Statistics and Probability Letters 145 (2019) 158–165

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Statistics and Probability Lettersjournal homepage: www.elsevier.com/locate/stapro

Another bias correction for asymmetric kernel density
estimation with a parametric start
Masayuki Hirukawa a, *, Mari Sakudo b,c,d,1,2,3
aRyukoku University, Japan
Development Bank of Japan, Japanc
Waseda University, Japand
Japan Economic Research Institute, Japanb

articlea b s t r a c t

info

Article history:
Received 13 April 2018
Received in revised form 23 July 2018
Accepted 2 September 2018
Available online 24 September 2018

This paper studies yet another semiparametric bias-corrected density estimation using
asymmetric kernels. The estimator can be obtained by making a multiplicative bias correction for the initial parametric model twice, and it is shown to establish rate improvement
when best implemented.
© 2018 Elsevier B.V. All rights reserved.

MSC:
62G7
62G20
Keywords:
Asymmetric kernel
Bias reduction
Boundary effectSemiparametric density estimation
Smoothing

1. Introduction
This paper aims at establishing rate improvement via yet another multiplicative bias correction (‘‘MBC’’) method for
univariate probability density estimation using asymmetric kernels. For a random variable X ∈ R drawn from a distribution
with an unknown density f , the MBC method considered throughout is based on the identity
f (x) ≡ g (x)

{

f (x)
g (x)

}

:= g (x) r (x) ,(1)

where g (x) is an initial density estimator and r (x) serves as a correction factor.
In what follows, the support of X is assumed to have a boundary; to be more specific, supp (X ) is assumed to be either[0, 1] or R+ . This type of data can be frequently observed in economics and finance. For example, recovery rates take values
between 0 and 1, whereas variables such as wages, incomes and insurance claims (or financial losses) are by construction
nonnegative. A convenient way of avoiding boundary bias due to kernel smoothing for such data is to employ asymmetric* Correspondence to: Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto 612-8577, Japan.

E-mail addresses: hirukawa@econ.ryukoku.ac.jp (M. Hirukawa), marisakudo@gmail.com (M. Sakudo).
1 Research Institute of Capital Formation, Development Bank of Japan, 9-7, Otemachi 1-chome, Chiyoda-ku, Tokyo 100-8178, Japan.

2 Waseda Institute of Political Economy, Waseda University, 6-1 Nishiwaseda 1-chome, Shinjuku-ku, Tokyo 169-8050, Japan.
3 Japan Economic Research Institute, 9-2, Otemachi 1-chome, Chiyoda-ku, Tokyo 100-0004, Japan.
https://doi.org/10.1016/j.spl.2018.09.002
0167-7152/© 2018 Elsevier B.V. All rights reserved.

M. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

159

Table 1
Functional forms of asymmetric kernels.
Kernel (j)Kj(x,b) (u)B (Chen, 1999)

KB(x,b) (u) = B{b/x+( 1,(1)−x)/b+1} 1 {u ∈ [0, 1]} .

MB (Chen, 1999)

KMB(x,b) (u) =ub/x 1−u (1−x)/b

ϱ (x)−1
ϱ (x)−1u b,0(1−u) b,1
1
B ϱb,0 (x),ϱb,1 (x)where ϱb,0 (x) ={{}ϱb (x)

for x ∈ [0, 2b),
for x ∈ [2b, 1]x/b{(1 − x) /b
ϱb,1 (x) = ϱ 1 − x)b(ϱb (x) = 2b2 + 5/2 −{u ∈ [0, 1]} ,for x ∈ [0, 1 − 2b], andfor x ∈ (1 − 2b, 1]√

4b4 + 6b2 + 9/4 − x2 − x/b.

ux/b exp(−u/b)

G (Chen, 2000)

KG(x,b) (u) = bx/b+1 Γ (x/b+1) 1 {u ≥ 0} .

