DBJ Discussion Paper Series, No.1209

Nonparametric Multiplicative Bias Correction for
Kernel-Type Density Estimation
Using Positive Data
Masayuki Hirukawa(Setsunan University)
And
Mari Sakudo(Development Bank of Japan)

December

2012

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 inorder to 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 views of the Institute.

白紙

Nonparametric Multiplicative Bias Correction for
Kernel-Type Density Estimation
Using Positive Data
Masayuki Hirukawa
Setsunan University

Mari Sakudoy
Development Bank of Japan
March 2012

Abstract
This paper improves convergence properties of asymmetric kernel density
estimators for nonnegative economic and …nancial variables via two classes ofmultiplicative bias correction (“MBC”) techniques. It is demonstrated that
under su¢ cient smoothness of the true density, each MBC technique reduces
the order of magnitude in bias, whereas the order of magnitude in varianceremains unchanged. Accordingly, the mean integrated squared error of each
MBC estimator achieves a faster convergence rate of O n 8=9 when best implemented, where n is the sample size. Furthermore, the estimator always
generates a nonnegative density estimate by construction. Plug-in smoothing
parameter choice methods are also proposed to implement the MBC estimators
using the Gamma and Modi…ed Gamma kernels. Finite sample performance ofthe estimators are examined via Monte Carlo simulations, and the estimators
are applied to estimating income distributions.
Keywords: Asymmetric kernel; boundary e¤ect; higher-order bias kernel;
income distribution.
JEL Classi…cation Codes: C13; C14.
MSC 2000 Codes: 62G7; 62G20.

Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka, 572-8508,
Japan; phone: (+81)72-839-8095; fax: (+81)72-839-8138; e-mail: hirukawa@econ.setsunan.ac.jp;
web: http://www.setsunan.ac.jp/~hirukawa/.
yResearch Institute of Capital Formation, Development Bank of Japan, 9-3, Otemachi 1-chome,
Chiyoda-ku, Tokyo 100-0004, Japan; phone (+81)3-3244-1241; fax (+81)3-3270-7084; e-mail: marisakudo@gmail.com.

1

Introduction

The aims of this paper are to improve convergence properties of kernel density estimators for nonnegative economic and …nancial variables via a couple of bias correction
techniques, and to apply the proposed estimators for estimating income distributions.
Although we focus primarily on income distributions, our proposal is expected to …twell with other distributions such as the loss distribution (= the distribution of apayment to the insured) in actuarial science and …nancial risk management, the distribution of important …nancial variables such as short-term interest rates,1 and eventhe baseline hazard in …nancial duration analysis.
To illustrate income distribution estimation, let fXi gni=1 be a random sample
drawn from a univariate distribution with density f . Income distributions are empirically characterized by two stylized facts, namely, (i) a natural boundary at theorigin (i.e. f has support on [0; 1)) and (ii) a mode near the boundary and a long tailwith sparse data. To capture these stylized facts, researchers often …t log-normal andPareto distributions for the region near the boundary and the right tail, respectively.
However, the imposition of a misspeci…ed model leads to inconsistent estimates andmisleading inference, as well as to disputable evaluations of inequality measures.
This motivates us to adopt nonparametric kernel methods for their ‡exibility incurve …tting. Nevertheless, standard kernel smoothing with symmetric kernels must
be modi…ed to accommodate the stylized facts. First, when the support of the density
has a boundary, boundary correction methods (e.g. Müller, 1991; Jones, 1993; Jonesand Foster, 1996; Zhang, Karunamuni and Jones, 1999; Hall and Park, 2002) should
1

CIR (Cox, Ingersoll and Ross, 1985) and Ahn and Gao’s (1999) inverse Feller processes areoften used as data generating processes of short-term interest rates. It is known that time-invariant
distributions of these processes are gamma and inverse gamma distributions, respectively. These
distributions possess the same stylized facts as described shortly.

1

be employed. Second, global smoothing with a single bandwidth may not work well.
If a short bandwidth is used to capture the mode near the origin, the density estimate
over the tail region tends to be wiggly. On the other hand, if a long bandwidth ischosen to preserve the shape of the tail part, the mode near the origin is considerablysmoothed away.

Variable bandwidth methods (e.g. Abramson, 1982; Terrell andScott, 1992) have been proposed as a remedy for this issue.
Recently, asymmetric kernel functions have emerged as a viable alternative toboundary correction methods.2 For an asymmetric kernel function Kj(x;b) ( ) indexed
by j that depends on a design point x > 0 and a smoothing parameter b > 0, thedensity estimator can be expressed asn
1X
f^j;b (x) =Kj(x;b) (Xi ) :
n i=1

(1)

Throughout, Kj(x;b) ( ) refers to the Gamma (“G”; Chen, 2000), Modi…ed Gamma(“MG”; Chen, 2000), Inverse Gaussian (“IG”; Scaillet, 2004), Reciprocal Inverse
Gaussian (“RIG”; Scaillet, 2004), Log-Normal (“LN”; Jin and Kawczak, 2003),3 andBirnbaum-Saunders (“BS”; Jin and Kawczak, 2003) kernels.

Functional forms ofthese kernels are presented in Table 1.
TABLE 1 AND FIGURE 1 ABOUT HERE
Asymmetric kernels can be viewed as a combination of a boundary correction
device and a ‘variable bandwidth’ method.
2

Because these kernels have support

Strictly speaking, asymmetric kernel functions should be referred to as kernel-type weighting
functions. In a slightly di¤erent context, Gouriéroux and Monfort (2006) and Jones and Henderson(2007) argue that unlike the cases of symmetric kernels, the roles of the data point X and the design
point x in asymmetric kernels are not exchangeable, which leads to lack of normalization in density
estimation using these kernels. Nevertheless, we follow the adopted convention in the literature forthese kernel-type functions.
3
Our de…nition of the Log-Normal kernel slightly di¤ers from the original one in Jin and Kawczak(2003). This de…nition
ensures that the leading variance term of the density estimator (1) becomespn 1 b 1=2 f (x) = (2 x) for a design point x > 0 so that x=b ! 1 as b ! 0.

2

on [0; 1), they eliminate boundary e¤ects by construction.

Besides, the kernels

have a number of important advantages in the analysis of economic and …nancial
data. First, shapes of the asymmetric kernels vary according to the position at whichsmoothing is made; in other words, the amount of smoothing changes in a locally
adaptive manner. Figure 1 plots shapes of six asymmetric kernels for four di¤erent
design points (x = 0:5, 1:0, 2:0, and 4:0), where the smoothing parameter value is…xed at b = 0:4.

The adaptive smoothing property in asymmetric kernels bears

some similarities to the variable bandwidth methods. However, a single smoothing
parameter su¢ ces for local adaptive smoothing, which makes asymmetric kernels
much more appealing in empirical work.

Second, asymmetric kernels achieve theoptimal rate of convergence (in mean integrated squared error (“MISE”) sense) within
the class of nonnegative kernel estimators. Third, unlike the case with symmetrickernels, the variances of asymmetric kernel estimators tend to decrease as the design
point moves away from the boundary. Note that the variance reduction for large x
is gained at the expense of increasing bias.
Provided that f is twice continuously di¤erentiable, for a design point x > 0 sothat x=b ! 1 as b ! 0, the leading bias and variance terms of the asymmetricon^kernel density estimator (1) can be approximated by Bias fj;b (x)a1;j (x; f ) bnoand V ar f^j;b (x)n 1 b 1=2 vj (x) f (x), where a1;j (x; f ) is a kernel-speci…c functionthat depends on x and derivatives of f (see Table 2 below for explicit forms), and8< 1=2 for j = G; M G; RIG
1
1
for j = LN; BS:(2)
vj (x) = p rj with rj =:2 x
3=2 for j = IG

Observe that vj (x) is proportional to x rj , which is the source of shrinking variancewith the position of x. This property is equivalent to the strategy of using longer

bandwidths over the tail region where the data are sparse, and it is particularly3advantageous for estimating income distributions.
Although asymmetric kernels are relatively new in the literature, several papers
report favorable evidence from applying them to empirical models in economics and…nance. A non-exhaustive list includes: (i) estimation of recovery rate distributions
on defaulted bonds (Renault and Scaillet, 2004), (ii) income distribution estimation(Bouezmarni and Scaillet, 2005; Hagmann and Scaillet, 2007); (iii) actuarial loss
distribution estimation (Hagmann and Scaillet, 2007; Gustafsson et al., 2009); (iv)hazard estimation (Bouezmarni and Rombouts, 2008); (v) regression discontinuity
design (Fé, 2010); (vi) realized integrated volatility estimation (Kristensen, 2010);
and (vii) estimation of di¤usion models (Gospodinov and Hirukawa, 2012).
This paper demonstrates that the convergence rate of the asymmetric kernel density estimator (1) can be improved via two classes of well-known bias correction techniques. To be more speci…c, each technique leads the bias convergence to be accelerated to O (b2 ) under su¢ cient di¤erentiability of f , while the order of magnitudeno1
in variance is maintained, i.e. the variance is still O nb1=2
if x=b ! 1, andno1
O nbrj +1=2
if x=b ! for some > 0 and the kernel-speci…c exponent rj givenin (2).

In the cases of symmetric second-order kernels, this kind of rate improve-ments can be typically achieved by employing higher-order kernels.4 To the best of
our knowledge, equivalent techniques are yet to be proposed for asymmetric kernels;
if any, they would generate negative density estimates over some parts of the support,
which cause di¢ culty in interpretation in practice. Instead, the rate improvement
is attained by applying two classes of multiplicative bias correction (“MBC”) techniques that are proposed by Terrell and Scott (1980) and Jones, Linton and Nielsen
4Jones and Foster (1993) provide an excellent review on the methods of generating higher-orderkernels from a given symmetric second-order kernel.

4

(1995) originally for symmetric kernel density estimation.5

For each of two MBC

estimators, the MISE takes the form of O b4 + n 1 b 1=2 .

Therefore, when best

implemented, each estimator can achieve the convergence rate of O n 8=9 in MISE,
which is faster than O n 4=5 , the MISE-optimal convergence rate within the classof nonnegative kernel estimators.

