Journal of Nonparametric Statistics

ISSN: 1048-5252 (Print) 1029-0311 (Online) Journal homepage: https://www.tandfonline.com/loi/gnst20

Family of the generalised gamma kernels: agenerator of asymmetric kernels for nonnegative
data
Masayuki Hirukawa & Mari Sakudo
To cite this article: Masayuki Hirukawa & Mari Sakudo (2015) Family of the generalised gamma
kernels: a generator of asymmetric kernels for nonnegative data, Journal of Nonparametric
Statistics, 27:1, 41-63, DOI: 10.1080/10485252.2014.998669
To link to this article: https://doi.org/10.1080/10485252.2014.998669

Published online: 28 Jan 2015.

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Journal of Nonparametric Statistics, 2015
Vol. 27, No. 1, 41–63, http://dx.doi.org/10.1080/10485252.2014.998669

Family of the generalised gamma kernels: a generator ofasymmetric kernels for nonnegative data
Masayuki Hirukawaa∗ and Mari Sakudob†
a Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka 572-8508, Japan;b Research Institute of Capital Formation, Development Bank of Japan, 9-7, Otemachi 1-chome,

Chiyoda-ku, Tokyo 100-8178, Japan(Received 19 June 2013; accepted 25 November 2014)
Unlike symmetric kernels, so far exploring asymptotics on asymmetric kernels has relied on diversified
approaches. This paper proposes a family of the generalised gamma (GG) kernels that is built on theprobability density function of the GG distribution [Stacy, E.W. (1962), ‘A Generalization of 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 modified gamma kernel [Chen, S.X. (2000), ‘Probability Density Function Estimation Using Gamma Kernels’, Annals of theInstitute of Statistical Mathematics, 52, 471–480]. Asymptotics on the kernels generated from the family
can be delivered by manipulating the conditions directly, as with symmetric kernels.
Keywords: asymmetric kernel; boundary correction; density estimation; kernel smoothing; generalised
gamma kernels
MSC 2010: 62G07; 62G20

1.

Introduction

Researchers and policy-makers are often interested in the distributions of nonnegative economic
and financial variables including incomes, wages, short-term interest rates, and insurance claims.
The distributions are empirically characterised by two stylised facts, namely, (i) a natural boundary at the origin and (ii) concentration of observations near the boundary and a long tail withsparse data. When their parametric specifications are found to be inappropriate, we resort to nonparametric density estimation. However, in estimating the distributions with symmetric kernels,
we must make two distinct modifications for the standard smoothing technique simultaneously.
For (i), boundary correction methods should be employed; see, for instance, Section 3 of Karunamuni and Alberts (2005) for a concise review of the methods. For (ii), adaptive smoothing suchas variable bandwidth methods (e.g. Abramson 1982) would be a remedy.
Recently, asymmetric kernels with support on [0, ∞) (e.g. Chen 2000; Jin and Kawczak 2003;
Scaillet 2004) have emerged as a viable alternative that can accommodate the stylised facts.
Although the kernels are relatively new in the literature, several papers report favourable evidence from applying them to empirical models in economics and finance; see, for instance,∗ Corresponding author. Email: hirukawa@econ.setsunan.ac.jp† Institute for Research in Contemporary Political and Economic Affairs, Waseda University, 1-6-1 Nishiwaseda,

Shinjuku-ku, Tokyo 169-8050, Japan.
c American Statistical Association and Taylor & Francis 2015
42

M. Hirukawa and M. Sakudo

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 stylised facts istheir property as a combination of a boundary correction device and adaptive smoothing that has
effect similar to the variable bandwidth methods.
A cumbersome aspect of asymmetric kernels, however, is that so far exploring their asymptotic
properties has relied on kernel-specific and thus diversified approaches. In contrast, symmetrickernels are built on a set of common conditions, and their asymptotic properties can be implied
straightforwardly by the conditions. Then, this paper proposes a new class of asymmetric kernels
that consists of a specific functional form and a set of common conditions. Our aim is that as longas an asymmetric kernel is generated from this class, we can deliver its asymptotic properties bymanipulating 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 the shape or functional form. While an apparent shape restriction (i.e. symmetry about the origin) is imposed onsymmetric kernels, there is no guidance on shapes of asymmetric kernels; actually, any shapes
are admissible as long as the kernels have support on [0, ∞). Recognising that most of the papers
on asymmetric kernels report superior finite sample performance of the modified gamma (MG)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 generalised gamma (GG) distribution (Stacy 1962) is chosen as the functional form. Combining the pdf with a set of common conditions, we finally define a family of the GG kernels. As
special cases, the GG kernels nest not only the MG kernel but also the newly proposed Weibull
and Nakagami-m kernels, 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 MG 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(MISE) within the class of nonnegative kernel density estimators. Third, the leading bias overthe interior region for each GG density estimator contains only the second-order derivative ofthe true density over the interior region, whereas the one for density estimators using some otherasymmetric kernels (e.g. the gamma kernel by Chen 2000) consists of terms involving the firstand second-order density derivatives. Fourth, the variance of the GG estimator tends to decreaseas the design point moves away from the boundary. This property is particularly advantageous
to estimating the distributions that have a long tail with sparse data. It is worth emphasising thatthese properties can be demonstrated by manipulating the common conditions directly.
This paper also demonstrates three additional properties of the GG density estimator. While
the first property, applicability of two multiplicative bias correction (MBC) techniques studied in
Hirukawa (2010) and Hirukawa and Sakudo (2014), is based on random sampling, the remaining two properties allow for dependent sampling. After the GG density estimator using weakly
dependent data are shown to admit the same first-order bias and variance approximations as thoseusing independent observations, it is proven that the density estimator is consistent even when
the true density becomes unbounded at the boundary. The gamma and MG kernels are known toshare all three properties, and thus it is demonstrated that all in all, the GG kernels are endowedwith the same attractive properties.
The remainder of this paper is organised as follows. Section 2 provides the definition of afamily of the GG kernels and introduces three special cases. Convergence properties of the GG
density estimator are also provided. Section 3 investigates three appealing properties of the GG
density estimator. Section 4 conducts Monte Carlo simulations to examine finite sample properties of the GG density estimator. Section 5 summarises the main results of the paper and suggests
some research extensions. Proofs of Theorems 1 and 2 are given in the Appendix. Besides,

Journal of Nonparametric Statistics

43

proofs of Theorems 4–6 and comprehensive simulation results are available on the first author’s
webpage.
∞This paper adopts the following notational conventions: (a) = 0 ya−1 exp(−y) dy (a > 0)dis the gamma function; and 1{·} signifies an indicator function. 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 → ∞. Lastly, in order to describe different 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 satisfies x/b → ∞ and x/b → κ for some 0 < κ < ∞ as
n → ∞, respectively.

2.
2.1.

A family of the GG kernels
Definition

Let Y be distributed by the GG distribution GG(α, β, γ ). Then, Y has the pdf
p(y; α, β, γ ) =γ yα−1 exp{−(y/β)γ }1{y ≥ 0}.
β α (α/γ )(1)

It is known that the mth uncentred moment of Y is given byE(Y m ) = β m

{(α + m)/γ }.
(α/γ )

(2)

We now provide the definition of a family of the GG kernels that consists of several common
conditions. Before proceeding, it could be beneficial to relate the definition to probability density estimation. Below we present a set of regularity conditions for asymmetric kernel density
estimation.
Assumption 1 The random sample {Xi }ni=1 is drawn from a univariate distribution with a pdf
f having support on [0, ∞).
Assumption 2

f is twice continuously differentiable.