MG (Chen, 2000)

KMG(x,b) (u) =uρb (x)−1 exp(−u/b)1bρb (x) Γ {ρ (x)}bwhere ρb (x) ={KNM (x,b) (u) =for x ≥ 2b.
for x ∈ [0, 2b)2uα−1 exp −{u/(β Γ (α/2)/Γ ((α+1)/2))}2{β Γ (α/2)/Γ ((α+1)/2)}α Γ (α/2)[

NM (Hirukawa and Sakudo, 2015){u ≥ 0} ,x/b(1/4) (x/b)2 + 1

1 {u ≥ 0} ,(x/b, x)()(1/4) (x/b)2 + 1, x2 /(4b) + b{
where (α, β) =]for x ≥ 2b
for x ∈ [0, 2b).

kernels. Let Kj(x,b) (·) be the asymmetric kernel indexed by j that depends on a design point x and a smoothing parameterb. We exclusively consider the beta (‘‘B’’ ) and modified beta (‘‘MB’’) kernels for supp (X ) = [0, 1] and the gamma (‘‘G’’),
modified gamma (‘‘MG’’) and Nakagami-m (‘‘NM’’) kernels for supp (X ) = R+ . Functional forms of these kernels arepresented in Table 1. Although our main focus is on the kernels, the results in this paper can be straightforwardly extended
to a wide variety of asymmetric kernels that have been proposed so far (e.g., Jin and Kawczak, 2003; Scaillet, 2004). Using arandom sample {Xi }ni=1 and the kernel j ∈ {B, MB, G, MG, NM }, we investigate the estimator of f of the form implied by theidentity (1){f˜j (x) := g (x) r̂j (x) := g (x)n

1 ∑ Kj(x,b) (Xi )ni=1g (Xi )}.

(2)

A few special cases of the estimator of this class have been already considered in the literature. First, if we take g (x) ≡ 1(i.e., (improper) uniform density), then f˜j (x) collapses to an ordinary asymmetric kernel density estimator (‘‘AKDE’’)nfˆj (x) =1∑nKj(x,b) (Xi ) .

(3)

i=1

Second, when g (x) belongs to a parametric family, f˜j (x) reduces to the Hjort and Glad (1995, ‘‘HG’’)-type semiparametric
MBC estimator f˜HG,j (x) studied by Hagmann and Scaillet (2007). Third, if g (x) is set equal to the AKDE (3), then f˜j (x) becomes
the Jones et al. (1995, ‘‘JLN’’)-type fully nonparametric MBC estimator f˜JLN ,j (x) examined by Hirukawa (2010) and Hirukawa
and Sakudo (2014).
The HG-MBC estimator has a O (b) bias in general, as the initial parametric model is typically misspecified. It becomes
unbiased up to the order considered only under( )correct specification of the model. In contrast, the JLN-MBC estimator
improves the bias convergence from O (b) to O b2 even in the worst-case scenario (i.e., under misspecification), provided
that the true density has sufficient smoothness. Then, within the framework of density estimation using standard symmetrickernels, Jones et al. (1999, ‘‘JSH’’) attempt to make the most of attractive(featuresin HG- and JLN-MBC by applying yet another)
MBC step to the HG-MBC estimator. The resulting estimator has a O b2 bias in general, and it becomes unbiased in thebest-case scenario (i.e., under correct specification).
This paper extends JSH’s proposal to asymmetric kernel density estimation. Because the JSH-MBC technique does notaffect( the order of magnitudein variance, the mean integrated squared error (‘‘MISE’’) of(our MBC)) estimator is in the form
−8/9
of O b4 + n−1 b(−1/2 . Therefore,
when
best
implemented,
theestimatorcanachieveaOn
MISE-convergence, which)
is faster than O n−4/5 , the optimal convergence rate in MISE within the class of nonnegative kernel estimators. Moreover,the estimator is a semiparametrically bias-corrected one with rate improvement. In this sense, this paper can be positioned
as a complement to the existing literature on semiparametric (e.g., Hagmann and Scaillet, 2007; Gustafsson et al., 2009) andnonparametric (e.g., Hirukawa, 2010; Hirukawa and Sakudo, 2014) MBC density estimations using asymmetric kernels.