Besides, two MBC estimators still maintain thesame attractive properties as the bias-uncorrected estimator (1) has. They are free ofboundary bias and always generate nonnegative density estimates everywhere by construction. Their variances still tend to decrease as the design point moves away fromthe boundary. Moreover, to implement the two classes of MBC estimators employing
the Gamma and Modi…ed Gamma kernels, this paper proposes plug-in methods ofchoosing the smoothing parameter b with a gamma density used as a reference.
In the closely related literature, Hagmann and Scaillet (2007) apply a semiparametric MBC technique in the spirit of Hjort and Jones (1996) (called local multiplicative bias correction (“LMBC”)) to asymmetric kernel density estimation. Gustafsson et al. (2009) propose another semi-parametric MBC technique called local
transformation bias correction (“LTBC”), which basically follows the idea of Rudemo(1991).

In both LMBC and LTBC, asymmetric kernels are employed at the bias

correction step after initial parametric density estimation. A key di¤erence is thatthe bias correction is made for the original data in LMBC and for the data transformed on the unit interval in LTBC. However, unlike the MBC estimators proposedin this paper, neither LMBC nor LTBC improves the bias in order of magnitude.
Furthermore, Chen (1999) proposes another asymmetric kernel (the Beta kernel),
which yields a boundary-bias-free density estimator on the unit interval. Two MBC
5

The forms of bias reductions achieved through MBC techniques are analogous between symmetric and asymmetric kernel cases. However, none of asymmetric kernels can be expressed in the formof K ( =b) =b, and thus it is worth emphasizing that mathematics and proof strategies for yielding
the results are totally di¤erent.

5techniques considered in this paper are shown to be applied for this kernel; see acompanion paper (Hirukawa, 2010) for details.
The remainder of this paper is organized as follows.

Section 2 introduces twoclasses of MBC density estimators and develops their asymptotic properties. Section
3 proposes plug-in methods of choosing the smoothing parameter b for Gamma kernelbased MBC estimators, and conducts Monte Carlo simulations to check …nite sample
properties of the estimators. Section 4 applies the MBC estimators for estimating
income distributions from the U.S. and Brazilian data sets. Section 5 summarizes
the main results of the paper. All proofs are given in the Appendix.
R1
This paper adopts the following notational conventions. ( ) = 0 y

1

exp ( y) dyd( > 0) denotes the gamma function. The expression ‘X = Y ’reads “A random variable X obeys the distribution Y .” Lastly, the expression ‘Xn

Yn ’is used whenever

Xn =Yn ! 1 as n ! 1.

2

MBC Estimators Using Asymmetric Kernels

2.1

De…nitions of Two MBC Estimators

Two MBC techniques considered in this paper are subclasses of higher-order bias kernel density estimation methods (Jones and Signorini, 1997), and these are originallydesigned for symmetric kernels. We now extend the techniques to density estimation
using asymmetric kernels with support on [0; 1).
In the spirit of Terrell and Scott (1980, abbreviated as “TS” hereafter), the …rst
class of MBC techniques constructs a multiplicative combination of two density estimators employing the same kernel but di¤erent smoothing parameters. Let f^j;b=c (x)be the density estimator using asymmetric kernel j and smoothing parameter b=c,where c 2 (0; 1) is some predetermined constant that does not depend on the design
6point x. Then, the TS-MBC asymmetric kernel density estimator can be de…ned asno 11 c n
o 1cc
f~T S;j (x) = f^j;b (x)f^j;b=c (x):(3)

TS originally proposed this method as an additive bias correction to the logarithms
of densities, and later it is generalized and reinterpreted as an MBC technique by
Koshkin (1988) and Jones and Foster (1993), respectively.
The second class of MBC techniques due to Jones, Linton and Nielsen (1995,
abbreviated as “JLN”hereafter) utilizes a single smoothing parameter b. In light ofnothe identity f (x) = f^j;b (x) f (x) =f^j;b (x) , the JLN-MBC asymmetric kernel density

estimator can be de…ned as

f~JLN;j (x) = f^j;b (x)(1 X Kj(x;b) (Xi )n i=1 f^j;b (Xi )
n

)

:

(4)

Recognize that the term inside the bracket is a natural nonparametric estimatorof the bias-correction term f (x) =f^j;b (x).

This MBC technique is also applied in

nonparametric regression (Linton and Nielsen, 1994), hazard estimation (Nielsen,
1998; Nielsen and Tanggaard, 2001), and spectral matrix estimation (Xiao and Linton,
2002; Hirukawa, 2006).6

Also observe that both f~T S;j (x) and f~JLN;j (x) always

generate non-negative density estimates everywhere by construction.

2.2

Asymptotic Properties of MBC Estimators

2.2.1

Asymptotic Results

To develop convergence properties of MBC estimators, we make the following assumptions:Because the correction term f (x) =f^b (x) reminds us of prewhitening in time series analysis, thisbias correction technique is referred to as “nonparametric prewhitening”in Xiao and Linton (2002)
and Hirukawa (2006).
6

7

Assumption 1. f has four continuous and bounded derivatives, and f (x) > 0 for
a given design point x > 0.

Assumption 2. The smoothing parameter b = bn satis…es b ! 0 and nbrj +5=2 ! 1
as n ! 1.
The smoothness condition on f in Assumption 1 is standard for consistency ofdensity estimators using fourth-order kernels, whereas the positivity of f (x) is required for MBC. Assumption 2 implies that the convergence rate of the smoothing
parameter b is slower than O n 1=(rj +5=2) . We require this condition to control
the order of magnitude in remainder terms when approximating the bias of eachof TS- and JLN-MBC estimators.

It will be shown shortly that the MSE-optimalsmoothing parameters for these estimators are b = O n 2=9 if the design point x
satis…es x=b ! 1, and that by = O n 1=(rj +9=2) if x=b !

for some> 0; theseconvergence rates are indeed within the required range.
The approximation to the bias of each of two MBC estimators is built on a fourthno^order Taylor expansion of E fj;b (x) . Under Assumptions 1 and 2, we haveon
E f^j;b (x) = f (x) + a1;j (x; f ) b + a2;j (x; f ) b2 + o b2 ;where a1;j (x; f ) and a2;j (x; f ) are kernel-speci…c functions that depend on the design
point x and derivatives of f . Using properties of the random variable corresponding
to each kernel, we can specify explicit forms of a1;j (x; f ) and a2;j (x; f ) as in Table 2.
TABLE 2 ABOUT HERE
This paper refers to the position of x as “interior x”if x=b ! 1, and “boundaryx” if x=b ! .

We now present two theorems on the approximations to bias andvariance terms of two MBC estimators.
8

Theorem 1. If Assumptions 1 and 2 hold, then the bias of the TS-MBC estimatorusing kernel j can be approximated bynoBias f~T S;j (x)

1
c (1

c)

pj (x) b2 :=

1
c (1

1
c) 2

a21;j (x; f )f (x)a2;j (x; f ) b2

for a1;j (x; f ) and a2;j (x; f ) in Table 2. For vj (x) and rj de…ned in (2), the varianceof the TS-MBC estimator can be approximated by
8nno < n 1 b 1=2 (c) vj (x) f (x) + o nb1=2
no
V ar f~T S;j (x) =: O nbrj +1=2 1
where

(c) =

1 + c5=2 (1 + c)1=2(1 + c)1=2 (11p
2 2c3=2c)2ofor interior x;

for boundary x:

Theorem 2. If Assumptions 1 and 2 hold, then the bias of the JLN-MBC estimatorusing kernel j can be approximated byno~Bias fJLN;j (x)qj (x) b2 :=f (x) a1;j (x; h) b2 ;where a1;j (x; h) is obtained by replacing f = f (x) in a1;j (x; f ) withh = h (x; f ) :=a1;j (x; f ):f (x)For vj (x) and rj de…ned in (2), the variance of the JLN-MBC estimator can beapproximated by
8nno < n 1 b 1=2 vj (x) f (x) + o nb1=2
no
V ar f~JLN;j (x) =: O nbrj +1=2 1

2.2.2

1

o

for interior x
for boundary x:

Discussions

Bias and Variance. Because the support of asymmetric kernels matches that of
the true density f , both TS- and JLN-MBC estimators are free of boundary bias.
More importantly, these estimators reduce the order of magnitude in bias from O (b)9to O (b2 ), while their variances are O

n

nb1=2

1

o

for interior x and Onnbrj +1=2

1ofor boundary x. Observe that orders of variances remain unchanged from those for thecorresponding bias-uncorrected estimator (1). We can also compare pj (x) and qj (x)in leading bias terms with the corresponding ones for symmetric second-order kernels.
As stated in Jones and Signorini (1997, Sections 3.2-3.3), when the symmetric kernels
are employed, the term corresponding to pj (x) for TS-MBC is a linear combinationof f 0000 (x) and ff 00 (x)g2 =f (x), and the term corresponding to qj (x) for JLN-MBC
is proportional to f (x) ff 00 (x) =f (x)g00 . The reason why pj (x) and qj (x) take more
complicated forms is that while odd-order moments of symmetric kernels are exactly
zero, those of asymmetric kernels around the design point x are often O (b) or O (b2 ).
As a result, extra density derivatives are included in pj (x) and qj (x).
The variance of JLN-MBC estimators is …rst-order asymptotically equivalent tothat of the corresponding bias-uncorrected estimator (1) for interior x.