Assumption 3

The smoothing parameter b(= bn > 0) satisfies b + (nb)−1 → 0 as n → ∞.

To generate a kernel from the pdf (1), we allow (α, β, γ ) to be a function of the design pointx > 0 and the smoothing parameter b, as in Chen (2000), Jin and Kawczak (2003) and Scaillet(2004). To put it another way, whenever we refer to the triplet (α, β, γ ), it should be interpreted
as a short-handed notation of (α, β, γ ) = (αb (x), βb (x), γb (x)) unless otherwise noted. Let thecrude GG kernel be KGG0 (u; x, b) := p(u; α, β, γ ). Given the random sample {Xi }ni=1 , we havethe density estimator fˆGG0 (x) = (1/n) ni=1 KGG0 (Xi ; x, b).
dLet ϑx = GG(α, β, γ ). Under Assumptions 2 and 3, a second-order Taylor expansion ofE{fˆGG0 (x)} around ϑx = x yields E{fˆGG0 (x)} = f (x) + E(ϑx − x)f  (x) + (1/2)E(ϑx − x)2 f  (x) +o{E(ϑx − x)2 }. It follows that unless E(ϑx ) = x exactly (at least for interior x), the leading biasof fˆGG0 would contain the term involving f  , which is less desirable. Although there are numerous choices of (α, β, γ ) that can achieve E(ϑx ) = x, we adopt the simplest resolution that weset β = x for interior x and employ the pdf of GG(α, β(α/γ )/ {(α + 1)/γ }, γ ) (which canbe obtained by the change of variable Z := [(α/γ )/ {(α + 1)/γ }]Y in Equation (1)) as thekernel. In the end, we reach the following definition of a family of the GG kernels.

44

M. Hirukawa and M. Sakudo

Definition 1 Let (α, β, γ ) = (αb (x), βb (x), γb (x)) ∈ R3+ be a continuous function of thedesign point x and the smoothing parameter b. For such (α, β, γ ), consider the pdf of
GG(α, β(α/γ )/ {(α + 1)/γ }, γ ), i.e.
KGG (u; x, b) =u
γ uα−1 exp[−{ β(α/γ )/ ((α+1)/γ}γ ]){β(α/γ )/ ((α + 1)/γ )}α (α/γ )1{u ≥ 0}.

(3)This pdf is said to be a family of the GG kernels if it satisfies each of the following conditions:
Condition 1.

β=x
for x ≥ C1 b,
ϕb (x) for x ∈ [0, C1 b)

where 0 < C1 < ∞ is some constant, the function ϕb (x) satisfies C2 b ≤ ϕb (x) ≤ C3 b forsome constants 0 < C2 ≤ C3 < ∞, and the connection between x and ϕb (x) at x = C1 b issmooth.
Condition 2. α, γ ≥ 1, and for x ∈ [0, C1 b), α satisfies 1 ≤ α ≤ C4 for some constant 1 ≤C4 < ∞. Moreover, connections of α and γ at x = C1 b, if any, are smooth.
Condition 3.
(α/γ )((α + 2)/γ )1 + (C5 /x)b + o(b)Mb (x) :==2{((α + 1)/γ )}O(1)for x ≥ C1 b
for x ∈ [0, C1 b)

for some constant 0 < |C5 | < ∞.
Condition 4.
(α/γ )(2α/γ )1 + o(1)=Hb (x) := 1/γ
2 ((α + 1)/γ )((2α − 1)/γ )O(1)for interior x.
for boundary x

Condition 5.
Ab,ν (x) :=∼γ ((α + 1)/γ )βVI (ν)(xb)(1−ν)/2
VB (ν)b1−νν−1{(ν(α − 1) + 1)/γ }ν (ν(α−1)+1)/γ {(α/γ )}2ν−1

for interior x,
for boundary xν ∈ R+ ,where constants 0 < VI (ν), VB (ν) < ∞ depend only on ν.
Conditions 1 and 2 form a legitimate kernel from the GG pdf. It follows from ϕb (x) = O(b)uniformly on [0, C1 b) in Condition 1 that the kernel is well defined in the vicinity of the origin.
While α ≥ 1 in Condition 2 also ensures boundedness of the kernel near the origin, γ ≥ 1 controls the tail-behaviour of the kernel and establishes an exponential rate of the tail decay. The
condition also allows for each of α and γ to be a piecewise function of (x, b) like β, where theconnection is made at x = C1 b; the common connection point simplifies asymptotic analyses
substantially. In addition, smooth connection requirements in Conditions 1 and 2 are inspired by
the construction of the MG kernel; see Chen (2000, p. 473) for details. Moreover, Condition 3

Journal of Nonparametric Statistics

45

and Conditions 4 and 5 are the requirements for valid approximations to the bias and variance of
the GG density estimatorn1
KGG (Xi ; x, b),fˆGG (x) =n i=1
respectively. Although all the common conditions appear to be high-level ones, it is not hard tofind a few special cases that satisfy them. Examples of the GG kernels are provided shortly.
2.2.
2.2.1.

Convergence properties of probability density estimators using the GG kernels
Local property

Bias-variance tradeoff. Before providing examples of the GG kernels, readers may wonder
whether the kernels are a truly legitimate one, that is, they can yield a consistent density estimator. To answer this question, we begin this section with presenting the theorem on approximations
to the bias and variance of fˆGG (x).
Theorem 1 Under Assumptions 1–3, the bias of fˆGG (x) can be approximated by Bias{fˆGG (x)} ∼B1 (x, f )b, where⎧⎨ C5 xf  (x) for x ≥ C b
1
2,
B1 (x, f ) =⎩for x ∈ [0, C1 b)
ξb (x)f (x)and ξb (x) = {ϕb (x) − x}/b = O(1). On the other hand, the variance of fˆGG (x) can be approximated by√(nb1/2 )−1 VI (2)f (x)/ x for interior x,ˆ
Var{fGG (x)} ∼for boundary x.
(nb)−1 VB (2)f (x)The theorem states that Conditions 1–5 do lead to familiar properties of asymmetric kernel
density estimators.1 By construction, fˆGG (x) is free of boundary bias and nonnegative everywhere. The bias of fˆGG (x) is O(b), and its variance is O{(nb1/2 )−1 } for interior x and O{(nb)−1 }for boundary x. Furthermore, as pointed out by Chen (2000) and Scaillet (2004), a unique featureof asymmetric kernel density estimators is that√ the variance coefficient decreases as x increases.
This is shared with fˆGG (x) in that VI (2)f (x)/ x decreases as the design point x moves away fromthe boundary. The shrinking variance coefficient reflects that more data points can be pooled tosmooth in areas with fewer observations. This property is particularly advantageous to estimating the distributions that have a long tail with sparse data, such as those of the economic andfinancial variables mentioned at the beginning of Section 1.
Mean-squared error (MSE). For interior x, the MSE of fˆGG (x) can be approximated by
MSE{fˆGG (x)} ∼ b2

C524

{xf  (x)}2 +VI (2) f (x)√ .
nb1/2 xThe smoothing parameter value that minimises the right-hand side of Equation (4) is√ 2/5
VI (2) 2/5 f (x)/ xb∗GG =
n−2/5 .
{xf  (x)}2
C52

(4)

(5)

Observe that the MSE-optimal smoothing parameter is O(n−2/5 ) = O(h∗2 ), where h∗ is the MSEoptimal bandwidth for density estimators using nonnegative symmetric kernels. Therefore, when

46

M. Hirukawa and M. Sakudo

best implemented, the approximation to the MSE becomes
MSE∗ {fˆGG (x)} ∼ 54 [C5 {VI (2)}2 ]2/5 [f  (x){f (x)}2 ]2/5 n−4/5 .

