance of f~T S;GG (x) can be approximatedbyon~Bias fT S;GG (x)noV ar f~T S;GG (x)
=

1
c (1
(

c)

1 B12 (x; f )B2 (x; f ) b2 ;c (1 c) 2
f (x)
n
o
p
1
for interior x(c) VI (2) f (x) = x + o nb1=2

p (x) b2 :=

nb1=2

1

O (nb) 1

1

for boundary x

where
(c) =

1 + c5=2 (1 + c)1=2(1 + c)1=2 (1p
2 2c3=2c)2:Moreover, the bias and variance of f~JLN;GG (x) can be approximated bynoBias f~T S;GG (x)on~V ar fJLN;GG (x)=q (x) b2 := f (x) B1 (x; h) b2 ;
n
(
p
1
1=2
nb
VI (2) f (x) = x + o nb1=2
O (nb)

11ofor interior x
for boundary x

where B1 (x; h) is obtained by replacing f in B1 (x; f ) with h = h (x; f ) := B1 (x; f ) =f (x).

As the theorem suggests, whether the two MBC techniques may accelerate thebias convergence from O (b) to O (b2 ) depends crucially on whether the second-ordernoterm in Bias f^GG (x) is O (b2 ). It is worth noting that Conditions 1-5 provide

no guidance on the order of magnitude in the second-order bias term. For instance,no
as indicated in the proof of Theorem 2, the second-order term in Bias f^W (x) is
17

;

;

O b3=2 . Because both MBC techniques merely improve the bias convergence up toO b3=2 , the theorem excludes such inferior cases.6
In contrast, for the MG and NM kernels, each MBC technique ameliorates theirbias convergence to O (b2 ). Explicit forms of p (x) and q (x) for the NM kernel are
8no2
x2 ff 00 (x)g
1
1 0
x 00
x2 000
>
f
(x)
+
f
(x)
+
f
(x)
for x 2b> 32 f (x)
8
2
3
4
>
>
)#
(
< "
b (x)+x=bb (x)+x=b +1
2
0
2
2
2
1 f b (x)f (x)gx 2
pN M (x) =;2 xb b (x) + xb
2b (x) + b(x)+x=b+1
2
f
(x)
>
b
>
>
2
>
:
for x 2 [0; 2b)8n 00 o00< f (x) x x f (x)for x 2b44 f (x)o00
nqN M (x) =;0: f (x) (x) b (x)f (x)
for
x
2
[0;
2b)
b
f (x)
where

b (x) can be found in Theorem 2; see Hirukawa and Sakudo (2014) for func-

tional forms of p (x) and q (x) for the MG kernel. It follows that MSEs of f~JLN;M G (x)and f~JLN;N M (x) for interior x can be approximated byno
M SE f~JLN;M G (x)onM SE f~JLN;N M (x)
=

1
16

x
f (x)
2

x f 00 (x)
2 f (x)

00 2

x
f (x)
4

x f 00 (x)
4 f (x)

00 2

f (x)

x
2

b4 +

1
f (x)
p p ;
1=2
nb 2
x

1
f (x)
p p
1=2
nb
2 x
p
2
00
x f 00 (x)
2 f (x)b4 + 1=2 p p :2 f (x)
nb 2
x
b4 +

The smoothing parameter values that minimize these approximations are
bJLN;M G =

1
22=3
1=3

bJLN;N M = 2

f (x)
p p
2
x

x
f (x)
2

x f 00 (x)
2 f (x)

00

4=9

f (x)
p p
2
x

x
f (x)
2

x f 00 (x)
2 f (x)00

4=9

n 2=9 ;
n 2=9 :

Observe that the relation bJLN;N M = 2bJLN;M G holds once again. Moreover, whenbest implemented, the optimal-MSEs of two estimators for interior x are …rst-order6noBecause the third-order term in Bias f^W (x) is O b2 , it would be possible in principle toimprove the bias convergence of this estimator from O (b) to O b2 by employing one of the bias
correction techniques twice. However, it is doubtful whether there is much gain in practice fromthe bias correction via iteration, and thus we do not consider such inferior cases any further.