While theLMBC density estimator by Hagmann and Scaillet (2007) also yields the same leading variance term, this technique does not reduce the bias in order of magnitude. In
contrast, when the JLN-MBC is applied for the density estimation using a symmetric second-order kernel, the leading variance term tends to be larger (although notin‡ated in order of magnitude) because the multiplier in the variance term involves
the roughness (or squared integral) of the ‘twiced’kernel (Stuetzle and Mittal, 1979).
Moreover, since the multiplier(c) in the variance for TS-MBC estimators is increas-

ing in c 2 (0; 1), ranging from 1 to 27/16, the variance of these estimators tends to belarger than that of the bias-uncorrected estimator (1) for interior x. Lastly (but notleast importantly), the asymptotic variances of both TS- and JLN-MBC estimatorsfor interior x are proportional to x rj , even after MBC is made. Therefore, MBC
estimators maintain the same appealing properties as the bias-uncorrected estimator10

has in estimating the distributions that have long tails with sparse data.
Mean Squared Error (“MSE”). For interior x, the MSEs of f~T S;j (x) and f~JLN;j (x)can be approximated bynop2j (x) 4
1
1=2
M SE f~T S;j (x) =(c) vj (x) f (x) + o b4 + n 1 b 1=2 ;2b + n b2
c (1 c)no
M SE f~JLN;j (x) = qj2 (x) b4 + n 1 b 1=2 vj (x) f (x) + o b4 + n 1 b 1=2 :The MSE-optimal smoothing parameters are
22bT S;j =c (1bJLN;j =vj (x) f (x)8qj2 (x)

c)

(c)

2=9

vj (x) f (x)8p2j (x)2=9n 2=9 ;

2=9

n 2=9 ;

which yield the optimal MSEsnof~T S;j (x)onM SE f~JLN;j (x)
M SE

where
(c) =

(

9
88=9

2=9

(c) pj (x) fvj (x) f (x)g8=9 n 8=9 ;9 2=9qj (x) fvj (x) f (x)g8=9 n 8=9 ;
8=98

1 + c5=2 (1 + c)1=2
c1=4 (1 + c)1=2 (1)8=9p
2 2c3=2c)9=4:Observe that the MSE-optimal smoothing parameters are O n 2=9 = O (h 2 ), where
h is the MSE-optimal bandwidth for fourth-order kernel estimators, or TS- or JLNMBC estimators using symmetric second-order kernels.

As a result, the optimal

MSEs of f~T S;j (x) and f~JLN;j (x) for interior x become O n 8=9 , as with MBC estimation using the second-order kernels. The convergence rate is faster than O n 4=5 ,the optimal convergence rate in the MSE of the corresponding bias-uncorrected estimator (1) for interior x. On the other hand, for boundary x, the MSEs of f~T S;j (x)and f~JLN;j (x) are O b4 + n 1 b (rj +1=2) , which yields the MSE-optimal smoothing
11

parameter by = O n 1=(rj +9=2) and the optimal MSE of O n 4=(rj +9=2) . The optimal convergence rate of MSEs is indeed faster than O n 2=(rj +5=2) , that of thebias-uncorrected estimator for boundary x.

Global Property. The undesirable convergence rates over boundary regions do nota¤ect the global properties of the MBC estimators. By the trimming argument in
Chen (2000), the MISEs of the MBC estimators areno~
M ISE fT S;j (x) =b4c2 (1 c)2

Z 1

p2j (x) dx
0
Z 1
1
1=2
vj (x) f (x) dx + o b4 + n 1 b 1=2 ;
+n b
(c)
0
Z 1
Z 1
n
o
4
2
1
1=2
~
M ISE fJLN;j (x) = b
qj (x) dx + n b
vj (x) f (x) dx + o b4 + n 1 b 1=2 ;00provided that p2j (x), qj2 (x), and vj (x) are integrable.7 The MISE-optimal smoothing
parameters are then given bybT S;j =c2 (1bJLN;j =(R 1c)2 (c)2=9vj (x) f (x) dx
0R1
8 0 qj2 (x) dx(R 1v (x) f (x) dx
0 Rj
1
8 0 p2j (x) dx

)2=9

)2=9

n 2=9 ;

n 2=9 :

Therefore, the optimal MISEs become
n
o
M ISE
f~T S;j (x)

n
o
M ISE
f~JLN;j (x)Furthermore, the multiplierminimized at c

9
88=9
9
88=9

(c)
Z 1

Z 1

2=9

p2j (x) dx

0

qj2 (x) dx

0

2=9

Z 1

Z 1

8=9

vj (x) f (x) dx

n 8=9 ;0vj (x) f (x) dx

8=9

n 8=9 :

0(c) in the optimal MISE for the TS-MBC estimator is

0:2636; this value is exclusively considered in subsequent analyses.

7

Throughout this paper, pM G (x) and qM G (x) refer to those for interior x (i.e. x
the integrated squared bias is considered.

122b), wheneverNormalization. Also observe that neither f~T S;j (x) nor f~JLN;j (x) integrates to one,
although the lack of normalization may be ignored in some applications. In general,
MBC leads to lack of normalization, even if symmetric second-order kernels are employed; see Section 2.2 of JLN, for example.

It is easy to see that the order ofmagnitude in the leading bias terms of renormalized TS- and JLN-MBC estimators
f~T S;j (x);f~TRS;j (x) = R 1~T S;j (x) dx
f0
f~JLN;j (x)R
f~JLN;j(x) = R 1
f~JLN;j (x) dx
0remains unchanged. Because
E
E

Z 1

0
Z 1

f~T S;j (x) dx

f~JLN;j (x) dx

0

Z 1
b2
= 1+
pj (x) dx + o b2 ;c (1 c) 0
Z 1
2
= 1+b
qj (x) dx + o b2 ;0

provided that pj (x) and qj (x) are integrable, the leading biases of f~TRS;j (x) andR
f~JLN;j(x) becomenoBias f~TRS;1 (x)

o
n
R
(x)
Bias f~JLN;1

1
c (1

c)

qj (x)Z 1

pj (x) dx b2 ;pj (x)0
Z 1
qj (x) dx b2 :0

The leading variances are una¤ected.

2.3

Further Bias Reduction via Iteration

In principle, further bias reduction is possible after the regularity conditions are properly strengthened.

Constructing a multiplicative combination of (s + 1) di¤erent

density estimators, we can extend the TS-MBC estimator to(s)
f~T S;j (x) =

s n
o s;r
Y
f^j;b=cr (x)
;
r=0

13

where c01, c1 ; : : : ; cs 2 (0; 1) are mutually di¤erent constants, and the exponent iss;r =( 1)s csr:sQ
(cp cr )p=0;p6=r

Similarly, the sth iterated JLN-MBC estimator can be de…ned as( n)
X Kj(x;b) (Xi )
1
(s)
(s
1)
f~JLN;1 (x) = f~JLN;j (x);(s 1)
n i=1 f~JLN;j
(Xi )
(0)

where f~JLN;j (x)f^j;b (x). Provided that the true density f is 2 (s + 1) times con-tinuously di¤erentiable, for each of these estimators, it can be shown that the orderof magnitude in bias is O (bs+1 ) while the order of magnitude in variance remains
unchanged from that of the corresponding bias-uncorrected estimator f^j;b (x). Inon(s)
particular, it can be shown that V ar f~JLN;j (x) is …rst-order asymptotically equio
o
n
n
(0)
valent to V ar f~JLN;j (x) = V ar f^j;b (x) for interior x. As a result, their optimal

MSEs are O n (4s+4)=(4s+5) and O n (2s+2)=(2s+3) for interior and boundary x, re-spectively. Accordingly, as the number of iterations increases, the global convergence
rate of iterated MBC estimators can be arbitrarily close to the parametric one whenbest implemented.

However, it is doubtful whether there is much gain in practicefrom these estimators, and thus we do not pursue this issue any further.

3

Finite Sample Performance

3.1

Monte Carlo Setup

We evaluate the …nite sample performance of two classes of MBC estimators via Monte
Carlo simulations. For estimators involving asymmetric kernels, we concentrate on
the Gamma and Modi…ed Gamma kernels due to their popularity in the literature.
This simulation study compares of the following …ve classes of estimators: (i) biasuncorrected estimators (1) [G, MG]; (ii) LMBC estimator with a gamma start and14

a log-linear correction factor in Hagmann and Scaillet (2007) [LMBC-G]; (iii) LTBC
estimator with the generalized Champernowne start8 and a log-linear correction factor
in Gustafsson et al. (2009) [LTBC-C]; (iv) TS-MBC estimators (3) [TS-MBC-G, TSMBC-MG]; and (vi) JLN-MBC estimators (4) [JLN-MBC-G, JLN-MBC-MG]. The
value of the constant c in each TS-MBC estimator is set equal to the MISE-optimal
c = 0:2636. Ten true distributions are considered, as listed in Table 3. All these
distributions are popularly chosen as models for the income distribution, the loss
distribution and the baseline hazard. For each distribution, 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 Gustafsson et al.
(2009), for each estimator f , we compute three performance measures, namely, theintegrated absolute deviation (“IAD”), the root integrated squared error (“RISE”),
and the root weighted integrated squared error (“RWISE”), where
IAD f (x)

=

Z 1

f (x)

f (x) dx;

0

RISE f (x)

=

sZ

1

f (x)

f (x)

f (x)

f (x)

2

dx;

0

RW ISE f (x)

=

sZ

1

2

x2 dx:0

IAD and RISE measure the error between the true and estimated densities with equal
weights across the support, whereas RWISE focuses on the …t of the tail part.
our reports, the integrals are approximated over the 500 points.
TABLE 3 ABOUT HERE
8

The density of the generalized Champernowne distribution (Buch-Larsen et al., 2005) isf (x; ; M; c) =with parameters> 0, M > 0 and c(x + c)1f(M + c)f(x + c) + (M + c)
0.

15

c g

2; x

2c g

0

In

3.2

Choices of Smoothing Parameters

Choosing the smoothing parameter b is an important practical issue.