(6)Observe that MSE∗ {fˆGG (x)} depends only on f (x) and not on x itself. The optimal MSE of fˆGG (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. On the other hand, for boundary x, MSE{fˆGG (x)} =†O(b2 + n−1 b−1 ), which yields the MSE-optimal smoothing parameter bGG = O(n−1/3 ) and the
optimal MSE of O(n−2/3 ).
2.2.2.

Global property

ˆThe
inferior rate in the optimal MSE∞√ of fGG (x) for boundary x does not affect its global property.
∞2
If 0 {xf (x)} dx and 0 {f (x)/ x} dx are both finite, then applying the trimming argument in
Chen (2000, p. 476) approximates the MISE of fˆGG (x) as
MISE{fˆGG (x)} ∼ b2

 ∞

C52
4

{xf  (x)}2 dx +0VI (2)
nb1/2

 ∞
0

f (x)
√ dx.
x

(7)

The smoothing parameter value that minimises the right-hand side of Equation (7) is
b∗∗
GG =



VI (2)
C52

2/5√
2/5   ∞0 {f (x)/ x} dx∞n−2/5 .
 (x)}2 dx
{xf
0

(8)

Therefore, when best implemented, the approximation to the MISE becomes5
MISE∗∗ {fˆGG (x)} ∼ [C5 {VI (2)}2 ]2/5
4



∞

{xf  (x)}2 dx

0

 ∞
0

4 1/5
f (x)n−4/5 .
√ dx
x

Note that O(n−4/5 ) is the optimal convergence rate of the MISE within the class of nonnegative
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 we consider a very
simple, Silverman’s (1986) rule-of-thumb type choice rule. Although in principle the rule is built
on the MISE (7) as the criterion, we make a couple of modifications. First, the unknown f is
replaced by a known reference density. For simplicity, we choose the pdf of G(μ, ω) (μ, ω > 0),that is, g(x) = xμ−1 exp(−x/ω)1{x ≥ 0}/{ωμ (μ)}, as the reference. Second, the criterion ismodified to the asymptotic weighted mean integrated squared error (AWMISE)AWMISE{fˆGG (x)} := b2

C52
4

 ∞
0

VI (2){xg (x)} w(x) dx + 1/2
nb


2

 ∞
0

g(x)
√ w(x) dx,
xwhere the weighting function w(x) ≥ 0 must be chosen to ensure finiteness of two integrals.
Given the specification of g(x), it turns out that w(x) = x3 fulfills this requirement. Then, theJournal of Nonparametric Statistics

47

AWMISE is simplified to


 2

1
VI (2)ω5/2 (μ + 5/2)
2 C5 Cμ (2μ)
ˆ
+ 1/2
,
AWMISE{fGG (x)} = b
4μ  2 (μ)
nb
(μ)
where
Cμ = 14 (μ − 2)2 (μ − 1)2 − (μ − 2)(μ − 1)2 (μ)
+ 12 (3μ − 4)(μ − 1)(μ) μ + 12 − (μ − 1)(μ) μ + 12 (μ + 1)
+ 14 (μ) μ + 12 (μ + 1) μ + 32 .
As a consequence, the AWMISE-optimal smoothing parameter is given by
‡‡

bGG =

 μ−1
2/5
4 VI (2)ω5/2 (μ)(μ + 5/2)
n−2/5 .
C52 Cμ (2μ)

In practice, parameters (μ, ω) need to be replaced by their method-of-moments estimates(μ̂, ω̂).2
It would be possible to choose b in a more sophisticated manner. For instance, replacing two
integrals in Equation (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.
2.3.

Examples of the GG kernels

This section introduces three special cases of the GG kernels. The sole reason why the threekernels are listed as examples is that for each of these kernels, approximations to the gammafunctions that appear in Mb (x), Hb (x) and Ab,ν (x) in Conditions 3–5 are readily available; see theproof of Theorem 2 in the Appendix for details.
It is worth emphasising that there could be many other examples belonging to this 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 the functional forms of (α, β, γ ) may not be unique, aslong 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 (α, β, γ ).
2.3.1.

ExamplesMG kernel. The MG kernel in Chen (2000) turns out to be an immediate example of the GG
kernels. Put⎧ x 
⎪for x ≥ 2b
⎨ b, x(α, β) =


x2
1 x 2
⎪
⎩
+ 1,
+b
for x ∈ [0, 2b)
4 b
4b

48

M. Hirukawa and M. Sakudo

and γ = 1 in Equation (3). Then, it collapses to
KGG (u; x, b) =uα−1 exp{−u/(β/α)}1{u ≥ 0},(β/α)α (α)(9)

which is the pdf of the gamma distribution G(α, β/α). Observe that α = ρb (x) in Chen (2000,
p. 473) and β/α = b. It follows that the above kernel finally reduces to the MG kernel
KMG(ρb (x),b) (u) =uρb (x)−1 exp(−u/b)1{u ≥ 0}.
bρb (x) {ρb (x)}Weibull kernel. To obtain the Weibull (W) kernel, let
⎧


⎪
2x
⎪
⎪
,x
⎨
b
(α, β) =
⎪
x2
1 x
⎪
⎪
+ 1,
+b
⎩
2 b
4b

for x ≥ 2b
for x ∈ [0, 2b)

and γ = α in Equation (3). Then, it becomesKGG (u; x, b) =αuα−1 exp[−{u/(β/ (1 + 1/α))}α ]1{u ≥ 0}.
{β/ (1 + 1/α)}α

Because the right-hand side is the pdf of the Weibull distribution W(α, β/ (1 + 1/α)), the W
kernel can be defined as3
KW(α,β/ (1+1/α)) (u)α=β/ (1 + 1/α)uβ/ (1 + 1/α)

α−1



u
exp −
β/ (1 + 1/α)α 

1{u ≥ 0}.

Nakagami-m kernel. Use exactly the same (α, β) as for the MG kernel but put γ = 2 inEquation (3). Then, it reduces to
KGG (u; x, b) =2uα−1 exp[−{u/(β(α/2)/ ((α + 1)/2))}2 ]1{u ≥ 0},{β(α/2)/ ((α + 1)/2)}α (α/2)

which is the pdf of the Nakagami-m distribution NM(α/2, (α/2)[β(α/2)/ {(α + 1)/2}]2 ) due
to Nakagami (1943, 1960).4 This distribution is frequently applied in telecommunications engineering as the distribution that can describe signal intensity of short-wave fading. In the end, theNakagami-m (NM) kernel is defined as
KNM(α/2,(α/2)[β(α/2)/ {(α+1)/2}]2 ) (u) =2.3.2.