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M. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

The remainder of this paper is organized as follows. Section 2 proposes the JSH-MBC estimator using asymmetric kernels
and develops its asymptotic properties. Section 3 conducts Monte Carlo simulations to examine finite-sample properties ofthe estimator. Section 4 concludes.
∫∞
This paper adopts the following notational conventions: Γ (a) = 0 ya−1 exp (−y) dy for a > 0 is the gamma function;∫1

B (p, q) = 0 yp−1 (1 − y)q−1 dy for p, q > 0 signifies the beta function; and 1 {·} denotes an indicator function. The
expression ‘Xn ∼ Yn ’ is used whenever Xn /Yn → 1 as n → ∞. Lastly, in order to describe different asymptotic propertiesof an asymmetric kernel estimator across positions of the design point x ∈ R+ (x ∈ [0, 1]), we denote by ‘‘interior x’’ and‘‘boundary x’’ a design point x that satisfies x/b → ∞ (x/b, (1 − x) /b → ∞) and x/b → κ (x/b → κ or (1 − x) /b → κ ) forsome 0 < κ < ∞ as n → ∞, respectively.
2. The JSH-MBC estimator
2.1. Definition

The JSH-MBC estimation for the unknown density f takes three steps. The first step fits a parametric model f (x; θ) tof (x) and estimates the parameter θ ∈ Θ ⊆ Rp by maximum likelihood (‘‘ML’’). Let θ̂ be the ML estimator of θ . However, theinitial parametric(model) is subject to misspecification, which tends to cause non-vanishing bias. Then, in the second step,x; θ̂ in (2) to obtain the HG-MBC estimator as

we put g (x) = f⎧n) ⎨1 ∑
K

⎫
⎬
Xi )
(
(
) .
f˜HG,j (x) := f x; θ̂ r̃j (x) := f x; θ̂⎩n⎭i=1 f Xi ; θ̂()(

j(x,b)The third and final step applies the MBC again to the HG-MBC estimator to accelerate the bias convergence. We define the
JSH-MBC estimator formally as{f˜JSH ,j (x) := f˜HG,j (x)n1 ∑ Kj(x,b) (Xi )ni=1

f˜HG,j (Xi )}.

2.2. Convergence properties
To describe convergence properties of the JSH-MBC estimator, we introduce additional notations. Let θ0 be the pseudo true
value that minimizes the Kullback–Leibler distance of f (x, θ) from the true f (x), where the word ‘‘pseudo’’ reflects possible
misspecification of the initial parametric model. We also denote f0 (·) := f (·, θ0 ) and r0 (·) := f (·) /f (·, θ0 ). Exploring theasymptotic properties of f˜JSH ,j (x) requires the following regularity conditions.
Assumption 1. {Xi }ni=1 are i.i.d. random variables drawn from a univariate distribution having a density f with supp (X )either on [0, 1] or R+ .
Assumption 2. For a given design point x ∈ supp (X ), f (x) , f0 (x) > 0, and r0 (x) has four continuous and boundedderivatives in the neighborhood of x.
Assumption 3. The smoothing parameter b = bn (> 0) satisfies b + nb3()−1→ 0 as n → ∞.

Assumption 4. (i) The parameter space Θ is a compact subset of Rp . (ii) f (x; θ) is continuous in θ for every x ∈ supp (X ).
(iii) θ0 is interior in Θ and uniquely minimizes the Kullback–Leibler distance. (iv) The( negative) log-likelihood of the sample{Xi }ni=1 uniformly converges in probability to the Kullback–Leibler distance. (v)√n θ̂ − θ0= Op (1).

Assumptions 1–3 are standard in the literature on MBC methods for density estimation using asymmetric kernels. Positivity of both f (x) and f0 (x) in Assumption 2 ensures finiteness of the asymptotic variance of the JSH-MBC estimator, whereas
the fourth-order smoothness( of r0)(x) is required for its bias approximation. Assumption 3 implies that the shrinkage rateof b must be slower than O n−1/3 . We require this condition to control the order of magnitude in remainder terms whenapproximating the bias in the third( step.
) It will be shown shortly( that )the MSE-optimal smoothing parameter for the JSHMBC estimator becomes b∗ = O n−2/9 for interior x and b† = O n−1/5 for boundary x; these convergence rates are indeed
within√ the required range. First four conditions of Assumption 4 establish (weak) consistency of θ̂ to θ0 . The final condition
on n-convergence reflects that what is required is the convergence rate of θ̂ , not its asymptotic distribution; interested
readers may consult, for example, White (1982) for sufficient conditions for asymptotic normality of θ̂ .
Assumptions 1–3 also suffice for the bias and variance approximations to the AKDE (3) on which our main result is built.
Specifically, the bias of fˆj (x) admits the expansion{}