To exped-

ite computations, as in Hirukawa (2010), we develop plug-in methods for TS- andJLN-MBC estimators that use a gamma density as a reference. The plug-in smoothing parameters for f~T S;M G (x) and f~JLN;G (x) (called “gamma-referenced smoothing
parameters” hereafter) are de…ned as the minimizers of asymptotic weighted mean
integrated squared errors (“AWMISEs”)
no^bGR T S = arg min AW M ISE f~T S;M G (x)
b
Z 1
Z
b4
n 1 b 1=2 (c) 1 g (x)2pp wT S (x) dx;= arg minp~M G (x) wT S (x) dx +b c2 (1
2
x
c)2 0
0
n
o
^bGR JLN = arg min AW M ISE f~JLN;G (x)b
Z 1Z
n 1 b 1=2 1 g (x)42pp wJLN (x) dx;= arg min bq~G (x) wJLN (x) dx +
b
2
x
0
0
where g (x) = x

1exp ( x= ) = f( )g is the density function for the gamma

distribution with parameters ( ; ), and p~M G (x) and q~G (x) can be obtained by replacing f (x) in pM G (x) and qG (x) with g (x). The weighting functions are chosen
as wT S (x) = x5 and wJLN (x) = x to ensure …niteness of integrals. The parameters( ; ) are replaced by their estimates^; ^via method of moments or maximum

likelihood (“ML”).9 Analytical expressions of ^bGR T S and ^bGR JLN , as well as ^bGR (=the gamma-referenced smoothing parameter for f^M G (x)), are given in the Appendix.
We do not pursue the gamma-referenced smoothing parameter for f~T S;G (x); since
extra terms are involved in pG (x), the minimizer of its AWMISE takes a much more
complicated form than ^bGR T S . On the other hand, although it is possible to derive9

ML estimates are used exclusively in this paper.

16

the gamma-referenced smoothing parameter for f~JLN;M G (x) in a similar way,10 ourpreliminary simulation results indicate that the formula frequently generates large
values and thus we do not advocate its use. From the viewpoint of practical relevance, ^bGR T S and ^bGR JLN are simply employed for f~T S;G (x) and f~JLN;M G (x) in our
simulations, respectively.

Similarly, ^bGR is chosen as the smoothing parameter forf^G (x).
Besides, a very simple formula is frequently applied in the literature (e.g. Gustafsson et al., 2009). From this viewpoint, a “rule-of-thumb” smoothing parameteris also considered for each estimator.

More precisely, we additionally employ: (i)^bROT 1 = ^ x n 2=5 for G, MG and LMBC-G11 ; (ii) ^bROT 2 = ^ u n 2=5 for LTBC-C; and(iii) ^bROT 3 = ^ x n 2=9 for TS-MBC-G, TS-MBC-MG, JLN-MBC-G, and JLN-MBCMG, where ^ x and ^ u are sample standard deviations of the original data and thetransformed data on the unit interval, respectively.
TABLE 4 AND FIGURE 2 ABOUT HERE

3.3

Simulation Results

Table 4 reports simulation averages and standard deviations of performance measures. The gamma-referenced method in general works better than the rule-of-thumbmethod for a given estimator, and it often substantially reduce the values of performance measures.12 Therefore, the table presents the results from the gamma-referenced
method for all estimators other than two semiparametric estimators (i.e. LMBC-G
10

Choosing x5 as the weighting function, we can derive the smoothing parameter as

arg min b
b

4

Z 1
0

n
2
5
q~M
G (x) x dx+

1

2

b 1=2
p

Z 1
0

g (x) 5p x dx =x

11(

4 9=2 ( + 9=2) ( )
p
2
2
4
(
1) (
2) (2 )

)2=9

n 2=9 :

Although Hagmann and Scaillet (2007) employ the cross validation method for LMBC-G, we
adopt the simple plug-in rule to speed up computations.
12
Exceptions are Distributions 3 and 4, but gains from the rule-of-thumb method are marginal.

17and LTBC-C); in fact, only the rule-of-thumb method is chosen for these estimators.
For each distribution, results are qualitatively similar across three sample sizes.
Comparing TS- and JLN-MBC estimators with their corresponding bias-uncorrected
estimators, we can see that JLN-MBC estimators often improve all three performancemeasures, whereas results are mixed for TS-MBC estimators. JLN-MBC outperforms
TS-MBC in most cases; only a few exceptions include RISEs for Distributions 5, 7.
This …nding is consistent with the simulation results reported in Jones and Signorini(1997). In addition, even when the two semiparametric estimators perform better interms of overall performance measures (i.e. IAD and RISE), JLN-MBC is competitive
with them in terms of the performance measure for the tail (i.e. RWISE); see theresults for Distributions 5, 6, 7, for instance.
Results do not con…rm that for a given estimator, employing the Modi…ed Gammakernel improves performance measures over the Gamma kernel. A rationale can befound in Figure 1.

The Gamma kernel put the maximum weight on the design

point x whereas the Modi…ed Gamma not, which explains why the Modi…ed Gamma
estimators may not outperform the Gamma estimators globally. In particular, TSMBC-MG consistently performs inferiorly to TS-MBC-G, except Distribution 5. The
poor performance of TS-MBC-MG can be attributed to the following two respects.
First, TS-MBC estimation relies on two smoothing parameters b and b=c. Controllingboth b and b=c is a cumbersome task.

Because 0 < c < 1, the density estimatorusing b=c tends to be oversmoothed, which is potentially a source of a large bias
in every TS-MBC estimator.

On the other hand, if we make b too short in orderto have a reasonable length of b=c, additional variability is introduced to the otherestimator using b due to undersmoothing. Second, when the Modi…ed Gamma kernel
is employed, these two smoothing parameters also play a role of determining the18

boundary region explicitly (e.g. [0; 2b) for the density estimator using b). Unless b
is short enough, there is a relatively small interior region for the density estimatorusing b=c. This aspect is also thought to worsen the performance measures of TSMBC-MG. In conclusion, when TS-MBC estimation is applied, it is desirable to use asmoothing parameter choice method that tends to provide a small value consistently.
Our experiment indicates that for a given distribution and a given sample size, ^bGR T S
on average yields a smaller value than ^bROT 3 does, which explains why the former ispreferable in general for TS-MBC estimation over the latter.
It is also worth remarking that performance of the two semiparametric estimators
depends on the shape of the distribution. Due to its gamma start, LMBC-G performs
well when the distribution looks like the gamma one (e.g. Distributions 1, 2, 7, 8).
On the other hand, because the generalized Champernowne distribution is designed
to capture the tail of the underlying distribution, the performance of LTBC-C isremarkable for medium- to heavy-tailed distributions (e.g. Distributions 5, 6, 9,
10). However, LMBC-G for Distributions 3 and 4 and LTBC-C for Distributions 1-4
perform inferiorly. To con…rm these …ndings, we now inspect average plots of theirdensity estimates.
Figure 2 presents average plots of density estimates from G, LMBC-G, LTBCC, TS-MBC-G, and JLN-MBC-G against the true density for selected distributions.
The plots are obtained from 1,000 Monte Carlo samples with sample size n = 100, andthe average for each estimator is taken over 1,000 replications for each grid. For each
distribution, the left and right panels correspond to the average plots over the region
near the origin and the tail part, respectively. Panels (a) and (d) indicate that LMBCG and LTBC-C can capture the shapes of Weibull and Pareto distributions nearly
perfectly, respectively. In contrast, as suggested by Panel (a), LTBC-C imprecisely

19

captures the shape near the origin of such a distribution as Distribution 1 or 2.
Moreover, Distributions 3 and 4 satisfy so-called the shoulder condition f 0 (0+ ) = 0.
In this case, local concavity of the true density around the origin tends to force the
gamma and generalized Champernowne starts to generate a spurious peak near theorigin (see Panel (b)), which leads to the poor performance of LMBC-G and LTBCC. Likewise, local convexity of the true density around the origin in Distribution 6
makes the gamma start for LMBC-G unbounded at zero. Panel (d) indicates thatthe unboundedness cannot be corrected even after the bias correction step.
We can also see from Figure 2 that if G underestimates (overestimates) the density,
JLN-MBC-G corrects the estimate in an upward (downward) direction, as reported in
Hirukawa (2006, 2010); such a bias correction mechanism is not obvious for TS-MBCG. Both of TS-MBC-G and JLN-MBC-G can trace out entire shapes of a variety
of distributions at a satisfactory level; the chances that they mistakenly capture theshapes would be low, unlike the two semiparametric estimators. While JLN-MBC-G
performs superiorly to TS-MBC-G in terms of the three performance measures, itappears that the latter is better at capturing heights of the peaks of distributions
than the former.

The …gure also demonstrates that both estimators can preserve

the shapes of tail parts equally well. Overall, simulation results indicate that MBC
estimators are viable alternatives to the two semiparametric estimators.

4

Applications to Income Data

4.1

Case #1: U.S. Data

In this section, the MBC density estimators are applied to a couple of income data
sets. Our …rst application focuses on the U.S. income data. We use the data set
studied in Abadie (2003), which includes 9,275 observations from the 1991 Survey of20Income and Program Participation (SIPP). The data set is available under the name“401ksubs.raw” on the web site for Wooldridge (2001), and the variable “inc” (=annual income, in thousands of dollars) is examined in this section.
Table 5 presents descriptive statistics of the data, including some inequality measures, where the data is converted to dollars. Although sample moments and quantiles
suggest a right-skewed distribution, it appears that the tail is not extremely heavy.
This is con…rmed by the Hill plot (= a plot of tail index estimates by Hill, 1975)of the data up until the 1,000th observation from the largest (unreported), whichindicates that estimates of the tail index roughly ranges from 4 to 5, except initialfew estimates.
TABLE 5 AND FIGURE 3 ABOUT HERE
The income distribution is estimated by G, LMBC-G, TS-MBC-G, and JLNMBC-G. The original data is converted to 105 dollars, and then the resulting density
estimates are back-transformed to the ones denominated in dollars.
step, we …t a gamma density g (x) = x

1exp ( x= ) = f

At the initial( )g to the converted

data by ML, and obtain parameter estimates ^ ; ^ = (3:158; 0:124). Then, plug-insmoothing parameters ^bGR , ^bROT 1 , ^bGR T S , and ^bGR JLN are chosen for G, LMBC-G,
TS-MBC-G, and JLN-MBC-G, respectively; their values are 0.0030, 0.0062, 0.0071,
and 0.0354. Given these numbers, we can draw Figure 3. For reference, the …gurepresents shapes around the mode (Panel (b)) and on the tail (Panel (c)), as wellas the entire shape (Panel (a)).