18

asymptotically equivalent, i.e.
no~
M SE fJLN;M G (x)no~M SE fJLN;N M (x)
9

28=3

3.2

x
f (x)
2

x f 00 (x)
2 f (x)00 2=9

f (x)
p p
2
x

8=9

n 8=9 :

Bias and Variance Approximations of GG Density Estimators Using Weakly Dependent Observations

The second property is concerned with estimating the marginal density from nonnegative time-series data. Examples include estimation of the distribution of important…nancial variables such as short-term interest rates or trading volumes, and even thebaseline hazard in …nancial duration analysis.
To allow for weakly dependent observations in GG density estimation, we replace
Assumptions 1 and 2 with the regularity conditions below. While similar conditions
can be found, for instance, in Bouezmarni and Rombouts (2010), they assume strong
mixing processes with an exponentially decaying mixing coe¢ cient and exclusively
study the MG kernel.7

We relax the mixing condition and prove that their result

holds for a broader class of asymmetric kernels which includes the MG kernel.
Assumption 1b. fXi g is a nonnegative, strictly stationary and strong mixing process with the mixing coe¢ cient(`) of size

(2r2) = (r2) for some r > 2.

Assumption 2b. Let f ( ) and fj ( ; ) be the marginal and joint densities of Xi and(Xi ; Xi+j ), respectively. Then, f is twice continuously di¤erentiable, and fj isuniformly bounded.
7

Carrasco and Chen (2002) provide the conditions that make GARCH processes stationary and-mixing with exponential decay, whereas Chen, Hansen and Carrasco (2010) discuss the conditions
that can establish -mixing with exponential decay in scalar di¤usion processes. In relation to thelatter, Feller’s square-root process and its inverse, which have been employed as models of shortterm interest rates by Cox, Ingersoll and Ross (1985) and Ahn and Gao (1999), respectively, areexamples of exponentially decaying -mixing processes. Since -mixing implies strong mixing, theirassumption can cover many important applications in economics and …nance.

19

The next theorem states that results in Theorem 1 are carried through even when
positive weakly dependent observations are used.

Theorem 4. Results in Theorem 1 still hold under Assumptions 1b, 2b and 3.

3.3

Convergence Properties of the GG Density Estimator for
the Density Unbounded at the Origin

As the third property we obtain convergence results of f^GG (x) when the true density
f (x) is unbounded at x = 0. Often a clustering of observations near the boundary
can be found in intraday trading volumes or realized volatilities, for instance.

Inthis case, it is highly likely that the true density has a pole at x = 0 (e.g. Malecand Schienle, 2014). Such shapes also appear in many other applications including
spectral densities of long memory processes. The following two theorems demonstrate
weak consistency and the relative convergence of f^GG (x) as x ! 0, provided that theassumption on the smoothing parameter is replaced by Assumption 3c.
Assumption 3c. The smoothing parameter b satis…es b + (nb2 )1! 0 as n ! 1.

Theorem 5. If the true density f (x) is unbounded at x = 0 and Assumptions 1bp
and 3c hold, then f^GG (0) ! 1.

Theorem 6. Suppose that the true density f (x) is unbounded at x = 0 and continuously di¤erentiable in the neighborhood of the origin. If Assumption 1b and 3c
also hold, thenf^GG (x) f (x) p!0f (x)as x ! 0.
FIGURE 2 ABOUT HERE
20

Bouezmarni and Scaillet (2005, p.399) focus exclusively on random samples and…nd that the property in Theorem 5 is peculiar to the G and MG kernels and notshared with other asymmetric kernels such as the Inverse Gaussian and Reciprocal
Inverse Gaussian kernels by Scaillet (2004). To illustrate the result from this theorem,
we prepare Figure 2, in which the W, NM, MG, and G density estimates based on 500
independent observations drawn from G(0:75; 1:25) are plotted. The …gure indicatesthat estimates using three examples of the GG kernels, as well as the G estimate, canindeed capture the shape of the density near the boundary reasonably well.