2(α/2)α/2
u2(α/2)−1[(α/2){β(α/2)/ ((α + 1)/2)}2 ]α/2 (α/2)

α/2
2
× exp −
u1{u ≥ 0}.
(α/2){β(α/2)/ ((α + 1)/2)}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 different design points (x =0, 1, 2, 4) at which the smoothing is performed. For reference, the gamma (G) kernel (Chen 2000)
is also drawn in each panel.5 It is worth noting that for all plotted functions, the value of theJournal of Nonparametric Statistics

(a)

(c)

Figure 1.

x=0

x=2

(b)

(d)

49
x=1

x=4

Shapes of the GG kernels when b = 0.2.

smoothing parameter is fixed at b = 0.2. When smoothing is made at the origin (Panel (a)), theNM kernel collapses to a half-normal 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
and becomes flatter; in other words, each kernel changes the amount of smoothing in a locally
adaptive manner. It is worth emphasising that unlike variable bandwidth methods (e.g. Abramson
1982), adaptive smoothing for these kernels is achieved by a single smoothing parameter, whichmakes them much more appealing in empirical work. We can also observe that the more distantthe design point is from the boundary, the closer the shapes become to a symmetric one; inparticular, 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 (α, β, γ ) for each of the three kernels
satisfies Conditions 1–2. Hence, to demonstrate Theorem 2, it suffices to check that Conditions
3–5 hold for each kernel.
Theorem 2 Let fˆj (x) be the probability density estimator using the kernel j ∈ {MG, W , NM }.
Then, under Assumptions 1–3, the bias and variance of fˆj (x) can be approximated byBias{fˆj (x)} ∼ B1,j (x, f )b and
√(nb1/2 )−1 VI,j (2)f (x)/ xˆ
Var{fj (x)} ∼(nb)−1 VB,j (2)f (x)for interior x,
for boundary x,

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

50

M. Hirukawa and M. Sakudo

B1,j (x, f )jC5,jx ≥ 2bx ∈ [0, 2b)

MG
W
NM

1
π /12
1/2

(1/2)xf  (x)(π 2 /24)xf  (x)(1/4)xf  (x)ξb (x)f  (x)ξb (x)f  (x)ξb (x)f  (x)2

VI,j (2)√1/(2 √π )
1/(2
√ 2)
1/ 2π

Moreover, ξb (x) = {(1/2)(x/b) − 1}2 = O(1) in this case.
It follows from Equation (5) and Theorem 2 that the MSE-optimal smoothing parameters of
three kernel density estimators for interior x are√ 2/5f (x)/ xn−2/5 ,2 π
{xf  (x)}2√ 2/52/5(12/π 2 )2
f (x)/ xn−2/5 ,b∗W =23/2
{xf  (x)}2√ 2/52/5
23/2
f (x)/ x∗
bNM = √
n−2/5 .
{xf  (x)}2πb∗MG =

2/5

1
√

∗∗
∗∗
Recognise that b∗NM = 2b∗MG holds; even b∗∗
NM = 2bMG is the case by Equation (8), where bNM and∗∗bMG are MISE-optimal smoothing parameters of the NM and MG kernels, respectively. Then, by
Equation (6), the optimal-MSEs of fˆMG (x) and fˆNM (x) for interior x have the relation

MSE∗ {fˆMG (x)} ∼ MSE∗ {fˆNM (x)}
∼

5
4

1
4π

2/5

[f  (x){f (x)}2 ]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 these kernels become first-order
asymptotically equivalent, and both kernels on [0, ∞) are in a sense equivalent to the Gaussian
kernel on (−∞, ∞). In contrast, when best implemented, the MSE of fˆW (x) for interior x can beapproximated by
MSE∗ {fˆW (x)} ∼

5
4

π2
96

2/5

[f  (x){f (x)}2 ]2/5 n−4/5 .

(11)Comparing the factors of Equation (10) and (11) reveal that (5/4){1/(4π )}2/5 ≈ 0.454178 . . .
and (5/4)(π 2 /96)2/5 ≈ 0.503178 . . . Therefore, we can see that fˆW (x) is slightly inefficient thanfˆMG (x) and fˆNM (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 those describedin Section 2.2. After demonstrating applicability of MBC techniques for independent observations, we show the remaining two properties for weakly dependent observations, namely, validityof the first-order bias and variance approximations in GG density estimation that are stated in
Theorem 1, and weak consistency of GG density estimators when the true density is unboundedJournal of Nonparametric Statistics

51

at the origin. It is known that the G and MG kernels possess all these properties. Therefore, thissection establishes that these appealing properties essentially inhere in the GG kernels.
3.1.

Nonparametric MBC for GG density estimation

We start with examining whether GG density estimators in general admit MBC techniques. While
Hagmann and Scaillet (2007) and Gustafsson, Hagmann, Nielsen, and Scaillet (2009) propose
semiparametric MBC procedures using asymmetric kernels, we concentrate on two classes of
nonparametric MBC methods studied in Hirukawa (2010) and Hirukawa and Sakudo (2014).
The first class of MBC estimators, originally proposed by Terrell and Scott (1980), is defined asf˜TS,GG (x) = {fˆGG,b (x)}1/(1−c) {fˆGG,b/c (x)}−c/(1−c) ,where fˆGG,b (x) and fˆGG,b/c (x) signify GG density estimators using smoothing parameters b andb/c, and c ∈ (0, 1) is some predetermined constant that does not depend on the design point x.
The second class of MBC estimators due to Jones, Linton, and Nielsen (1995) is implied by theidentity f (x) = fˆGG,b (x){f (x)/fˆGG,b (x)} and defined as
 n
 KGG (Xi ; x, b)
˜fJLN,GG (x) = fˆGG,b (x) 1.
n i=1 fˆGG,b (Xi )Both f˜TS,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 and 3 asfollows. 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 satisfies b + (nb3 )−1 → 0 as n → ∞.

The next theorem refers to acceleration in bias convergence via the bias correction methods.
The proof is similar to the ones for Theorems 1 and 2 of Hirukawa and Sakudo (2014), and this
it is omitted.
Theorem 3 If Assumptions 1, 2a and 3a hold, and E{fˆGG (x)} admits the expansion E{fˆGG (x)} =f (x) + B1 (x, f )b + B2 (x, f )b2 + o(b2 ), where B1 (x, f ) (which can be found in Theorem 1) andB2 (x, f ) are kernel-specific functions depending on x and derivatives of f , then the biases off˜TS,GG (x) and f˜JLN,GG (x) can be approximated byBias{f˜TS,GG (x)} ∼1
1 1p(x)b2 :=
c
c 2





B21 (x, f )− B2 (x, f ) b2 ,f (x)Bias{f˜JLN,GG (x)} ∼ q(x)b2 := −f (x)B1 (x, h)b2 ,where B1 (x, h) is obtained by replacing f in B1 (x, f ) with h = h(x, f ) := B1 (x, f )/f (x).6 In
addition, the variance of each estimator is O{(nb1/2 )−1 } for interior x and O{(nb)−1 } forboundary x.

52

M. Hirukawa and M. Sakudo

The main point of the theorem is that under sufficient differentiability of f , the bias convergence of each MBC estimator is accelerated from O(b) to O(b2 ) with the order of magnitudein variance left unchanged. Whether the bias convergence may speed up depends crucially on
whether the second-order term in Bias{fˆGG (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, as
indicated in the proof of Theorem 2, the second-order term in Bias{fˆW (x)} is O(b3/2 ). Because
both MBC techniques merely improve the bias convergence up to O(b3/2 ), the theorem excludes
such inferior cases.7 In contrast, for the MG and NM kernels, each MBC technique ameliorates
their bias convergence to O(b2 ). Explicit forms of p(x) and q(x) for the NM kernel are⎧ 2 x x2 
1 1 
x {f (x)}2⎪⎪−
f (x) + f (x) + f (x)⎪⎪⎪32 f (x)
8 2
3
4
⎪
⎪
⎪