The …gure suggests that while estimates from G

and TS-MBC-G are almost indistinguishable, the ones from the remaining two are
considerably di¤erent, in particular, in terms of the location and height of the mode;see Panel (b) for details. Moreover, a closer look at Panel (c) …nds that the wiggle

21

over the tail generated by G is substantially smoothed away by JLN-MBC-G; thiskind of smoothing mechanism is not observed in TS-MBC-G.

4.2

Case #2: Brazilian Data

Our second application employs the Brazilian income data in 1990. The large micro data set (n = 71; 523) is collected via the annual National Household Survey(PNAD) by the Brazilian Statistical O¢ ce, and it has been also analyzed in Hagmann and Scaillet (2007).

Using similar data sets, Cowell, Ferreira and Litch…eld(1998) investigate the dynamics of the Brazilian income distribution in 1980s. The
data considered is monthly per capita denominated in 1990 cruzeiros. Analyzing thisdata set is of particular interest, because Brazil has the eighth largest GDP in theworld and faces a strong inequality in the income distribution.
Table 6 presents descriptive statistics of the data.

The income distribution isempirically known to have a single sharp peak near the origin and a long right tail.
Indeed, sample moments and quantiles reveal an extremely right-skewed distribution.
The Hill plot of the data up until the 8,000th observation from the largest suggeststhat estimates of the tail index roughly range from 1.6 to 3, except initial few estimates. Inequality measures much larger than those from the U.S. data also re‡ect thelong, thick tail.
TABLE 6 AND FIGURE 4 ABOUT HERE
As in the previous section, the income distribution is estimated by G, LMBC-G,
TS-MBC-G, and JLN-MBC-G. Again, the original data is converted to 105 cruzeiros,and then the resulting density estimates are back-transformed to the ones denominated in cruzeiros. Fitting a gamma density g (x) = x
the converted data yields ML estimates1

exp ( x= ) = f( )g to^ ; ^ = (0:887; 0:588). Computing values
22

of smoothing parameters ^bGR , ^bROT 1 , ^bGR T S , and ^bGR JLN as 0.0050, 0.0104, 0.0246,
and 0.0533, we can obtain Figure 4. Although the value of ^ implies that unlike theU.S. case, the data would hint an income distribution unbounded at zero, the estimate
from LMBC-G does not con…rm this gesture. Because of quite a few observations,
it seems that there is a consensus on the shape among estimators, except that of themode. G generates the highest peak. JLN-MBC-G substantially smooths out themode, and as a result, the peak is lowest. LMBC-G and TS-MBC-G yield a similar
shape around the mode, and the height of the peak is in-between. On the other hand,a careful investigation of Panel (c) reveals that G and LMBC-G generate wiggles over
the tail, whereas two MBC estimators smooth them out thoroughly.

5Conclusion

This paper has demonstrated that two well-known MBC techniques designed originally for symmetric kernels can be applied to density estimation using asymmetrickernels that have support on [0; 1). Under su¢ cient smoothness of the true density,
both bias reduction methods are shown to improve the order of magnitude in bias
from O (b) to O (b2 ), while the order of magnitude in variance remains unchanged.
Two classes of MBC density estimators are by construction nonnegative, and establish
a faster convergence rate of O n 8=9 in MSE for the interior part when best implemented, as with symmetric second-order kernels. Monte Carlo simulations indicate
superior performance of JLN-MBC estimators in particular, compared to corresponding bias-uncorrected estimators.

The MBC estimators are applied to estimating

income distributions from the U.S. and Brazilian data.

23

A

Appendix

The proof of each theorem requires kernel-speci…c arguments, which include Taylor
expansions and properties of the random variable corresponding to the kernel. To
save space, we present only the proofs when the Gamma kernel is employed.

A.1

Proof of Theorem 1

This proof closely follows the one for Theorem 1 of Hirukawa (2010).

Bias. Applying the …rst equation on p.489 of Hirukawa (2010), we can approximateon
E f~T S;G (x) = f (x)+

1
c (1

1
c) 2

a21;G (x; f )f (x)a2;G (x; f ) b2 +o b2 +O n 1 b 1 ;where the remainder term O (n 1 b 1 ) = o (b2 ) by Assumption 2.
Variance. The variance of f~T S;G (x) can be expressed ason
V ar f~T S;G (x)
h
n
o
1
^
V ar fG;b (x)=(1 c)2

n
o
n
oi
2
^
^
^
2cCov fG;b (x) ; fG;b=c (x) + c V ar fG;b=c (x) + O n 1 :For interior x, a similar argument to the one in the proof for Theorem 1 of Hirukawa(2010) yieldsnoV ar f^G;b (x) = n 1 b 1=2 vG (x) f (x) + o n 1 b 1=2 ;noV ar f^G;b=c (x) = n 1 b 1=2 c1=2 vG (x) f (x) + o n 1 b 1=2 ;
p 1=2
n
o
2c
vG (x) f (x) + o n 1 b 1=2 :Cov f^G;b (x) ; f^G;b=c (x) = n 1 b 1=2(1 + c)1=2
Then, the result immediately follows.
n
o
n
o
^
^
On the other hand, for boundary x, each of V ar fG;b (x) , V ar fG;b=c (x) andno
Cov f^G;b (x) ; f^G;b=c (x) is at most O (nb) 1 . Therefore, the stated order of mag-nitude is also established.

24

A.2

Proof of Theorem 2This proof closely follows the one for Theorem 2 of Hirukawa (2010). To approximate
the variance, the following two lemmata are required:

Lemma 1. Suppose that Assumptions 1 and 2 hold. Let
1 X Kj(x;b) (Xi ) Kj(Xi ;b) (u):n i=1f 2 (Xi )n

2Kj(x;b) (u)& j (u) =
f (u)
Then,
& j (u)wherej =for some

8
>
>
>
>
>
<
>
>
>
>
>
:

1
2
1(

1=2

1
1=2

2

j (u) =j

Kj(x;b) (u);f (u)if u=b ! 1
if u=b !
if ub ! 0
if ub !

1for j = G; M G; RIG
for j = LN; BS
for j = IG

> 0.

Lemma 2. Suppose that Assumptions 1 and 2 hold. Let X > 0 be drawn from the
distribution with density f (x) having support on [0; 1). Then, the transformationj ( ) de…ned in Lemma 1 can be approximated by

E

2
j (X)

=

b 1=2 vj (x) =f (x) + o b 1=2
O b (rj +1=2)

for interior x;
for boundary x

where vj (x) and rj are de…ned in Theorem 1 .
A.2.1

Proof of Lemma 1

It su¢ ces to demonstrate that
1 X KG(x=b+1;b) (Xi ) KG(Xi =b+1;b) (u)n i=1f 2 (Xi )n

KG(x=b+1;b) (u)
f (u)

25

1
1=2

if u=b ! 1:
if u=b !

LetThen, for an arbitrarily large n, we can approx-

G (y) := KG(x=b+1;b) (y) =f (y).

imate (1=n)

Pn

i=1 KG(x=b+1;b) (Xi ) KG(Xi =b+1;b) (u) =f

Z 1

KG(x=b+1;b) (X1 ) KG(X1 =b+1;b) (u)f 2 (X1 )=n yuy=b exp ( u=b)
= exp
log u
by=b+1 (y=b + 1)
b

u
b

E

2

(Xi ) byuy=b exp ( u=b)
(y)
dy:
G
by=b+1 (y=b + 1)

0

(A1)

Now,
y
log b
b

log y

log

y o
;
b

where
y
b

log

=

y
b

1
2

y
b

log y

1
2

y 1
b
+ log 2 ++ O b3
b 2
12y

log b

by Stirling’s approximation. Then, the right-hand side of (A1) can be approximated
by
b 1=2
p
2

Z 1
0

G (y)

p

y

exp

y
(log ub

By a change of variable w = (ulog y)uybb+ O b3
12y

dy:

(A2)

y) =b1=2 and Taylor expansions for the logarithm

and exponential functions, (A2) can be further rewritten as
Z u=b1=2
b1=2 w
1
G u
p
1=2
2
1
(u b1=2 w)
u b1=2 wbexp
log u log u b1=2 w
b 1=2 w
+ O b3 dwb
12 (u b1=2 w)
Z u=b1=2
b1=2 w
1
w2
G u
= p
exp
1=2
2u
2
1
(u b1=2 w)
b1=2 w3
bw4
b
bw6
1
+
+ O b3=2 dw:(A3)
6u2
12u3 12u 72u4
p
Take another change of variable v = w= u, and let ( ) and ( ) denote the standard
normal density and distribution functions, respectively. Then, (A3) reduces to
Z pu=b
1=2
p
u
1=2
p
(v)
u
b
uv
G
u b1=2 uv
1
b1=2 v 3
bv 4
b
bv 6
p
1
+
+ O b3=2 dv
12u 12u 72u
6 u
KG(x=b+1;b) (u)1
if u=b ! 1:
1=2
if u=b !
f (u)26A.2.2

Proof of Lemma 2

Pick = b1
E

for some 2 (0; 1=2). Then, by the trimming argument in Chen (2000),
Z 1
Z 1
Z
KG(2x=b+1;b=2) (u)22+du + O b;
G (u) f (u) du = Ab (x)G (X) =
f (u)
0
0

where
b 1 (2x=b + 1)
Ab (x) = 2x=b+1 22(x=b + 1)

(

b 1=2
p
x1=2
(2 +1)b 1
22 +1 2 ( +1)2if x=b ! 1:

if x=b !

The stated result follows from recognizing that
Z 1
KG(2x=b+1;b=2) (u)du = E f 1 ( x ) = f 1 (x) + o (1) ;
f (u)
0
where

d
x = G (2x=b + 1; b=2).

A.2.3

Proof of Theorem 2

Bias. Write h (x) = a1;G (x; f ) =f (x).