4

Finite Sample Performance

4.1

Setup

This simulation study examines accuracy of GG density estimators.

We compare…nite sample performance of three GG estimators using the W, NM and MG kernels
with that of three other density estimators, namely, (i) the density estimator using yet
another asymmetric kernel, (ii) the one obtained via back-transforming a symmetrickernel density estimate based on the log-transformed data, and (iii) a symmetrickernel density estimator in the original scale with a proper boundary correction.
For (i), because of its popularity in empirical studies, the G kernel is chosen as theasymmetric kernel that does not belong to the GG kernels.

For (ii) and (iii), thelog-transformed (“LT”) density estimator using the Gaussian kernel and the local
linear (“LL”) density estimator using the Epanechnikov kernel (e.g. Jones, 1993) arechosen, respectively.
For each distribution in the list below, 1; 000 data sets of sample size n = 100,
200 or 500 are simulated. All density estimates are evaluated on an equally spaced
grid of 500 points over the interval [0; 5]. Following Chen (2000) and Scaillet (2004),

21

as the performance measure for each estimator f , we compute the root integratedqR
2
1
squared error (“RISE”) RISE f (x) =f (x) f (x) dx. In our report,0
the integral is approximated over the 500 points. The smoothing parameter b (forW, NM, MG, and G) or the bandwidth h (for LT and LL) is chosen as the minimizerof the (approximated) RISE, again as in Chen (2000) and Scaillet (2004). Lastly, six
distributions below are considered as truths, because of their popularity in parametricmodelling of income distributions, actuarial loss distributions and baseline hazards.
Distribution 5, proposed by Buch-Larsen et al. (2005) for modelling distributions of
insurance payments and operational risks, is known to have a Pareto-type tail. This
distribution is adopted in this simulation study, from the viewpoint that it is thought
to mimic the shape of income distributions well. Distribution 6, which is unboundedat the origin, is prepared to investigate the property stated in Theorem 5. Figure 3
presents the shapes of these densities.

1. Gamma:

f (x) = x 1 exp ( x= ) 1 fx
( ; ) = (1:5; 1) :2. Weibull:f (x) = ( = ) (x= ) 1 exp f (x= ) g 1 fx 0g ;( ; ) = (1:5; 1:5) :pexp(x)2 = (2 2 ) 1 fx 0g ;f (x) = 2= 2
( ; ) = (0; 1:5) :pf (x) = 1= x 2
exp(log x)2 = (2 2 ) 1 fx 0g ;( ; ) = (0; 0:75) :3. Half-Normal:

4. Log-Normal:

0g = f5. GeneralizedChampernowne:

f (x) = (x + c) 1 f(M + c)
c g 1 fx2
f(x + c) + (M + c)
2c g ;( ; M; c) = (2:5; 0:5; 0:25) :6. Gamma with Pole:f (x) = x 1 exp ( x= ) 1 fx
( ; ) = (0:75; 1:25) :0g = f

FIGURE 3 AND TABLE 1 ABOUT HERE
22( )g ;

0g( )g ;

4.2

Simulation Results

Table 1 presents averages and standard deviations of RISEs and averages of tuning
parameter values (i.e. smoothing parameter values for W, NM, MG, and G, andbandwidth lengths for LT and LL) over 1; 000 Monte Carlo replications. For each
distribution, results are qualitatively similar across three sample sizes. A quick inspection reveals that overall the MG estimator performs better than the G estimator;
only exceptions can be found in Distribution 1 for n = 100 and Distribution 6 for all
sample sizes. This is congruous with the results in Chen (2000) and Scaillet (2004).
Each of …rst three distributions is chosen so that one of the GG estimators canhave clear advantage.