⎨ 
ξb (x)+x/b
ξb (x)+x/b

2

2
+ 1)
pNM (x) = 1 {ξb (x)f (x)} − ξb (x) + x ( 2 )( 2⎪2ξ(x)+x/b+1
b
2
⎪
f (x)
b
((
))
⎪
⎪
⎪
x
 x 2  2
⎪
⎪
⎪
− 2
ξb (x) +⎩bb⎧  
x x f (x)⎪⎪for x ≥ 2b,⎨−f (x)4
4 f (x)qNM (x) =ξ (x)f  (x) ⎪⎪⎩−f (x)ξb (x) b
for x ∈ [0, 2b),f (x)for x ≥ 2b,

for x ∈ [0, 2b),where ξb (x) can be found in Theorem 2; see Hirukawa and Sakudo (2014) for functional formsof p(x) and q(x) for the MG kernel.
To provide the variance approximation of each MBC estimator, we must additionally specify
the functional form of (α, β, γ ) = (αb (x), βb (x), γb (x)). For example, for each of the MG and NM
kernels,Var{f˜TS,GG (x)} ∼ (nb1/2 )−1 λ(c)VI (2)
√
√
1/2
−1
f (x)/ x and Var{f˜JLN,GG (x)} ∼ (nb ) VI (2)f (x)/ x for interior x, where√(1 + c5/2 )(1 + c)1/2 − 2 2c3/2
λ(c) =(1 + c)1/2 (1 − c)2
is increasing in c ∈ (0, 1) with limc→0 λ(c) = 1 and limc→1 λ(c) = 27/16. In other words,
Var{f˜TS,MG (x)} and Var{f˜TS,NM (x)} at interior design points tend to inflate slightly after the bias
correction, whereas Var{f˜JLN,MG (x)} and Var{f˜JLN,NM (x)} are first-order asymptotically equivalent
to Var{fˆMG (x)} and Var{fˆNM (x)}, respectively.
As a consequence, we can find an interesting relation between f˜JLN,MG (x) and f˜JLN,NM (x) once
again. The MSEs of these estimators for interior x can be approximated by
MSE{f˜JLN,MG (x)} ∼ −f (x)x2 2

b4 +

f (x)
1
√ √ ,
nb1/2 2 π x

 2
f (x)
1
x f  (x) 
b4 + 1/2 √ √
4 f (x)nb
2π x√
   2
2 f (x)x x f (x)1b4 + 1/2 √ √ .
−f (x)=
16
2 2 f (x)nb 2 π xx
MSE{f˜JLN,NM (x)} ∼ −f (x)4

x f  (x)
2 f (x)Journal of Nonparametric Statistics

53

The smoothing parameter values that minimise these approximations are −4/9
f (x)x x f  (x) 
n−2/9 ,f (x)√ √22/3 2 π x
2 2 f (x) −4/9
f (x)x x f  (x) 
n−2/9 .
b∗JLN,NM = 21/3 √ √f (x)
2 2 f (x)2 π xb∗JLN,MG =1



Observe that the relation b∗JLN,NM = 2b∗JLN,MG holds once again. Moreover, when best implemented, the optimal-MSEs of two estimators for interior x are first-order asymptoticallyequivalent, that is,MSE∗ f˜JLN,MG (x) ∼ MSE∗ {f˜JLN,NM (x)}
∼

3.2.

9
28/3



f (x)
√ √
2 π x

8/9
f (x)

x
2



x f  (x)
2 f (x) 2/9

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 timeseries data. Examples include estimation of the distribution of important financial variables suchas short-term interest rates or trading volumes, and even the baseline hazard in financial 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, forinstance, in Bouezmarni and Rombouts (2010), they assume strong mixing processes with an
exponentially decaying mixing coefficient and exclusively study the MG kernel.8 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 {Xi } is a nonnegative, strictly stationary and strong mixing process with themixing coefficient α() of size −(2r − 2)/(r − 2) 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 differentiable, and fj is uniformly bounded.
The next theorem states that results in Theorem 1 are carried through even when positive
weakly dependent observations are used.
Theorem 4
3.3.

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

Convergence properties of the GG density estimator for the density unbounded at theoriginAs the third property we obtain convergence results of fˆGG (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 realised volatilities, for instance. In this case, it is highly likely that thetrue density has a pole at x = 0 (e.g. Malec and Schienle 2014). Such shapes also appear in many

54

M. Hirukawa and M. Sakudo

Figure 2.

GG density estimates when the true distribution is G(0.75, 1.25).

other applications including spectral densities of long memory processes. The following two theorems demonstrate weak consistency and the relative convergence of fˆGG (x) as x → 0, provided
that the assumption on the smoothing parameter is replaced by Assumption 3c.
Assumption 3c

The smoothing parameter b satisfies b + (nb2 )−1 → 0 as n → ∞.

Theorem 5 If the true density f (x) is unbounded at x = 0 and Assumptions 1b and 3c hold,pthen fˆGG (0) → ∞.
Theorem 6 Suppose that the true density f (x) is unbounded at x = 0 and continuously
differentiable in the neighbourhood of the origin. If Assumptions 1b and 3c also hold, then fˆ (x) − f (x)  p

 GG
→0



f (x)
as x → 0.
Bouezmarni and Scaillet (2005, p. 399) focus exclusively on random samples and find that theproperty in Theorem 5 is peculiar to the G and MG kernels and not shared with other asymmetrickernels 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 figure indicates that estimates using three examples of the GG kernels, as well as the G
estimate, can indeed capture the shape of the density near the boundary reasonably well.

4.
4.1.

Finite sample performanceSet-upThis simulation study examines accuracy of GG density estimators. We compare finite sample performance of three GG estimators using the W, NM and MG kernels with that of threeJournal of Nonparametric Statistics

55

other density estimators, namely, (i) the density estimator using yet another asymmetric kernel,(ii) the one obtained via back-transforming a symmetric kernel density estimate based on the logtransformed (LT) data, and (iii) a symmetric kernel density estimator in the original scale with aproper boundary correction. For (i), because of its popularity in empirical studies, the G kernel
is chosen as the asymmetric kernel that does not belong to the GG kernels. For (ii) and (iii), theLT density estimator using the Gaussian kernel and the local linear (LL) density estimator using
the Epanechnikov kernel (e.g. Jones 1993) are chosen, respectively.
For each distribution in the list below, 1000 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), as the performance measure for each estimator f¯ , we compute the root integrated squared error (RISE) RISE{f¯ (x)} =∞{f¯ (x) − f (x)}2 dx. In our report, the integral is approximated over the 500 points. The
0smoothing parameter b (for W, NM, MG, and G) or the bandwidth h (for LT and LL) is chosen as the minimiser of 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, Nielsen, Guillén, and Bolancé (2005) for modelling distributionsof 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 theshape of income distributions well. Distribution 6, which is unbounded at the origin, is prepared to investigate the property stated in Theorem 5. Figure 3 presents the shapes of these
densities.
(1)

Gamma:(2)

Weibull:(3)

Half-Normal:(4)

Log-Normal:(5)

Generalised Champernowne:(6)

Gamma with Pole:

4.2.

Simulation results

f (x) = xα−1 exp(−x/β)1{x ≥ 0}/{β α (α)},(α, β) = (1.5, 1).
f (x) = (α/β)(x/β)α−1 exp{−(x/β)α }1{x ≥ 0},(α, β) = (1.5,√ 1.5) .
f (x) = {2/( 2π σ )} exp{−(x − μ)2 /(2σ 2 )}1{x ≥ 0},(μ, σ ) = (0, 1.5)
√ .
f (x) = {1/(x 2π σ )} exp{−(log x − μ)2 /(2σ 2 )}1{x ≥ 0},(μ, σ ) = (0, 0.75) .
f (x) = α(x + c)α−1 {(M + c)α − cα }1{x ≥ 0}
×{(x + c)α + (M + c)α − 2cα }−2 ,(α, M , c) = (2.5, 0.5, 0.25).
f (x) = xα−1 exp(−x/β)1{x ≥ 0}/{β α (α)},(α, β) = (0.75, 1.25).