By the procedures in Section A.2.1 of

Hirukawa (2010, pp.490-491) and properties of gamma random variables, we canrecognize that
o
n
n
o
x
E f~JLN;G (x) = f (x) f (x) h0 (x) + h00 (x) b2 +o b2 = f (x) f (x) a1;G (x; h) b2 +o b2 :2
Variance. Following the procedures in Section A.2.2 of Hirukawa (2010, p.492), we
can obtain
1 X KG(x=b+1;b) (Xi )
f (x)
n i=1
f (Xi )nf~JLN;G (x)where

1 X KG(x=b+1;b) (Xi )n i=1
f (Xi )n(2f^G;b (Xi )f (Xi ))(

2f^G;b (Xi )f (Xi ));1X=
& G (Xi )n i=1nfor & G ( ) de…ned in Lemma 1. Lemma 1 implies that & G (Xi ) can be approximatedby an iid random variableno
V ar f~JLN;G (x)G (Xi ), and thus

11f 2 (x) V ar f G (X1 )g = f 2 (x) EnnFinally, applying Lemma 2 completes the proof.
27

2
G (X1 )+O n 1:

A.3

Formulae for Gamma-Referenced Smoothing Parameters

The analytical expression of ^bGR T S is^bGR T S = c2 (12=9c)2 (c)(4

9=2( + 9=2) ( )p
16 CT S ( ) (2 ))2=9

n 2=9 ;where
2

CT S ( ) =

3
1
(
2)2
(
1)2
36
2
1
3
1
(
2)
(
1)2 ( )
+
6
2
2
1
3
1
+ (
2)
(
1) ( )+( + 1)
9221
311)2 ( )+
( + 1)
+
+ (
4
2
2
1
1
3
(
1) ( )+( + 1)+( + 2)
3
2
2
1
1
3
5
+
+ ( )
( + 1)+( + 2)
+
9
2
2
2

:

On the other hand, ^bGR JLN takes a much simpler form. It is given by^bGR JLN =

(

4

5=2

( + 1=2) ( )
p
4
(2 )

)2=9

n 2=9 :

Moreover, the gamma-referenced smoothing parameter for f^M G (x) is de…ned asno^bGR = arg min AW M ISE f^M G (x)bZZb2 1 2 00
n 1 b 1=2 1 g (x)2pp w (x) dx;= arg minx fg (x)g w (x) dx +
b
4 0
2
x
0
where the weighting function w (x) is chosen as w (x) = x3 to ensure …niteness of
integrals. It follows that ^bGR can be expressed as^bGR =(

4

5=2( + 5=2) ( )p
8 C ( ) (2 )
28

)2=5

n 2=5 ;

whereC( ) =

1
(
4

2)2 (

1)2

(

1) ( )

+

(
1
2

2) (
( + 1) +

1
(
1) (3
2
1
+
( + 1)
21)2 ( ) +4) ( )

1
( )
4

+

3
2

+

1
2

:

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31

Table 1: Functional Forms of Asymmetric Kernels
Kernel (j)Functional Form (u 0)G
KG(x=b+1;b) (u) = ux=b exp ( u=b) = bx=b+1 (x=b + 1)
MG
KM G( b (x);b) (u) = u b (x) 1 exp ( u=b) = b b (x) f b (x)g ;x=b
for x 2bwhere b (x) =2(1=4) (x=b) + 1 for x 2 [0; 2b)1uIG
KIG(x;1=b) (u) = p2 1bu3 exp
2 + ux
2bx x
x b
u
RIG
KRIG(1=(x b);1=b) (u) = p21 bu exp
2 + xu b
n 2b x bo(log u log x)2
LN
KLN (log x;b) (u) = up12 b exp
2b
n
o
1=2
3=2
1
x
x
1 u
BS
+ u
exp
2 + uxKBS (b1=2 ;x) (u) = 2xp2 b u
2b x
Table 2: Explicit Forms of a1;j (x; f ) and a2;j (x; f )Kernel (j)
G
MG
IG
RIG
LN
BS

a1;j (x; f )f (x) + x2 f 00 (x)
x 00
f (x)
for x 2b2;0b (x) f (x) for x 2 [0; 2b)where b (x) = b (x) xb = O (1)
x3 00
f (x)
2
x 00
f (x)
2
2
x 0
f (x) + x2 f 00 (x)22x 0
f (x) + x2 f 00 (x)2
0a2;j (x; f )2
5f (x) + 6 xf 000 (x) + x8 f 0000 (x)2x 000
f (x) + x8 f 0000 (x)for x 2b3
2
1x
00
b (x) + b (x) + b f (x) for x 2 [0; 2b)200

6x5 000f (x) + x8 f 0000 (x)22
1 00
f (x) + x2 f 000 (x) + x8 f 0000 (x)24x 0
f (x) + 78 x2 f 00 (x) + 43 x3 f 000 (x) + x8 f 0000 (x)84
3 2 00x f (x) + 34 x3 f 000 (x) + x8 f 0000 (x)4Table 3: True Distributions Considered in Monte Carlo Simulations
Distribution
Density Function f (x) ; x 0
1. Gamma
x 1 exp ( x= ) = f( )g ; ( ; ) = (1:5; 1) :1
2. Weibull( = ) (x=n )expof (x= ) g ; ( ; ) = (1:5; 1:5) :3. Half-Normal

4. Half-Logisitic
5. Log-Normal

6. Pareto
7. Burr
8. Generalized Gamma
9. Log-Normal andPareto Mixture 1
10. Log-Normal andPareto Mixture 2(x )2
p 2 exp; ( ; ) = (0; 1:5) :
2 2
2
2
x
x
2
exp n s= 1o+ exp;(ss(log x )21pexp; ( ; ) = (0; 0:75) :
2 2
x 2
+1

; s) = (0; 1) := (x + ) ; ( ; ) = (1; 2) :x 1 = (1 + x ) +1 ; ( ; ) = (1:5; 2:5) :xh 1 exp f n(x= ) g = oi ( = ) ; ( n; ; ) = (5;o 2; 2:5) :2(log x )exp+ (1 p) (x+ ) +1 ;2 2
0:5; 1; 2) : nh(p; ; ; ; n) = (0:7; 0;
oi
o
2
(log
x
)
p xp12 exp
+
(1
p)
;
+1
2
2
(x+ )
(p; ; ; ; ) = (0:3; 0; 0:5; 1; 2) :

p

p1
x 2

32

Table 4: Averages and Standard Deviations of Performance Measures(Distributions 1 and 2)

Estimator

IAD

n = 100
RISE

RWISE

G

0.02581 0.03816 0.04787(0.00861) (0.01227) (0.01737)
MG
0.02651 0.03858 0.05253(0.00927) (0.01192) (0.02090)
LMBC-G
0.02348 0.03422 0.04386(0.00846) (0.01301) (0.01666)
LTBC-C
0.02877 0.04478 0.05020(0.00954) (0.01417) (0.01917)
TS-MBC-G
0.02622 0.03940 0.04893(0.00904) (0.01318) (0.01879)
TS-MBC-MG
0.02708 0.04084 0.05058(0.00920) (0.01271) (0.01959)
JLN-MBC-G
0.02322 0.03647 0.04108(0.00862) (0.01302) (0.01646)
JLN-MBC-MG
0.02403 0.03845 0.04127(0.00825) (0.01043) (0.01663)
G

0.02678 0.03985 0.04856(0.00931) (0.01384) (0.01819)
MG
0.02755 0.03964 0.05342(0.00949) (0.01307) (0.02065)
LMBC-G
0.02484 0.03697 0.04420(0.00919) (0.01421) (0.01692)
LTBC-C0.02955 0.04596 0.05182(0.01036) (0.01569) (0.02007)
TS-MBC-G
0.02761 0.04143 0.05005(0.00986) (0.01502) (0.01946)
TS-MBC-MG
0.02834 0.04209 0.05229(0.00985) (0.01440) (0.02022)
JLN-MBC-G
0.02387 0.03650 0.04137(0.00983) (0.01490) (0.01738)
JLN-MBC-MG
0.02404 0.03846 0.03970(0.00863) (0.01203) (0.01654)

IAD

n = 200
RISE

RWISE

Distribution #1: Gamma
0.02015 0.03058 0.03668(0.00652) (0.00966) (0.01281)
0.02044 0.03026 0.03974(0.00685) (0.00911) (0.01516)
0.01771 0.02600 0.03289(0.00620) (0.00948) (0.01228)
0.02202 0.03509 0.03776(0.00681) (0.01017) (0.01336)
0.01981 0.03085 0.03579(0.00675) (0.01018) (0.01348)
0.02039 0.03167 0.03697(0.00671) (0.00927) (0.01400)
0.01779 0.02948 0.03040(0.00648) (0.00995) (0.01213)
0.01857 0.03154 0.03039(0.00605) (0.00754) (0.01215)
Distribution #2: Weibull
0.02082 0.03137 0.03795(0.00646) (0.00996) (0.01295)
0.02102 0.03047 0.04102(0.00677) (0.00932) (0.01511)
0.01876 0.02805 0.03391(0.00610) (0.00970) (0.01155)
0.02264 0.03579 0.03999(0.00716) (0.01145) (0.01409)
0.02059 0.03129 0.03740(0.00666) (0.01039) (0.01349)
0.02098 0.03145 0.03882(0.00675) (0.00983) (0.01416)
0.01830 0.02861 0.03161(0.00671) (0.01050) (0.01199)
0.01827 0.03019 0.02968(0.00584) (0.00811) (0.01126)

33

IAD

n = 500
RISE

RWISE

0.01464 0.02307 0.02624(0.00419) (0.00662) (0.00823)
0.01460 0.02195 0.02827(0.00430) (0.00596) (0.00960)
0.01234 0.01837 0.02300(0.00388) (0.00630) (0.00774)
0.01564 0.02593 0.02645(0.00425) (0.00654) (0.00846)
0.01391 0.02302 0.02428(0.00428) (0.00695) (0.00855)
0.01426 0.02307 0.02511(0.00421) (0.00587) (0.00884)
0.01267 0.02278 0.02071(0.00423) (0.00694) (0.00772)
0.01345 0.02467 0.02070(0.00380) (0.00472) (0.00760)
0.01503 0.02288 0.02766(0.00451) (0.00682) (0.00910)
0.01491 0.02160 0.02944(0.00448) (0.00610) (0.00997)
0.01313 0.01969 0.02420(0.00411) (0.00636) (0.00825)
0.01605 0.02599 0.02825(0.00447) (0.00690) (0.00909)
0.01422 0.02188 0.02577(0.00456) (0.00691) (0.00918)
0.01441 0.02179 0.02662(0.00443) (0.00620) (0.00939)
0.01302 0.02102 0.02239(0.00474) (0.00718) (0.00877)
0.01291 0.02225 0.02061(0.00382) (0.00497) (0.00793)(continued)