As expected, W and MG, as well as G, perform well forDistributions 1 and 2, whereas NM dominates for Distribution 3. Poor performanceof W, MG and G for Distribution 3 could be attributed to their di¢ culty in capturing
the shape of the pdf near the origin. The pdf is decreasing and satis…es the shoulder
condition f 0 (0) = 0. Accordingly, the data suggest concavity of the density in thevicinity of the origin. As a consequence, these kernels tend to misinterpret the local
concavity as an indication of a mode over the strictly positive region. We can also
…nd that LL exhibits comparable performance to the GG estimators, whereas LT isconsistently outperformed.
The GG estimators perform satisfactorily even for Distribution 4, which is advantageous to LT, or for Distribution 5, which could be unfavorable to all due to its
Pareto-type tail. As predicted by Theorem 5, the GG estimators also exhibit reasonably good performance for Distribution 6. In contrast, LL and LT are dominatedfor Distributions 4 and 6 and Distribution 5, respectively.
A rationale as to why the LT estimator f^LT (x) does not necessarily perform
well (although it is popularly applied in empirical works) may be found in its bias
23

property. Following Wand, Marron and Ruppert (1991), we can approximate the bias
and variance of the estimator asno^Bias fLT (x)noV ar f^LT (x)1f (x) + 3xf 0 (x) + x2 f 00 (x) h2 ;2p(nh) 1 f (x) = (2 x) if x=h ! 1
p:1(nh2 ) f (x) = (2) if x=h ! > 0

While f^LT (x) shares the shrinking variance property as the design point moves awayfrom the boundary (which comes from the fact that the variance coe¢ cient is proportional to f (x) =x) with asymmetric kernel estimators, its leading bias takes a lessfavorable form in that it contains three terms involving the true density and its …rst
two derivatives.
We also make a few remarks on tuning parameters b and h. First, both hLT andhLL appear to be systematically long. Large values of the former may be attributed
to the fact that log-transformed observations tend to spread out toward the large(in magnitude) negative side for each distribution.

Second, for each combinationof the sample size and distribution (except Distribution 6), we can …nd a consistentordering of bG , bM G , bW , and bN M from the smallest to the largest. Third, whether
bN M becomes roughly twice the size of bM G as predicted in Section 2.3.2 depends
crucially on distributions and sample sizes. The derivation of (7) (and thus (8)) isbuilt on the trimming argument for the boundary region. Therefore, whether …nite
sample results support the relation relies on how fast the shrinkage of b is and/or how
small the weight that the distribution puts near the boundary is. While it is hard to…nd the relation bN M

2bM G in Table 1, in unreported simulations where the truth

is GG (5; 2; 2:5) (that puts a very small weight over the boundary region), we obtain(bN M ; bM G ) = (0:0884; 0:0452) ; (0:0658; 0:0334) ; (0:0460; 0:0232) for n = 100; 200; 500
so that the relation is con…rmed for each sample size.

24

5

Conclusion

Unlike symmetric kernels, exploring asymptotics on asymmetric kernels has reliedon kernel-speci…c, diversi…ed approaches.

To pursue a uni…ed approach in their

asymptotics, this paper has proposed a new class of asymmetric kernels that is built
on the GG pdf and a few common conditions and thus referred to as a family of
the GG kernels. The family can generate asymmetric kernels that share appealing
properties with the MG kernel. Asymptotics on kernels belonging to the family canbe delivered by manipulating the conditions directly, as with symmetric kernels. As
special cases, the family encompasses the W and NM kernels, as well as the MG kernel.
This paper also demonstrates three additional properties of the GG density estimator
including applicability of MBC and weak consistency for the density unbounded at
the origin.

Monte Carlo simulations indicate good …nite sample properties of GG

density estimators.
There are two possible research extensions. First, choice methods of the smoothing parameter b in GG density estimation need to be further investigated. Promising
candidates include a more sophisticated plug-in method like the solve-the-equationplug-in method by Sheather and Jones (1991) and the cross-validation method. In
particular, to the best of our knowledge, no equivalent method to the former has
ever been proposed for asymmetric kernel density estimation.
of-…t tests can be built on the GG kernels.