Bias fˆj (x) = a1,j (x, f ) b + a2,j (x, f ) b2 + o b2 ,( )

M. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

161

Table 2
Explicit forms of a1,j (x, f ) and vj (x).
Kernel (j)a1,j (x, f )B(1 − 2x) f (1) (x) + (x/2) (1 − x) f (2) (x)⎧for x ∈ [0, 2b)⎨ςb (x) f (1) (x)MB⎩(x/2) (1 − x) f (2) (x)−ςb (1 − x) f (1) (x)f (1) (x) + (x/2) f (2) (x)

G

{
MG

(x/2) f (2) (x)ξb (x) f (1) (x){NM(x/4) f (2) (x)ξb (x) f (1) (x)for x ∈ [2b, 1 − 2b]for x ∈ (1 − 2b, 1]

vj (x)
{ √ √

}

{ √ √

}

1/ 2 π1/ 2 πx (1 − x)x (1 − x)( √ √ )

1/ 2 πx( √ √ )

for x ≥ 2b
for x ∈ [0, 2b)

1/ 2 πx

for x ≥ 2b
for x ∈ [0, 2b)

(√

√ )

1/

2π

x

Note: ςb (x) = (1 − x) {ϱb (x) − x/b} /[1 + b {ϱb (x) − x/b}] for ϱb (x) given in Table 1,and ξb (x) = {(1/2) (x/b) − 1}2 .

where a1,j (x, f ) and a2,j (x, f ) are kernel-specific functions that depend on the design point x and derivatives of f . The
variance of fˆj (x) can be approximated by

{(
)
(
)
}
n−1 b−1/2 vj (x) f (x) + o n−1 b−1/2
ˆ( −1 −1 )Var fj (x) =for interior x

{

O n

b

for boundary x.

Table 2 presents explicit forms of a1,j (x, f ) and vj (x). Notice that the table does not provide a2,j (x, f ), because the dominantbias term of {f˜JSH ,j (x}) is free of this coefficient, as will be seen shortly. We also omit the analytical expression of the dominantterm in Var fˆj (x) for boundary x and report only its order of magnitude in the sense that the inferior rate does not affect
the global property of f˜JSH ,j (x).
The theorem below documents the main result of this paper.
Theorem 1. If Assumptions 1–4 hold, then the bias of f˜JSH ,j (x) for j ∈ {B, MB, G, MG, NM } can be approximated as{}

Bias f˜JSH ,j (x) = −f (x) a1,j x, hj (x, r0 ) b2 + o b2 ,

{

}

( )

where a1,j x, hj {(x, r0 ) can} be obtained{}by replacing f = f (x) in a1,j (x, f ) given in Table 2 with hj (x, r0 ) := a1,j (x, r0 ) /r0 (x).

{}In addition, Var f˜JSH ,j (x) ∼ Var fˆj (x) regardless of the position of x.

Proof. To save space, we only consider the case for j = G. The theorem can be immediately established by combining theproof of Theorem 2 in Hirukawa and Sakudo (2014) with Eqs. (4.15) and (4.16) of Hagmann and Scaillet (2007). ■A few remarks are in order. First, Theorem 1 explains how the JSH-MBC estimator succeeds appealing bias properties
of(the) HG- and JLN-MBC estimators. It is demonstrated that the bias convergence in f˜JSH ,j (x) is accelerated from O (b) toO b2 under sufficient smoothness of f , even when the initial parametric model is misspecified. This is also the property of
JLN-MBC. Indeed, the JLN-MBC estimator may be viewed as a special case of the JSH-MBC estimator; the former correspondswith the latter when the (improper) uniform density is chosen as the initial parametric model, to be more precise. On theother hand, if the parametric proposal is close enough to (coincides with) the truth, then r0 (x) ≈ 1 (r0 (x) ≡ 1) so thatf˜JSH ,j (x) has a very small leading bias coefficient (is unbiased up to the order considered). This property is what HG- and
JSH-MBC have in common.
Second, the theorem also states that the variance of f˜JSH ,j (x) is first-order asymptotically equivalent to those of fˆj (x)and f˜JLN ,j (x). In contrast, while the variance of the original JSH-MBC estimator using a symmetric kernel is still first-order
asymptotically equivalent to that of the JLN-MBC estimator using the same kernel, it is larger than that of the ordinary, biasuncorrected density estimator using the same kernel. This is because the dominant variance term of the original JSH-MBC
estimator involves the roughness of the ‘twiced’ kernel generated by the symmetric kernel (Stuetzle and Mittal, 1979).
Third, we should briefly mention the local and global properties of f˜JSH ,j (x). Its mean squared error (‘‘MSE’’) is of thefollowing form:{ ()}
O b4 + n−1 b−1/2
˜()
MSE fJSH ,j (x) =O b4 + n−1 b−1{