Table 1 presents averages and standard deviations of RISEs and averages of tuning parametervalues (i.e. smoothing parameter values for W, NM, MG, and G, and bandwidth lengths for LT andLL) over 1000 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 andDistribution 6 for all sample sizes. This is congruous with the results in Chen (2000) and Scaillet(2004).
Each of first three distributions is chosen so that one of the GG estimators can have clear
advantage. As expected, W and MG, as well as G, perform well for Distributions 1 and 2, whereas
NM dominates for Distribution 3. Poor performance of W, MG and G for Distribution 3 could be56

Figure 3.

M. Hirukawa and M. Sakudo

Shapes of true densities for Monte Carlo simulations.

attributed to their difficulty in capturing the shape of the pdf near the origin. The pdf is decreasing
and satisfies the shoulder condition f  (0) = 0. Accordingly, the data suggest concavity of thedensity in the vicinity 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 find that LL
exhibits comparable performance to the GG estimators, whereas LT is consistently 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 predictedby Theorem 5, the GG estimators also exhibit reasonably good performance for Distribution 6.
In contrast, LL and LT are dominated for Distributions 4 and 6 and Distribution 5, respectively.
A rationale as to why the LT estimator fˆLT (x) does not necessarily perform well (although it
is popularly applied in empirical works) may be found in its bias property. Following Wand,
Marron, and Ruppert (1991), we can approximate the bias and variance of the estimator as
1{f (x) + 3xf  (x) + x2 f  (x)}h2 ,2√(nh)−1 f (x)/(2 π x) if x/h → ∞,Var{fˆLT (x)} ∼√(nh2 )−1 f (x)/(2 π η) if x/h → η > 0.

Bias{fˆLT (x)} ∼While fˆLT (x) shares the shrinking variance property as the design point moves away from theboundary (which comes from the fact that the variance coefficient is proportional to f (x)/x) withasymmetric kernel estimators, its leading bias takes a less favourable form in that it contains
three terms involving the true density and its first two derivatives.
We also make a few remarks on tuning parameters b and h. First, both hLT and hLL appear to besystematically long. Large values of the former may be attributed to the fact that LT observations
tend to spread out toward the large (in magnitude) negative side for each distribution. Second,
for each combination of the sample size and distribution (except Distribution 6), we can finda consistent ordering of bG , bMG , bW , and bNM from the smallest to the largest. Third, whetherbNM becomes roughly twice the size of bMG as predicted in Section 2.3.2 depends cruciallyon distributions and sample sizes. The derivation of Equation (7) (and thus Equation (8)) isbuilt on the trimming argument for the boundary region. Therefore, whether finite 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 find the relation bNM ≈ 2bMG in Table 1,

Journal of Nonparametric Statistics
Table 1.

Averages of performance measures and tuning parameter values.
n = 100

GG
Non-GG

GG
Non-GG

GG
Non-GG

GG
Non-GG

GG
Non-GG

GG
Non-GG57

n = 200

n = 500

RISE

b or hRISEb or hRISEb or h

W
NM
MG
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
1.0272(1) Gamma
0.0294 (0.0081)
0.0306 (0.0076)
0.0289 (0.0088)
0.0290 (0.0098)
0.0348 (0.0114)
0.0302 (0.0088)

0.17010.18610.1105
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
0.09750.06830.06010.31490.5152

W
NM
MG
G
LT
LL

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
0.11370.37300.8234(2) Weibull
0.0297 (0.0090)
0.0307 (0.0088)
0.0286 (0.0092)
0.0294 (0.0103)
0.0368 (0.0110)
0.0294 (0.0092)

0.12280.1382
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
0.25840.5098

W
NM
MG
G
LT
LL

0.0274 (0.0113)
0.0251 (0.0120)
0.0303 (0.0117)
0.0327 (0.0112)
0.0586 (0.0191)
0.0256 (0.0119)(3) Half-Normal
0.3445 0.0225 (0.0083)
0.3809 0.0207 (0.0088)
0.2662 0.0245 (0.0087)
0.1750 0.0262 (0.0087)
0.4735 0.0457 (0.0140)
1.4774 0.0203 (0.0094)

0.27450.31520.20390.13250.39971.25980.0172 (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
0.32241.0193

W
NM
MG
G
LT
LL

0.0429 (0.0153)
0.0447 (0.0152)
0.0416 (0.0158)
0.0458 (0.0150)
0.0401 (0.0166)
0.0482 (0.0147)(4) Log-Normal
0.0830 0.0332 (0.0110)
0.0932 0.0343 (0.0108)
0.0624 0.0324 (0.0114)
0.0535 0.0360 (0.0108)
0.3207 0.0315 (0.0122)
0.4415 0.0381 (0.0106)

0.06540.07490.04800.03900.27820.36830.0242 (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

W
NM
MG
G
LT
LL

0.0477 (0.0169)
0.0477 (0.0161)
0.0504 (0.0190)
0.0504 (0.0195)
0.0700 (0.0246)
0.0513 (0.0183)(5) Generalised Champernowne
0.1324 0.0391 (0.0126) 0.0922
0.1448 0.0390 (0.0122) 0.1033
0.0881 0.0403 (0.0141) 0.0618
0.0676 0.0403 (0.0151) 0.0485
0.4451 0.0544 (0.0177) 0.3807
0.4850 0.0413 (0.0137) 0.3618

0.0295 (0.0101)
0.0294 (0.0099)
0.0298 (0.0106)
0.0301 (0.0107)
0.0394 (0.0117)
0.0308 (0.0101)

0.05470.06370.03880.03180.31180.2637

W
NM
MG
G
LT
LL

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

0.0359 (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 Pole
0.0858 0.0494 (0.0136)
0.0855 0.0523 (0.0128)
0.0803 0.0500 (0.0131)
0.0571 0.0494 (0.0157)
0.7389 0.0498 (0.0213)
0.5280 0.0549 (0.0125)

0.0591
0.0586
0.05540.03890.62980.3905

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

in unreported simulations where the truth is GG(5, 2, 2.5) (that puts a very small weight over theboundary region), we obtain (bNM , bMG ) = (0.0884, 0.0452), (0.0658,0.0334), (0.0460,0.0232)
for n = 100, 200, 500 so that the relation is confirmed for each sample size.
5.

Conclusion

Unlike symmetric kernels, exploring asymptotics on asymmetric kernels has relied onkernel-specific, diversified approaches. To pursue a unified approach in their asymptotics, this58

M. Hirukawa and M. Sakudo

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 can be 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 finite sample properties of GG density estimators.
There are two possible research extensions. First, choice methods of the smoothing parameterb in GG density estimation need to be further investigated. Promising candidates include a moresophisticated plug-in method like the solve-the-equation plug-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.
Second, goodness-of-fit tests can be built on the GG kernels. Fernandes and Grammig (2005)
and Fernandes, Mendes, and Scaillet (2011) obtain favorable finite-sample results from applying
the G kernel to specification testing for duration models and testing for symmetry, respectively.
Combining the GG kernels with these testing procedures appear to be another good applicationof the kernels. These extensions are currently under authors’ investigation and will be addressedin separate papers.