Second, goodness-

Fernandes and Grammig (2005) andFernandes, Mendes and Scaillet (2011) obtain favorable …nite-sample results fromapplying the G kernel to speci…cation testing for duration models and testing forsymmetry, respectively.

Combining the GG kernels with these testing procedures

appear to be another good application of the kernels. These extensions are currently
under authors’investigation and will be addressed in separate papers.
25

A

Appendix

In order to approximate the gamma function, we frequently refer to the following
well-known formulae:
Stirling’s formula (“SF”).
p(z + 1) =2 z z+1=2 e z 1 +1
1++O z 32
12z 288z

as z ! 1:

Series expansion of the log gamma function (“SELG”).
log (1 + z) =1X( 1)k (k) kz for jzj < 1;
z+
k
k=2

P
= limn!1 ( nk=1 k 1 log n) = 0:5772156649 : : :Pk(k > 1) is the Riemann zeta function.
is Euler’s constant, and (k) = 1
n=1 n

where (only in this context)

Legendre duplication formula (“LDF”).
(z)

A.1

1
z+
2

=

p
22z 1

(2z) for z > 0:

Proof of Theorem 1

Bias. Letdx = GG ((1=2) E ( xx)2 f 00 (x) + o E ( x;( = ) = f( + 1) = g ; ). Then, a second-order Taylornonoexpansion of E f^GG (x) around x = x yields E f^GG (x) = f (x)+E ( x x) f 0 (x)+
E( m
x ) =

In particular, E ( x ) =
E

2
x

mf

x)2 . It follows from (2) that( = )gm 1 f( + m) = g:[ f( + 1) = g]m(by construction) and=2( = ) f( + 2) = g=[ f( + 1) = g]2Using Conditions 1 and 3, we have, for x
C5 xb + o (b), and thus E ( x

Mb (x) :C1 b, E ( x ) = x and Ex) = 0 and E ( x
26

2

x)2

2
x

= x2 +

C5 xb. As a consequence,noBias f^GG (x)(C5 =2) xf 00 (x) b. On the other hand, for x 2 [0; C1 b), E ( x ) ='b (x) = O (b) and x = O (b) hold, and thus E ( xb (x) b,E( xx) = [f'b (x)Moreover, it follows from E 2x
n
o
2
2
0
^
x) = O (b ). Therefore, Bias fGG (x)b (x) f (x) b.
where

b (x)

xg =b] b :== O (b2 ) that

= O (1).

noVariance. As usual, we consider the approximation V ar f^GG (x) = (1=n) fE (Ki2 ) + O (1)g.

A straightforward calculation yields, for 2 R+ ,
28
n
9 1
(
+1 =
<
6
KGG (u; x; b) = 4
n
(
1)+1
:
;
uf (

1)+1g 1

exp

h

u

o

o2

3
7

15

( )=( 1= ( +1 ))oi ( 1)+1 n

n
+1
= 1=
8
0
<
pdf of GG @ (= Ab; (x):no^Hence, V ar fGG (x)

1)+1

(

1) + 1;
1=1)+1

o 1 fu+1(1=n) Ab;2 (x) E ff (& x )g, whered& x = GG 2

1;

21=

( = )
;
f( + 1) = g

By the mean-value theorem, E ff (& x )g = f (x)+E (& x

0g

19
=
A
;
:
;

:

x) f 0 (& x ) for some & x joining& x and x. It follows from (2) that
E (& x ) =21=( = ) (2 = )f( + 1) = g f(21) = g= Hb (x) :Then, Condition 4, together with Condition 1, implies that E (& xE ff (& x )gx) = o (1) so thatf (x) regardless of the position of x. Finally, Condition 5 establishes

the approximation to the variance.