for interior x
for boundary x

.

∗
−2/9
†
−1/5
Substituting the MSE-optimal smoothing parameter
into the corresponding MSE leads( −8/9 ) b = (O−4n/5 ) and b = O n
to the best-possible MSE convergence of O n
and O nfor interior and boundary x, respectively. Moreover, the

(

)

(

)

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M. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

inferior MSE convergence for boundary x does not affect the global property of f˜JSH ,j (x). By the trimming argument by Chen
(1999, p.136) and Chen (2000, p.476), the MISE of f˜JSH ,j (x) is}{(})} E vj (X )({2
4
MISE f˜JSH ,j (x) = b E f (X ) a1,j X , hj (X , r0 ) +
+
o
b4 +
1/2
{

nb

1

)

nb1/2

provided that two expectations are finite, where a1,j for j = MB and for j ∈ {MG, NM } refers to the one for x ∈ [2b, 1 − 2b]and for x ≥ 2b, respectively. The MISE-optimal smoothing parameter that minimizes two dominant terms is given by

[
b

∗∗

=

]2/9

E vj (X )}{8E f (X ) a21,j X , hj (X , r0 )

{

)}

(

n−2/9 .

Therefore, the optimal MISE becomes{}

MISE ∗∗ f˜JSH ,j (x) ∼}]8/9 −8/9)}]1/9 [ {
(
9 [ {
n
.
E vj ( X )E f (X ) a21,j X , hj (X , r0 )88/9

It is noteworthy that the best possible convergence rate of JSH-MBC is n4/9 , which is the best-possible nonparametric one
under fourth-order smoothness of the unknown density; see Stone (1980) for more details.
3. Finite-sample performance
3.1. Monte Carlo setup
We conduct a small Monte Carlo study to assess finite-sample properties of the JSH-MBC estimator. Among all asymmetrickernels, the gamma kernel is exclusively considered due to its popularity. The simulation study compares the following four
estimators as variants of (2): (i) the ordinary AKDE [G]; (ii) the JLN-MBC estimator [JLN]; (iii) the HG-MBC estimator with agamma start [HG]; and (iv) the JSH-MBC estimator with a gamma start [JSH].
Four distributions below are chosen as the truth. All these distributions are popularly chosen as models for income
distributions, loss distributions, baseline hazard, and stochastic production frontier analysis.
xα−1 exp (−x/β) /{β α Γ (α)} , (α, β) = (1.5, 1) .
α−1
α
(α/β)
{
( (x/β))} exp {−(x/β) } , (α, β) = (1.5, 1.5) .

1. Gamma
2. Weibull
3. Half-Normal

2/

4. Generalized Gamma

√

α−1

γx

2πσ

exp −(x − µ)2 / 2σ 2

{

γ

(

)}

, (µ, σ ) = (0, 1.5) .

α

exp {−(x/β) } /{β Γ (α)} , (α, β, γ ) = (2, 2, 2.5) .