Table 4: Continued (Distributions 3 and 4)

Estimator
G
MG
LMBC-G
LTBC-CTS-MBC-G
TS-MBC-MG
JLN-MBC-GJLN-MBC-MG

G
MG
LMBC-G
LTBC-CTS-MBC-G
TS-MBC-MG
JLN-MBC-GJLN-MBC-MG

IAD

n = 100
RISE

RWISE

IAD

n = 200
RISE

RWISE

0.02480 0.03708 0.04472(0.00874) (0.01305) (0.01696)
0.02669 0.03971 0.05043(0.00937) (0.01491) (0.01951)
0.02603 0.04441 0.04184(0.00945) (0.01690) (0.01576)
0.02744 0.04260 0.04651(0.01028) (0.01679) (0.01786)
0.02511 0.03809 0.04434(0.00974) (0.01547) (0.01781)
0.02768 0.04308 0.04782(0.01031) (0.01773) (0.01895)
0.02208 0.03267 0.03936(0.00942) (0.01434) (0.01711)
0.02162 0.03251 0.03660(0.00937) (0.01481) (0.01587)

Distribution #3: Half-Normal
0.01946 0.02914 0.03530(0.00627) (0.00964) (0.01205)
0.02062 0.03085 0.03915(0.00682) (0.01084) (0.01400)
0.01983 0.03372 0.03258(0.00659) (0.01177) (0.01134)
0.02091 0.03267 0.03541(0.00748) (0.01302) (0.01240)
0.01891 0.02851 0.03366(0.00690) (0.01110) (0.01238)
0.02061 0.03192 0.03610(0.00736) (0.01262) (0.01341)
0.01686 0.02489 0.03051(0.00693) (0.01059) (0.01275)
0.01629 0.02465 0.02782(0.00670) (0.01080) (0.01134)

0.02397 0.03452 0.04481(0.00844) (0.01216) (0.01677)
0.02532 0.03644 0.05041(0.00916) (0.01400) (0.01972)
0.02457 0.04005 0.04194(0.00912) (0.01596) (0.01599)
0.02685 0.04093 0.04633(0.01018) (0.01645) (0.01793)
0.02387 0.03453 0.04457(0.00944) (0.01429) (0.01786)
0.02618 0.03931 0.04731(0.01003) (0.01683) (0.01887)
0.02093 0.03019 0.03826(0.00912) (0.01354) (0.01654)
0.02110 0.03027 0.03790(0.00917) (0.01394) (0.01624)

Distribution #4: Half-Logistic
0.01874 0.02710 0.03530(0.00620) (0.00911) (0.01235)
0.01959 0.02834 0.03913(0.00675) (0.01028) (0.01455)
0.01865 0.03025 0.03251(0.00642) (0.01114) (0.01149)
0.02043 0.03118 0.03571(0.00704) (0.01145) (0.01277)
0.01792 0.02584 0.03368(0.00681) (0.01040) (0.01275)
0.01945 0.02916 0.03556(0.00727) (0.01212) (0.01366)
0.01588 0.02292 0.02923(0.00663) (0.01010) (0.01182)
0.01589 0.02295 0.02873(0.00663) (0.01026) (0.01169)

34

IAD

n = 500
RISE

RWISE

0.01387 0.02115 0.02534(0.00408) (0.00623) (0.00827)
0.01458 0.02225 0.02780(0.00441) (0.00700) (0.00959)
0.01379 0.02383 0.02310(0.00409) (0.00730) (0.00759)
0.01485 0.02355 0.02554(0.00566) (0.01110) (0.00888)
0.01276 0.01951 0.02304(0.00437) (0.00687) (0.00849)
0.01392 0.02183 0.02473(0.00464) (0.00787) (0.00919)
0.01167 0.01729 0.02157(0.00430) (0.00657) (0.00826)
0.01117 0.01722 0.01913(0.00418) (0.00685) (0.00733)
0.01339 0.01969 0.02539(0.00409) (0.00597) (0.00851)
0.01390 0.02049 0.02785(0.00441) (0.00670) (0.01005)
0.01291 0.02130 0.02292(0.00402) (0.00687) (0.00787)
0.01412 0.02195 0.02493(0.00459) (0.00773) (0.00884)
0.01209 0.01770 0.02296(0.00436) (0.00650) (0.00880)
0.01315 0.01998 0.02422(0.00460) (0.00759) (0.00942)
0.01086 0.01587 0.02013(0.00419) (0.00635) (0.00781)
0.01088 0.01605 0.01964(0.00423) (0.00658) (0.00783)(continued)

Table 4: Continued (Distributions 5 and 6)

Estimator

IAD

n = 100
RISE

RWISE

G

0.03241 0.05338 0.04884(0.00966) (0.01530) (0.01658)
MG
0.03106 0.04690 0.05380(0.01100) (0.01652) (0.01996)
LMBC-G
0.02534 0.03937 0.04252(0.00900) (0.01403) (0.01656)
LTBC-C
0.02853 0.04721 0.04542(0.01065) (0.01764) (0.01837)
TS-MBC-G
0.03126 0.05085 0.04983(0.00977) (0.01557) (0.01822)
TS-MBC-MG
0.02949 0.04496 0.05132(0.01008) (0.01584) (0.01903)
JLN-MBC-G
0.03121 0.05656 0.04158(0.00866) (0.01397) (0.01441)
JLN-MBC-MG
0.03482 0.06361 0.04563(0.00899) (0.01327) (0.01575)
0.0132684
G

0.03696 0.09475 0.03309(0.01109) (0.03207) (0.01076)
MG
0.04168 0.08360 0.04994(0.01352) (0.02722) (0.01791)
LMBC-G
0.02515 0.07257 0.02849(0.00880) (0.03237) (0.01114)
LTBC-C
0.02604 0.06495 0.02601(0.00999) (0.02871) (0.01059)
TS-MBC-G
0.03173 0.07971 0.03069(0.01225) (0.03562) (0.01146)
TS-MBC-MG
0.05906 0.11707 0.06751(0.02082) (0.03067) (0.03707)
JLN-MBC-G
0.03053 0.07616 0.02850(0.00915) (0.02600) (0.01046)
JLN-MBC-MG
0.04037 0.10432 0.03655(0.00977) (0.02352) (0.01086)

IAD

n = 200
RISE

RWISE

Distribution #5: Log-Normal
0.02537 0.04239 0.03763(0.00682) (0.01121) (0.01165)
0.02373 0.03623 0.04115(0.00781) (0.01206) (0.01410)
0.01947 0.03074 0.03215(0.00614) (0.00957) (0.01144)
0.02163 0.03626 0.03420(0.00681) (0.01170) (0.01196)
0.02392 0.04030 0.03679(0.00671) (0.01104) (0.01230)
0.02200 0.03393 0.03793(0.00705) (0.01109) (0.01283)
0.02491 0.04682 0.03147(0.00619) (0.01018) (0.01012)
0.02809 0.05344 0.03517(0.00653) (0.00997) (0.01092)
Distribution #6: Pareto
0.02954 0.07801 0.02542(0.00807) (0.02363) (0.00785)
0.03193 0.06493 0.03754(0.00892) (0.01938) (0.01140)
0.01993 0.06229 0.02121(0.00669) (0.02597) (0.00800)
0.01969 0.04940 0.01987(0.00662) (0.01927) (0.00745)
0.02543 0.06773 0.02275(0.00912) (0.02749) (0.00815)
0.05020 0.10405 0.05380(0.01541) (0.02371) (0.02462)
0.02441 0.06376 0.02138(0.00680) (0.01967) (0.00731)
0.03325 0.09053 0.02823(0.00713) (0.01784) (0.00734)

35

IAD

n = 500
RISE

RWISE

0.01832 0.03127 0.02675(0.00484) (0.00805) (0.00794)
0.01695 0.02625 0.02916(0.00537) (0.00868) (0.00948)
0.01401 0.02261 0.02250(0.00442) (0.00719) (0.00786)
0.01551 0.02641 0.02410(0.00428) (0.00738) (0.00749)
0.01715 0.03044 0.02480(0.00465) (0.00776) (0.00809)
0.01503 0.02342 0.02559(0.00474) (0.00790) (0.00835)
0.01857 0.03661 0.02201(0.00434) (0.00698) (0.00686)
0.02108 0.04207 0.02498(0.00462) (0.00751) (0.00736)
0.02173 0.05865 0.01828(0.00522) (0.01539) (0.00527)
0.02217 0.04523 0.02593(0.00479) (0.01149) (0.00631)
0.01529 0.05188 0.01496(0.00508) (0.02065) (0.00546)
0.01395 0.03484 0.01420(0.00707) (0.01671) (0.00679)
0.01921 0.05443 0.01584(0.00602) (0.01829) (0.00535)
0.04019 0.08893 0.03954(0.00924) (0.01636) (0.01288)
0.01831 0.04988 0.01510(0.00446) (0.01330) (0.00494)
0.02544 0.07398 0.02011(0.00442) (0.01149) (0.00451)(continued)

Table 4: Continued (Distributions 7 and 8)

EstimatorGIAD

n = 100
RISE

RWISE

0.02962 0.06226 0.03435(0.00909) (0.02063) (0.01080)
MG
0.03053 0.06159 0.03867(0.01007) (0.02013) (0.01288)
LMBC-G
0.02567 0.05274 0.03060(0.00903) (0.02188) (0.01027)
LTBC-C0.02984 0.06927 0.03104(0.01032) (0.02484) (0.01171)
TS-MBC-G
0.02899 0.06260 0.03413(0.00951) (0.02243) (0.01134)
TS-MBC-MG
0.02967 0.06355 0.03543(0.00984) (0.02114) (0.01196)
JLN-MBC-G
0.02798 0.06759 0.02809(0.00898) (0.02098) (0.00965)
JLN-MBC-MG
0.02765 0.06627 0.02792(0.00905) (0.01481) (0.01076)