27

A.2

Proof of Theorem 2

A.2.1

The MG Kernel

Condition 3. Observe that Mb (x) = 1 + 1= . For x

2b, Mb (x) = 1 + (1=x) b sothat C5;M G = 1. On the other hand, for x 2 [0; 2b),= (x=b)2 =4 + 1 = O (1), andthus Mb (x) = O (1) holds.

Condition 4. Substituting(2 ) 1 yields Hb (x) == x=b into Hb (x) = 1

1 + O (b) = 1 + o (1) for interior x. On the other hand, for boundary x,
regardless of whether= x=b (when x = O (b) and x= O (1)= (x=b)2 =4 + 12b) or(when x = O (b) and x 2 [0; 2b)). It follows that Hb (x) = O (1).
Condition 5. Because = = b regardless of the position of x, we havef (Ab; (x) = b1(

1) + 1g
:
( )g
p
1=2
2
e

1)+1 f

For interior x, SF yields, as = x=b ! 1, ( )p2 f (1)g ( 1)+1=2 e ( 1) . Then,
Ab; (x)

b1

(
1=2

p

1)(1

:

The result immediately follows from de…ning VI;M G ( ) :=recognizing that (

1)(1

)=2

=

(1

)=2

(1

ary x, the result is established by de…ning
8f ( 1)+1g< ( 1)+1 f ( )g
VB;M G ( ) :=(n4 2 +1)
:
2 +1
2
4

A.2.2

4

+1

1= )(1o1) + 1g)=2
1

2

and f (

)=2

n

1=2

p

(x=b)(1if x=b !

2

if x=b !

2 (0; 2)

2

)=2

1

o 1

and

. For bound-:

The W Kernel(1 + 2= ) = f (1 + 1= )g2 . For x 2b, we
p2b=x
1. Then, by SELG
may pick an arbitrarily small b > 0 so that j2= j =Condition 3. Observe that Mb (x) =28

and (2) =
1+
1+

2

=6, two gamma functions admit the following approximations:2
1= exp log

1+

= exp log

1+

2

2

=1

1

=1

+
+

( 2 =3) + 2 2
2

3

+O

( 2 =12) + ( 2 =2)
2

;
3

+O

:

Applying a geometric series expansion to the approximation to f (1 + 1= )g 2 …nallydelivers Mb (x)1 + ( 2 =6) = 2 = 1 + f 2 = (12x)g b so that C5;W =other hand, for x 2 [0; 2b),2=12. On the= (x=b) =2 + 1 = O (1), and thus Mb (x) = O (1) is thecase.

Condition 4. In this case, Hb (x) = 21= (11= ) (1 + 1= ) (1

1

1= )

. It

is easy to see that Hb (x) = O (1) for boundary x. On the other hand, for interiorx, 21= = exp f(1= ) log 2g = 1 + O b1=2 , and (11= ) (1 + 1= ) (11= ) =

1 + O b1=2 by SELG. Therefore, Hb (x) = 1 + O b1=2 = 1 + o (1) holds.

Condition 5. We have1

1+ 1

Ab; (x) =For interior x, because
f + (1

)= g

1

+1

:

+1

=

p
2x=b ! 1, we may approximate+(1( ) and)=. In addition, ( = )(1 + 1= )
1

= 2(

1)=2

and thusAb; (x)

2 2

1

( )

1

1

(xb) 2 := VI;W ( ) (xb) 2 :For boundary x, the result is established by de…ning
8
n
o 1
1
1
1+ p1
+p
>
>
2
2
2
2
<
if x=b !
1+p2
VB;W ( ) :=
n
o 1
1
2
>
f (1+ +2)g
f + 2(1+2 ) g+2)>: 2( 2 +4
if x=b !
2(1
)
+
+2

29

2
:
2 (0; 2)1,(xb)(1)=2,

A.2.3

The NM Kernel

Condition 3. Observe that Mb (x) = ( =2) [ ( =2) = f( + 1) =2g]2 . For x 2b,p
2
it follows from LDF that Mb (x) = ( =2) 2 1 f ( =2)g2 = f( )g . Next, SF
implies that, as= x=b ! 1,
( ) =

p

2

=

p

2

2

1=2

1+

e

=2 1=2

e

2

1
2
;
+O
12
1
=2
2
1+
+O
6

(A1)
(A2)