For each distribution, 1000 data sets of sample size n = 100, 200 or 500 are simulated. All density estimates areevaluated on an equally spaced grid of 500 points over the interval [0, 5].
f¯ (x) is evaluated⏐{ Performance} ∫ ∞ ⏐ of an estimatorby three measures, namely, the integrated absolute deviation (‘‘IAD’’) IAD f¯ (x) = 0 ⏐f¯ (x) − f (x)⏐ dx, the root integrated

√∫

}2
{
}
∞ {¯
f (x) − f (x) dx, and the integrated absolute bias (‘‘IAB’’) IAB f¯ (x) =
0
⏐
∫∞⏐ {
}
⏐E f¯ (x) − f (x)⏐ dx. In our reports, the integrals are approximated over the 500 points. As regards choices of smoothingsquared error (‘‘RISE’’) RISE f¯ (x)

{

}

=

0

parameters, our pilot simulations indicate that plug-in methods such as the one in Hirukawa and Sakudo (2014) do not work
well. Then, we turn to the simple rule of thumb as in Gustafsson et al. (2009). Specifically, smoothing parameter values are
set equal to (i) b̂ROTA = σ̂x n−2/5 for G and HG and (ii) b̂ROTB = σ̂x n−2/9 for JLN and JSH, where σ̂x is the standard deviation ofeach Monte Carlo sample.
3.2. Results and guidance for practitioners
Table 3 presents simulation averages and standard deviations of IAD and RISE as well as IAB. The results are mixed. It
is no surprise that HG outperforms others in the Gamma case, because the gamma start correctly specifies the truth. In
the remaining three cases, the gamma start is misspecified. When the deviation of the truth from the gamma start is small(i.e., the Weibull case), HG still performs best and JSH follows. On the other hand, when the deviation is substantial (i.e., theHalf-Normal and Generalized Gamma cases), G and JLN appear to have advantage.
JSH does not outperform HG in terms of IAD or RISE except for the Generalized Gamma case. Most possibly, the extra MBC
step from HG to JSH generates additional higher-order terms that can be omitted in its first-order asymptotics but may be
non-negligible in finite samples. However, as a positive side of the MBC step, we can find that JSH often yields the smallest
IAB.
Because of the mixed results, practitioners may wonder which estimator(s) to employ. Then, following Hagmann and
Scaillet (2007, Section 4.6), we attempt to provide guidance on choosing among the fully parametric ML, fully nonparametricM. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

163

Table 3
Averages and standard deviations of performance measures.
IAD

RISE

n

Distribution

Estimator

Ave

SD

Ave

SD

IAB

100

Gamma

G
JLN
HG
JSH

0.02583
0.02337
0.02187
0.02391(0.00875)(0.00834)(0.00852)(0.00900)

0.03893
0.03879
0.03092
0.03340(0.01203)(0.01264)(0.01229)(0.01263)

0.017450.015610.00087
0.00259

Weibull

G
JLN
HG
JSH

0.02666
0.024800.022620.02464(0.00957)(0.01009)(0.00914)(0.00945)

0.039700.03859
0.03275
0.03520(0.01386)(0.01539)(0.01322)(0.01349)

0.01830
0.017880.005480.00495

Half-Normal

G
JLN
HG
JSH

0.02397
0.021850.02418
0.02574(0.00859)(0.00926)(0.00963)(0.00971)

0.03504
0.031750.040220.04143(0.01223)(0.01383)(0.01552)(0.01551)

0.01349
0.01165
0.009910.00813

Generalized Gamma

G
JLN
HG
JSH

0.031680.03096
0.03818
0.03149(0.01121)(0.01087)(0.04458)(0.03271)

0.041480.04206
0.05704
0.04676(0.01471)(0.01447)(0.07169)(0.06469)

0.026110.026120.021810.01581

Gamma

G
JLN
HG
JSH

0.02035
0.018400.01646
0.01797(0.00662)(0.00634)(0.00610)(0.00645)

0.03191
0.03284
0.02334
0.02521(0.00943)(0.00973)(0.00878)(0.00909)

0.01397
0.012860.000390.00153

Weibull

G
JLN
HG
JSH

0.02100
0.01952
0.01730
0.01868(0.00688)(0.00711)(0.00612)(0.00638)

0.03191
0.031230.02518
0.02687(0.01026)(0.01112)(0.00905)(0.00933)

0.014610.01450
0.00437
0.00330

Half-Normal

G
JLN
HG
JSH

0.01895
0.016850.018800.01964(0.00625)(0.00694)(0.00684)(0.00699)

0.027770.024440.03153
0.03190(0.00921)(0.01037)(0.01109)(0.01109)0.01049
0.00925
0.007900.00568