Acknowledgements
We would like to thank the editor Irène Gijbels, an associate editor and two anonymous referees for their comments thathave substantially improved this paper. We are also grateful to Jin Seo Cho, Benedikt Funke, Bruce Hansen, Susumu Imai,
Yoshihide Kakizawa, Yoshihiko Nishiyama, Katsumi Shimotsu, Nabil Zougab, and participants 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 andsuggestions. The views expressed herein and those of the authors do not necessarily reflect the views of the Development
Bank of Japan.

Conflict of interest disclosure statement
No potential conflict of interest was reported by the authors.

Funding
The first author gratefully acknowledges financial support from Japan Society of the Promotion of Science [grant number
23530259].

Notes
1. Conditions 1–5 even establish approximations to the bias and variance of nonparametric regression estimators (e.g.
local constant and LL estimators) using the GG kernels.
2. Simulation results of GG density estimators with the rule-of-thumb smoothing parameters plugged in are available
on the first author’s webpage.
3. This definition differs from the one given in Kuruwita, Kulasekera, and Padgett (2010, Table 1), who constructtheir Weibull kernel from a different motivation. Apparently, their definition
KW (u; x, b) =tends to be unbounded near the origin.

   1  u 1/b−1
u 1/b
exp −
1{u ≥ 0}bx x
x

Journal of Nonparametric Statistics

59

4. The pdf of NM(μ, ω) (μ ≥ 1/2, ω > 0) isp(y; μ, ω) =

5.
6.

7.

8.

 μ 
2μμ 2μ−1
exp − y2 1{y ≥ 0}.
yωμ (μ)ω

It is widely recognised that the Nakagami-m distribution was first proposed in Nakagami (1960). In reality, however,the distribution originates as early as in Nakagami (1943).
This kernel does not belong to the GG kernels, because it can be obtained by setting (α, β) = (x/b + 1, x + b) inEquation (9).
Because an error is found in the bias approximation for the Terrell and Scott (1980)-type MBC estimator in
Hirukawa (2010) and Hirukawa and Sakudo (2014), we have corrected the error here. We thank Dr Nabil Zougab
for pointing it out.
Because the third-order term in Bias{fˆW (x)} is O(b2 ), it would be possible in principle to improve 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 from the bias correction via iteration, and thus we do not consider
such inferior cases any further.
Carrasco and Chen (2002) provide the conditions that make GARCH processes stationary and β-mixing withexponential decay, whereas Chen, Hansen, and Carrasco (2010) discuss the conditions that can establish β-mixing
with exponential decay in scalar diffusion processes. In relation to the latter, Feller’s square-root process and its
inverse, which have been employed as models of short-term interest rates by Cox, Ingersoll, and Ross (1985) and
Ahn and Gao (1999), respectively, are examples of exponentially decaying β-mixing processes. Since β-mixing
implies strong mixing, their assumption can cover many important applications in economics and finance.

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Appendix
In order to approximate the gamma function, we frequently refer to the following well-known formulae:
Stirling’s formula (SF).
(z + 1) =



√
1
1
−3
2π zz+1/2 e−z 1 +
+
O(z
)
+
12z
288z2

as z → ∞.

Series expansion of the log gamma function (SELG).
log (1 + z) = −γ z +∞(−1)k ζ (k)
k=2

k

zk

for |z| < 1,

where (only in this context) γ = limn→∞ ( nk=1 k −1 − log n) = 0.5772156649 . . . is Euler’s constant, and ζ (k) =∞ −k(k > 1) is the Riemann zeta function.
n=1 nLegendre’s duplication formula (LDF).
(z) z +

A.1.

1
2

√

=

π
(2z)
22z−1

for z > 0.

Proof of Theorem 1dBias. Let θx = GG(α, β(α/γ )/ {(α + 1)/γ }, γ ). Then, a second-order Taylor expansion of E{fˆGG (x)} around θx = x
ˆyields E{fGG (x)} = f (x) + E(θx − x)f  (x) + (1/2)E(θx − x)2 f  (x) + o{E(θx − x)2 }. It follows from Equation (2) thatE(θxm ) = β m{(α/γ )}m−1 {(α + m)/γ }.
[{(α + 1)/γ }]m

In particular, E(θx ) = β (by construction) andE(θx2 ) = β 2

(α/γ ){(α + 2)/γ }= β 2 Mb (x).
[{(α + 1)/γ }]2Using Conditions 1 and 3, we have, for x ≥ C1 b, E(θx ) = x and E(θx2 ) = x2 + C5 xb + o(b), and thus E(θx − x) = 0
and E(θx − x)2 ∼ C5 xb. As a consequence, Bias{fˆGG (x)} ∼ (C5 /2)xf  (x)b. On the other hand, for x ∈ [0, C1 b), E(θx ) =ϕb (x) = O(b) and x = O(b) hold, and thus E(θx − x) = [{ϕb (x) − x}/b]b := ξb (x)b, where ξb (x) = O(1). Moreover, it
follows from E(θx2 ) = O(b2 ) that E(θx − x)2 = O(b2 ). Therefore, Bias{fˆGG (x)} ∼ ξb (x)f  (x)b.

Journal of Nonparametric Statistics

61

2 (X ; x, b)} + O(1)]. A straightforward
Variance. As usual, we consider the approximation Var{fˆGG (x)} = (1/n)[E{KGGicalculation yields, for ν ∈ R+ ,
γ ((α + 1)/γ ) ν−1
{(ν(α − 1) + 1)/γ }νKGG(u; x, b) =βν (ν(α−1)+1)/γ {(α/γ )}2ν−1·

γ u{ν(α−1)+1}−1 exp[−{ β(α/γ )/(ν 1/γu ((α+1)/γ )) }γ ][β(α/γ )/{ν 1/γ ((α + 1)/γ )}]ν(α−1)+1 {(ν(α − 1) + 1)/γ }β(α/γ )= Ab,ν (x) · pdf of GG ν(α − 1) + 1, 1/γ,γ.
ν ((α + 1)/γ )1 {u ≥ 0}

Hence, Var{fˆGG (x)} ∼ (1/n)Ab,2 (x)E{f (ςx )}, wheredςx = GG 2α − 1,β(α/γ ),γ
21/γ {(α + 1)/γ }.

By the mean-value theorem, E{f (ςx )} = f (x) + E(ςx − x)f  (ς̄x ) for some ς̄x joining ςx and x. It follows from Equation(2) that(α/γ )(2α/γ )= βHb (x).
E(ςx ) = β 1/γ2 {(α + 1)/γ }{(2α − 1)/γ }Then, Condition 4, together with Condition 1, implies that E(ςx − x) = o(1) so that E{f (ςx )} ∼ f (x) regardless of theposition of x. Finally, Condition 5 establishes the approximation to the variance.

A.2.

Proof of Theorem 2

A.2.1.