:

Then, we …nd that the approximation of Mb (x) takes a very simple form Mb (x) =1 + (2 ) 1 + O (2) = 1 + (2x) 1 b + o (b) so that C5;N M = 1=2. On the other hand,= (x=b)2 =4 + 1 = O (1), and thus Mb (x) = O (1) holds.

for x 2 [0; 2b),1=2) ( =2) ( ) = 21=2 f( + 1) =2g ( + 1=2) is

Condition 4. Hb (x) = (O (1) for boundary x because of the same reason as in the proof for the MG kernel.
p
For interior x, using LDF and then (2 )
2 (2 )2 1=2 e 2 as = x=b ! 1,
as well as (A1) and (A2), yields
Hb (x) =

1 3
2 2
1=2
2

2

2

( )

2

1
= 1 + O (b) = 1 + o (1) :
2

1

(2 )

Condition 5. We have1Ab; (x) =

2

1

(

1)
2

n

1

+1
2

(
2

2

+ 12

2

1)
1

+ 21

o

:

For interior x, by LDF, Ab; (x) reduces toAb; (x) =

1

2

1

(

1)
2

+ 12

2(

1)(

1)+ (

Next, by SF, as

f (

and

1) =2g (f (= x=b ! 1,p
1) =2g2 f (p( ) f ( )g

1)g

30

1) 1

p
1) 1=2

1

n

(

f (
o

1)g
3

1)

2

2

2 f (

1)g (

e

(

1)=2

.

2

:

1) 1=2e(

1)

Substituting theseapproximations, as well as (A1) and (A2), we …nally deduce that
Ab; (x)Moreover

1

(1

)=2

= (xb)(1

Ab; (x)

1

1

2

)=2

1
p
1=2 (
)

e =2
p
1=2 (
)

(

1

1= ) (and (11 (xb)

1

1

1

1)=2

1)
2

:
e

31

, and thus

1:= VI;N M ( ) (xb) 2 :2

For boundary x, the result is established by de…ning
81+1
1 f ( 2 )g>f ( 21)+1 g
2>
if x=b !
>
(
1)+1
2
1
<
2
f
(
2 )g
n
o 1
VB;N M ( ) :=2
+1
( 8 2 + 12 )
>
1
8
8
>
n
o2 1
>
if x=b !
2+ 1: 2 +4
2
8
2
+1
8

=2

2

2
:
2 (0; 2)

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33

Figure 1: Shapes of the GG Kernels When b = 0:2(a) x = 0(b) x = 1(c) x = 2(d) x = 4

Figure 2: GG Density Estimates When the True Distribution Is G (0:75; 1:25)

Figure 3: Shapes of True Densities for Monte Carlo Simulations

34

Table 1: Averages of Performance Measures and Tuning Parameter Values
n = 100
RISE
b or hn = 200
RISE
b or hn = 500
RISE
b or h

1. Gamma
GG

W
NM
MG
Non-GG G
LT
LL

0.0356 (0.0098)
0.0368 (0.0091)
0.0362 (0.0112)
0.0358 (0.0125)
0.0441 (0.0157)
0.0368 (0.0116)

0.27780.30070.17120.14040.4434
1.02720.0294 (0.0081)
0.0306 (0.0076)
0.0289 (0.0088)
0.0290 (0.0098)
0.0348 (0.0114)
0.0302 (0.0088)

GG

0.0374 (0.0119)
0.0385 (0.0116)
0.0367 (0.0127)
0.0368 (0.0140)
0.0470 (0.0154)
0.0367 (0.0127)

0.18700.20900.1272
0.11370.37300.82340.0297 (0.0090)
0.0307 (0.0088)
0.0286 (0.0092)
0.0294 (0.0103)
0.0368 (0.0110)
0.0294 (0.0092)