Generalized Gamma

G
JLN
HG
JSH

0.02503
0.024900.03559
0.03213(0.00823)(0.00790)(0.03404)(0.03047)

0.03291
0.03374
0.05467
0.05121(0.01090)(0.01046)(0.05543)(0.05663)

0.02072
0.021450.023140.01944

Gamma

G
JLN
HG
JSH

0.01500
0.01346
0.01151
0.01245(0.00436)(0.00435)(0.00384)(0.00409)

0.02467
0.02665
0.016530.01774(0.00665)(0.00707)(0.00584)(0.00609)

0.01033
0.009830.000260.00070

Weibull

G
JLN
HG
JSH

0.015410.014310.012360.01314(0.00483)(0.00509)(0.00416)(0.00429)

0.023810.023880.018070.01907(0.00716)(0.00767)(0.00604)(0.00617)

0.010810.01106
0.003290.00190

Half-Normal

G
JLN
HG
JSH

0.01358
0.01179
0.013390.01370(0.00409)(0.00437)(0.00436)(0.00446)

0.020240.01709
0.02306
0.02275(0.00601)(0.00649)(0.00733)(0.00725)

0.007450.00656
0.005620.00331

Generalized Gamma

G
JLN
HG
JSH

0.017840.01860
0.03006
0.02937(0.00549)(0.00547)(0.02062)(0.01961)

0.02373
0.02526
0.047170.04706(0.00723)(0.00718)(0.03393)(0.03474)

0.014630.016200.02236
0.02122200

500

Note: ‘‘Ave’’ and ‘‘SD’’ are simulation averages and standard deviations, respectively.

(i.e., G and JLN) and semiparametric (i.e., HG and JSH) estimators. Four curves in Fig. 1 are the plots of{ √ √ ( )}⎧⎪log r̃G (x) + n−1 b−1/2 / 4 π xf x; θ̂
⎪
⎪
⎪
⎪
√
⎪
⎪
√ √ ( )⎪⎪
n−1/2 b−1/4 / 2 π xf x; θ̂⎨{( )}Z (x) =⎪
log r̃G (x) + n−1 b−1 Γ (2κ + 1) / 22(κ+1) Γ 2 (κ + 1) f x; θ̂
⎪
⎪
⎪
⎪
√
⎪
{
( )}
⎪
⎪
⎪
⎩
n−1/2 b−1/2 Γ (2κ + 1) / 22κ+1 Γ 2 (κ + 1) f x; θ̂for interior x

for boundary x

164

M. Hirukawa, M. Sakudo / Statistics and Probability Letters 145 (2019) 158–165

Fig. 1. Plots of Z (x) for JSH-MBC estimation.

constructed from 500 random draws of the four distributions we consider. When the parametric start coincides with thetruth, Z (x) is approximately distributed as N (0, 1) for each x, and the curve should move within ±1.96 about 95% of thetime. Based on a visual inspection of the figure, we suggest the followings:
1. If the curve is within ±1.96 (Gamma), then we do not need to go beyond ML.
2. If some parts of the curve lie outside ±1.96 but it always stays near ±1.96 (Weibull), then we rely on a semiparametricestimator (i.e., either HG or its bias-corrected version JSH).
3. If the curve considerably deviates from ±1.96 (Half-Normal and Generalized Gamma), then we resort to a fully
nonparametric estimator (i.e., either G or its bias-corrected version JLN).

4. Conclusion
This paper has extended yet another semiparametric MBC technique originally proposed by JSH to asymmetric kernel
density estimation.
( ) It is demonstrated that the JSH-MBC estimator smoothed by asymmetric kernels accelerates the bias
from O (b) to O b2 in general under sufficient smoothness of the true density, whereas the order of magnitude in varianceremains()unchanged. It follows that when best implemented, the MISE of the estimator achieves a faster convergence rate ofO n−8/9 . Monte Carlo simulations confirm nice bias properties of the JSH-MBC estimator.
Acknowledgments
We would like to thank an anonymous referee for his/her constructive comments and suggestions, which substantially
improve the quality of the paper. This research was supported, in part, by the grant from Japan Society for the Promotion of
Science (Grant Number 15K03405). The views expressed herein and those of the authors do not necessarily reflect the viewsof the Development Bank of Japan.
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