The MG kernel

Condition 3. Observe that Mb (x) = 1 + 1/α. For x ≥ 2b, Mb (x) = 1 + (1/x)b so that C5,MG = 1. On the other hand, forx ∈ [0, 2b), α = (x/b)2 /4 + 1 = O(1), and thus Mb (x) = O(1) holds.
Condition 4. Substituting α = x/b into Hb (x) = 1 − (2α)−1 yields Hb (x) = 1 + O(b) = 1 + o(1) for interior x. On
the other hand, for boundary x, α = O(1) regardless of whether α = x/b (when x = O(b) and x ≥ 2b) or α = (x/b)2 /4 +1 (when x = O(b) and x ∈ [0, 2b)). It follows that Hb (x) = O(1).
Condition 5. Because β/α = b regardless of the position of x, we have
{ν(α − 1) + 1}.
ν ν(α−1)+1 {(α)}ν
√√
For interior x, SF yields, as α = x/b → ∞, (α) ∼ 2πα α−1/2 e−α and {ν(α − 1) + 1} ∼ 2π {ν(α −ν(α−1)+1/2−ν(α−1)e. Then,1)}
Ab,ν (x) = b1−ν

b1−ν (α − 1)(1−ν)/2.
√
ν 1/2 ( 2π )ν−1√
The result immediately follows from defining VI,MG (ν) := {ν 1/2 ( 2π)ν−1 }−1 and recognising that (α − 1)(1−ν)/2 =(1−ν)/2(1−ν)/2(1−ν)/2(1 − 1/α)∼ (x/b). For boundary x, the result is established by definingα⎧{ν(κ − 1) + 1}x⎪⎪
if → κ ≥ 2⎨ ν(κ−1)+1bν
{(κ)}ν
.
VB,MG (ν) :=x
((ν/4)κ 2 + 1)⎪⎪
if → κ ∈ (0, 2)⎩ (ν/4)κ 2 +1bν{(κ 2 /4 + 1)}νAb,ν (x) ∼A.2.2.

The W kernel

Condition 3. Observethat Mb (x) = (1 + 2/α)/{(1 + 1/α)}2 . For x ≥ 2b, we may pick an arbitrarily small b > 0 so√that |2/α| = | 2b/x| ≤ 1. Then, by SELG and ζ (2) = π 2 /6, two gamma functions admit the following approximations:


2
2
2γ
(π 2 /3) + 2γ 2 1++ O(α −3 ),= exp log  1 +
=1−
+
α
α
α
α2


1
γ
(π 2 /12) + (γ 2 /2)1+ O(α −3 ).
= exp log  1 +
=1− +
 1+
α
α
α
α2

62

M. Hirukawa and M. Sakudo

Applying a geometric series expansion to the approximation to {(1 + 1/α)}−2 finally delivers Mb (x) ∼ 1 +(π 2 /6)/α 2 = 1 + {π 2 /(12x)}b so that C5,W = π 2 /12. On the other hand, for x ∈ [0, 2b), α = (x/b)/2 + 1 = O(1), andthus Mb (x) = O(1) is the case.
Condition 4. In this case, Hb (x) = {21/α (1 − 1/α)(1 + 1/α)(1 − 1/α)}−1 . It is easy to see that Hb (x) = O(1) for
boundary x. On the other hand, for interior x, 21/α = exp{(1/α) log 2} = 1 + O(b1/2 ), and (1 − 1/α)(1 + 1/α)(1 −1/α) = 1 + O(b1/2 ) by SELG. Therefore, Hb (x) = 1 + O(b1/2 ) = 1 + o(1) holds.
Condition 5. Ab,ν (x) reduces toAb,ν (x) =

α
β

ν−1

{(1 + 1/α)}ν−1 (ν + (1 − ν)/α).
ν ν+(1−ν)/α√For interior x, because α = 2x/b → ∞, we may approximate (1 + 1/α) ∼ 1, {ν + (1 − ν)/α} ∼ (ν) andν+(1−ν)/αν∼ ν . In addition, (α/β)ν−1 = 2(ν−1)/2 (xb)(1−ν)/2 , and thusνAb,ν (x) ∼ 2(ν−1)/2

(ν)(xb)(1−ν)/2 := VI,W (ν)(xb)(1−ν)/2 .
ννFor boundary x, the result is established by defining√√
2 (ν−1)/2 {(1 + 1/ 2κ)}ν−1 (ν + (1 − ν)/ 2κ)√ν ν+(1−ν)/ 2κ
VB,W (ν) :=κ
⎪2(κ + 2) ν−1 {(1 + 2/(κ + 2))}ν−1 {ν + 2(1 − ν)/(κ + 2)}⎪⎩
κ2 + 4ν ν+2(1−ν)/(κ+2)⎧⎪⎪⎨

A.2.3.

x→ κ ≥ 2,bx
if → κ ∈ (0, 2).
bif

The NM kernel

2
Condition 3. Observe that
√ Mb (x) = (α/2)[(α/2)/ {(α + 1)/2}] . For x ≥ 2b, it follows from LDF that Mb (x) =(α/2)[2α−1 {(α/2)}2 /{ π(α)}]2 . Next, SF implies that, as α = x/b → ∞,√
12π α α−1/2 e−α 1 ++ O(α −2 ) ,12α α  √  α α/2−1/21
e−α/2 1 += 2π
+ O(α −2 ) .
226α
(α) =(A1)(A2)

Then, we find that the approximation of Mb (x) takes a very simple form Mb (x) = 1 + (2α)−1 + O(α −2 ) = 1 +(2x)−1 b + o(b) so that C5,NM = 1/2. On the other hand, for x ∈ [0, 2b), α = (x/b)2 /4 + 1 = O(1), and thus Mb (x) =O(1) holds.
x because of theCondition 4. Hb (x) = (α − 1/2)(α/2)(α)/[21/2 {(α + 1)/2}(α + 1/2)] is O(1) for boundary√
same reason as in the proof for the MG kernel. For interior x, using LDF and then (2α) ∼ 2π(2α)2α−1/2 e−2α as
α = x/b → ∞, as well as Equations (A1) and (A2), yields
Hb (x) =α − (1/2) 23α−2 {(α/2)}2 (α)
1∼1−= 1 + O(b) = 1 + o(1).
21/2
π
(2α)
2α

Condition 5. In this case,Ab,ν (x) = β 1−ν2ν−1ν ν(α−1)/2+1/2{((α + 1)/2)}ν−1 {ν(α − 1)/2 + 1/2}.
{(α/2)}2ν−1

For interior x, by LDF, Ab,ν (x) reduces toAb,ν (x) = β 1−ν√( π )ν {(α)}ν−1 {ν(α − 1)}.
ν ν(α−1)/2+1/2 2(ν−1)(α−1)+ν(α−1)−1 {ν(α − 1)/2}{(α/2)}3ν−2
2ν−1

√
√
Next, by SF, as α = x/b → ∞, {ν(α − 1)} ∼ 2π {ν(α − 1)}ν(α−1)−1/2 e−ν(α−1) and {ν(α − 1)/2} ∼ 2π{ν(α −ν(α−1)−1/2−ν(α−1)/2e. Substituting these approximations, as well as Equations (A1) and (A2), we finally deduce
1)/2}Journal of Nonparametric Statistics

63

that
Ab,ν (x) ∼ β 1−ν α (ν−1)/2eν/2√
ν 1/2 ( π )ν−1

1−

1
α

ν(α−1)/2.

Moreover β 1−ν α (1−ν)/2 = (xb)(1−ν)/2 and (1 − 1/α)ν(α−1)/2 ∼ e−ν/2 , and thusAb,ν (x) ∼1(xb)(1−ν)/2 := VI,NM (ν)(xb)(1−ν)/2 .
√ν 1/2 ( π )ν−1

For boundary x, the result is established by defining

VB,NM (ν) :=

⎧
⎪
⎪
⎨
⎪
⎪
⎩

2
κ

ν−1

8
κ2 + 4{((κ + 1)/2)}ν−1 {(ν(κ − 1) + 1)/2}
ν (ν(κ−1)+1)/2 {(κ/2)}2ν−1
ν−1{(κ 2 /8 + 1)}ν−1 ((ν/8)κ 2 + 1/2)
ν (ν/8)κ +1/2 {(κ 2 /8 + 1/2)}2ν−12x→ κ ≥ 2,bx
if → κ ∈ (0, 2).
bif