GG

0.0274 (0.0113)
0.0251 (0.0120)
0.0303 (0.0117)
0.0327 (0.0112)
0.0586 (0.0191)
0.0256 (0.0119)

0.34450.38090.26620.17500.47351.47740.0225 (0.0083)
0.0207 (0.0088)
0.0245 (0.0087)
0.0262 (0.0087)
0.0457 (0.0140)
0.0203 (0.0094)

GG

0.0429 (0.0153)
0.0447 (0.0152)
0.0416 (0.0158)
0.0458 (0.0150)
0.0401 (0.0166)
0.0482 (0.0147)

0.08300.09320.06240.05350.32070.44150.0332 (0.0110)
0.0343 (0.0108)
0.0324 (0.0114)
0.0360 (0.0108)
0.0315 (0.0122)
0.0381 (0.0106)

GG

0.0477 (0.0169)
0.0477 (0.0161)
0.0504 (0.0190)
0.0504 (0.0195)
0.0700 (0.0246)
0.0513 (0.0183)

0.13240.14480.0881
0.06760.44510.4850GG

0.0617 (0.0181)
0.0652 (0.0168)
0.0627 (0.0171)
0.0614 (0.0202)
0.0639 (0.0304)
0.0650 (0.0161)

0.08580.08550.08030.05710.73890.52800.17010.18610.1105
0.09620.38200.75240.0221 (0.0057)
0.0232 (0.0055)
0.0211 (0.0059)
0.0220 (0.0066)
0.0252 (0.0074)
0.0234 (0.0061)

0.0897
0.09750.06830.06010.31490.51520.12280.1382
0.09150.08130.31870.66000.0214 (0.0058)
0.0222 (0.0058)
0.0204 (0.0060)
0.0218 (0.0069)
0.0267 (0.0073)
0.0217 (0.0060)

0.08090.0911
0.06340.0526
0.25840.50980.0172 (0.0056)
0.0158 (0.0059)
0.0184 (0.0060)
0.0193 (0.0057)
0.0329 (0.0085)
0.0147 (0.0064)

0.19360.23540.13800.0911
0.32241.01930.0242 (0.0080)
0.0245 (0.0078)
0.0238 (0.0081)
0.0263 (0.0075)
0.0232 (0.0082)
0.0282 (0.0073)

0.0471
0.05620.03340.02610.23100.2937

2. Weibull
W
NM
MG
Non-GG G
LT
LL

3. Half-Normal
W
NM
MG
Non-GG G
LT
LL

0.27450.31520.20390.13250.39971.2598

4. Log-Normal
W
NM
MG
Non-GG G
LT
LL

0.06540.07490.04800.03900.27820.3683

5. Generalized Champernowne
W
NMMG
Non-GG G
LT
LL

0.0391 (0.0126)
0.0390 (0.0122)
0.0403 (0.0141)
0.0403 (0.0151)
0.0544 (0.0177)
0.0413 (0.0137)

0.09220.10330.06180.04850.38070.36180.0295 (0.0101)
0.0294 (0.0099)
0.0298 (0.0106)
0.0301 (0.0107)
0.0394 (0.0117)
0.0308 (0.0101)

0.05470.06370.03880.03180.31180.26370.0359 (0.0091)
0.0380 (0.0088)
0.0368 (0.0098)
0.0363 (0.0107)
0.0361 (0.0140)
0.0438 (0.0099)

0.03800.03730.03570.02420.51380.2552

6. Gamma with PoleW
NMMG
Non-GG G
LT
LL

0.0494 (0.0136)
0.0523 (0.0128)
0.0500 (0.0131)
0.0494 (0.0157)
0.0498 (0.0213)
0.0549 (0.0125)

0.0591
0.0586
0.05540.03890.62980.3905

Note: Numbers in parentheses are simulation standard deviations of RISEs. “b orh” denotes simulation averages of the values of smoothing parameters b for W, NM,
MG, and G, or the lengths of bandwidths h for LT and LL.

35