IAD

n = 200
RISE

RWISE

Distribution #7: Burr
0.02309 0.04983 0.02657(0.00656) (0.01557) (0.00773)
0.02342 0.04778 0.02949(0.00731) (0.01475) (0.00932)
0.01920 0.03949 0.02315(0.00609) (0.01503) (0.00717)
0.02246 0.05283 0.02344(0.00682) (0.01663) (0.00805)
0.02200 0.04951 0.02535(0.00652) (0.01629) (0.00785)
0.02235 0.04894 0.02631(0.00688) (0.01466) (0.00832)
0.02246 0.05801 0.02149(0.00642) (0.01607) (0.00670)
0.02230 0.05736 0.02115(0.00662) (0.01026) (0.00771)

IAD

n = 500
RISE

RWISE

0.01668 0.03685 0.01906(0.00463) (0.01097) (0.00531)
0.01649 0.03360 0.02077(0.00482) (0.00987) (0.00601)
0.01328 0.02719 0.01626(0.00416) (0.00971) (0.00499)
0.01573 0.03752 0.01653(0.00438) (0.01039) (0.00525)
0.01553 0.03695 0.01730(0.00465) (0.01156) (0.00534)
0.01560 0.03518 0.01789(0.00454) (0.00930) (0.00545)
0.01704 0.04786 0.01536(0.00461) (0.01150) (0.00475)
0.01696 0.04745 0.01488(0.00441) (0.00679) (0.00502)

Distribution #8: Generalized Gamma
0.02875 0.03846 0.09656
0.02226 0.03010 0.07596
0.01584 0.02175 0.05447(0.01022) (0.01360) (0.03499) (0.00722) (0.00982) (0.02597) (0.00467) (0.00654) (0.01697)
MG
0.02857 0.03792 0.09891
0.02217 0.02972 0.07775
0.01575 0.02141 0.05541(0.01014) (0.01329) (0.03695) (0.00725) (0.00966) (0.02746) (0.00469) (0.00647) (0.01782)
LMBC-G
0.02314 0.03110 0.07623
0.01755 0.02367 0.05814
0.01242 0.01688 0.04148(0.00960) (0.01307) (0.03207) (0.00674) (0.00923) (0.02373) (0.00443) (0.00619) (0.01567)
LTBC-C0.03100 0.04358 0.10924
0.02398 0.03344 0.08365
0.01752 0.02418 0.05973(0.01038) (0.01573) (0.03970) (0.00726) (0.01121) (0.02827) (0.00526) (0.00843) (0.01993)
TS-MBC-G
0.02925 0.04048 0.10045
0.02174 0.03015 0.07524
0.01486 0.02074 0.05137(0.00996) (0.01426) (0.03649) (0.00694) (0.01006) (0.02629) (0.00449) (0.00658) (0.01695)
TS-MBC-MG
0.02946 0.04065 0.10183
0.02190 0.03027 0.07628
0.01496 0.02078 0.05198(0.00995) (0.01416) (0.03692) (0.00698) (0.01001) (0.02670) (0.00453) (0.00657) (0.01727)
JLN-MBC-G
0.02655 0.03656 0.09626
0.02084 0.02892 0.07698
0.01516 0.02121 0.05743(0.00916) (0.01174) (0.03084) (0.00650) (0.00850) (0.02366) (0.00427) (0.00572) (0.01587)
JLN-MBC-MG
0.02451 0.03376 0.08451
0.01906 0.02652 0.06676
0.01346 0.01903 0.04819(0.00803) (0.01056) (0.02825) (0.00544) (0.00727) (0.02036) (0.00349) (0.00492) (0.01369)(continued)G

36

Table 4: Continued (Distributions 9 and 10)

Estimator

IAD

n = 100
RISE

RWISE

IAD

n = 200
RISE

RWISE

IAD

n = 500
RISE

RWISE

Distribution #9: Log-Normal and Pareto Mixture 1
G
0.03273 0.05493 0.04978
0.02607 0.04448 0.03868
0.01891 0.03297 0.02751(0.01038) (0.01681) (0.01766) (0.00760) (0.01239) (0.01276) (0.00533) (0.00901) (0.00855)
MG
0.03537 0.05600 0.05766
0.02749 0.04423 0.04380
0.01978 0.03232 0.03122(0.01237) (0.01790) (0.02441) (0.00893) (0.01346) (0.01648) (0.00581) (0.00913) (0.01030)
LMBC-G
0.03328 0.06093 0.04671
0.02685 0.04991 0.03695
0.02002 0.03783 0.02727(0.01140) (0.02112) (0.01650) (0.00871) (0.01555) (0.01286) (0.00615) (0.01105) (0.00896)
LTBC-C0.03090 0.05722 0.04487
0.02391 0.04620 0.03344
0.01746 0.03567 0.02362(0.00972) (0.01691) (0.01659) (0.00705) (0.01245) (0.01206) (0.00444) (0.00839) (0.00757)
TS-MBC-G
0.03104 0.05286 0.04583
0.02422 0.04263 0.03410
0.01772 0.03244 0.02368(0.01042) (0.01708) (0.01773) (0.00768) (0.01268) (0.01269) (0.00514) (0.00879) (0.00815)
TS-MBC-MG
0.03383 0.05579 0.05325
0.02566 0.04319 0.03909
0.01837 0.03131 0.02766(0.01372) (0.01991) (0.03012) (0.00944) (0.01442) (0.01771) (0.00593) (0.00944) (0.01049)
JLN-MBC-G
0.03527 0.06194 0.04853
0.02959 0.05328 0.03941
0.02295 0.04296 0.02937(0.00969) (0.01597) (0.01479) (0.00705) (0.01155) (0.01082) (0.00477) (0.00800) (0.00740)
JLN-MBC-MG
0.03411 0.05945 0.04931
0.02798 0.05017 0.03921
0.02178 0.04032 0.02968(0.01075) (0.01732) (0.01826) (0.00801) (0.01301) (0.01341) (0.00524) (0.00851) (0.00864)
Distribution #10: Log-Normal and Pareto Mixture 2
0.03035 0.06028 0.03903
0.02370 0.04917 0.02978
0.01731 0.03698 0.02168(0.01012) (0.01911) (0.01361) (0.00693) (0.01391) (0.01028) (0.00438) (0.00973) (0.00692)
MG
0.03179 0.05814 0.04909
0.02465 0.04576 0.03715
0.01776 0.03323 0.02613(0.00951) (0.01718) (0.02032) (0.00634) (0.01277) (0.01326) (0.00418) (0.00889) (0.00815)
LMBC-G
0.02630 0.05372 0.03569
0.02016 0.04180 0.02725
0.01504 0.03072 0.02023(0.00941) (0.02555) (0.01273) (0.00677) (0.01884) (0.00958) (0.00442) (0.01170) (0.00667)
LTBC-C
0.02809 0.05777 0.03439
0.02109 0.04409 0.02572
0.01512 0.03192 0.01841(0.01008) (0.02282) (0.01344) (0.00704) (0.01654) (0.00955) (0.00466) (0.01093) (0.00654)
TS-MBC-G
0.02691 0.05156 0.03683
0.02074 0.04218 0.02737
0.01574 0.03360 0.01978(0.01020) (0.01898) (0.01428) (0.00668) (0.01328) (0.01039) (0.00407) (0.00896) (0.00679)
TS-MBC-MG
0.03684 0.07109 0.04938
0.02996 0.06054 0.03705
0.02337 0.04991 0.02660(0.01412) (0.02206) (0.03264) (0.00898) (0.01555) (0.01737) (0.00528) (0.01103) (0.00834)
JLN-MBC-G
0.02603 0.05059 0.03504
0.02097 0.04268 0.02719
0.01669 0.03516 0.02042(0.00727) (0.01314) (0.01240) (0.00491) (0.00960) (0.00923) (0.00332) (0.00683) (0.00633)
JLN-MBC-MG
0.02824 0.06236 0.03374
0.02415 0.05516 0.02671
0.02040 0.04744 0.02101(0.00670) (0.01166) (0.01234) (0.00460) (0.00788) (0.00877) (0.00335) (0.00533) (0.00579)
G

Note : Numbers in parentheses are simulation standard deviations.

37

Table 5: Descriptive Statistics of the U.S. Income Data
Descriptive statistics:
Quantiles:
Inequality measures:
Sample size
9,275
5% 12,837
Coe¢ cient of variation
10% 15,300
Gini coe¢ cientMean
39,255
Standard deviation 24,090
25% 21,660
Theil index
50% 33,288
Mean log deviation
Skewness
1.60
Kurtosis
6.68
75% 50,163
Minimum
10,008
90% 70,878
Maximum
199,041
95% 88,074

0.61
0.32
0.16
0.17

Table 6: Descriptive Statistics of the Brazilian Income Data
Descriptive statistics:
Quantiles:
Inequality measures:
Sample size
71,5235%
4,800
Coe¢ cient of variation
Mean
52,18310%
6,056
Gini coe¢ cient
25% 12,166
Theil index
Standard deviation
90,661
Skewness
11
50% 26,142
Mean log deviation
Kurtosis319
75% 57,000
Minimum2
90% 116,130
Maximum
5,011,000
95% 183,000
Figure 1: Shapes of Asymmetric Kernels When b = 0:4(a) x = 0.5(b) x = 1.0(c) x = 2.0(d) x = 4.0

38

1.74
0.59
0.68
0.66

Figure 2: Average Plots of Density Estimates for Selected Distributions (n = 100)(a) Weibull [Distribution 2]:<< near the origin >><< on right tail >>(b) Half-Normal [Distribution 3]:<< near the origin >><< on right tail >>(c) Log-Normal [Distribution 5]:<< near the origin >><< on right tail >>(d) Pareto [Distribution 6]:<< near the origin >><< on right tail >>39

Figure 3: U.S. Income Distribution(a) Overall(b) Near the Origin(c) On Right Tail

Figure 4: Brazilian Income Distribution(a) Overall(b) Near the Origin(c) On Right Tail40