compiles/8f0013ef0f2b9c6f92df168d9a909d7c/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 7, No. 1, 2014, 22-36

ISSN 1307-5543 – www.ejpam.com

Empirical Likelihood Ratio Based Goodness-of-Fit Test for the

Generalized Lambda Distribution

Wei Ning

Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH,

USA

Abstract. In this paper, we propose a goodness-of-fit test based on the empirical likelihood method for

the generalized lambda distribution (GLD) family. Such a nonparametric test approximates the optimal

Neyman-Pearson likelihood ratio test under the unknown alternative distribution scenario. The p-value

of the test is approximated through the simulations due to the dependency of the test statistic on the

data. The test is applied to the roller data set and the pollen data set to illustrate the testing procedure

for the sufficiency of the GLD fittings.

2010 Mathematics Subject Classifications: 62G05, 62G10, 62G20

Key Words and Phrases: Generalized lambda distribution; Empirical likelihood; Nonparametric;

Goodness-of-fit test.

1. Introduction

Modeling the skewed or the heavy tailed data is an important issue in statistical data

fitting. There are extensive distribution families proposed by many researchers to achieve

this goal. The generalized lambda distribution (GLD) family was originally introduced by

Tukey [20], who proposed an one-parameter lambda distribution. Tukey’s lambda distribution

was generalized, for the purpose of generating random variables for Monte Carlo simulation

studies, to the four parameters GLD proposed by Ramberg and Schmeiser [13, 14]. Ramberg

et al. [15] developed a four-parameter system with the tables for fitting a wide variety of

curve shapes. Since the early 1970s, the GLD has been applied to fitting phenomena in many

fields of endeavor with continuous probability density functions (pdf). The GLD family with

four parameters λ1, λ2, λ3, λ4, which is denoted as GLD(λ1,λ2,λ3,λ4), has a probability

density function

f (x) =
λ2

λ3 yλ3−1+λ4(1− y)λ4−1
, at x =Q(y), (1)

Email address: wning@bgsu.edu

http://www.ejpam.com 22 c© 2014 EJPAM All rights reserved.



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 23

where Q(y) is the percentile function defined as

Q(y) = λ1+
yλ3 − (1− y)λ4

λ2

,

where 0 ≤ y ≤ 1, λ1 and λ2 are location and scale parameters respectively. Karian and

Dudewicz [4] gave the first four moments of GLD(λ1,λ2,λ3,λ4) with λ3 >−1/4, λ4 > −1/4

as

α1 = µ= E(X ) = λ1+
A

λ2

,

α2 = σ
2 = E[(X −µ)2] =

B− A2

λ2
2

,

α3 = E(X − E(X ))3/σ3 =
C − 3AB+ 2A3

λ3
3σ

3
,

α4 = E(X − E(X ))4/σ4 =
D− 4AC + 6A2B− 3A4

λ4
2σ

4

(2)

where

A=
1

1+λ3

−
1

1+λ4

,

B =
1

1+ 2λ3

+
1

1+ 2λ4

− 2β(1+λ3, 1+λ4),

C =
1

1+ 3λ3

−
1

1+ 3λ4

− 3β(1+ 2λ3, 1+λ4) + 3β(1+λ3, 1+ 2λ4),

D =
1

1+ 4λ3

+
1

1+ 4λ4

− 4β(1+ 3λ3, 1+λ4) + 6β(1+ 2λ3, 1+ 2λ4)

− 4β(1+λ3, 1+ 3λ4),

and β(·, ·) is a Beta function. The GLD family is known for its high flexibility on approaching

many well-known distributions and ability to fit the data sets with different shapes, especially

with those with heavy tails. There has been further extensive work done in this field. For

example, Karian and Dudewicz [4] provided the tabulated tables for the method of moment

and the percentile method which are used to estimate the parameters of the GLD. King and

MacGillivray [6] and Lakhany and Massuer [7] considered a definite fit to the data set by

maximizing the goodness of fit. Su [16] used a discretized method to fit GLD to the empirical

data. Su [17] derived the estimation procedure by using the maximum likelihood method.

Asquith [1] provided L-moments and TL-moments for the GLD. Fournier et al. [2] proposed

a new estimation method by combining the method of moment and the percentile method.

Ning et al. [8] considered the fitting problem involving the mixture of two GLDs and as a

result made the comparisons to the other mixture distribution families. Ning and Gupta [9]

proposed a GLD change point model to detect the change points for the DNA copy number.

Su et al. [19] proposed a GLD based calibration model to achieve more flexible data fitting

comparing the classic normal calibration model and the skew normal calibration model. For



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 24

the other recent work related to the GLD family and its applications, the readers are referred

to Karian and Dudewicz [5].

As for data fitting, the extent to which how well the proposed distribution family can fit

the data is an important issue. Goodness-of-fit test is the test that is always used to check the

sufficiency of the data fitting by a given distribution. Neyman-Pearson lemma indicates that

the likelihood ratio test (LRT) is the uniformly powerful (UMP) test for the hypotheses:

H0 : f = f0 versus H1 : f = f1 with f0 and f1 both known. However, the alternative distribu-

tion is usually not known in practice. Recently, Vexler and Gurevich [22] and Vexler et al. [23]

constructed goodness-of-fit test based on the empirical likelihood method to approximate the

optimal Neyman-Pearson likelihood ratio test with an unknown alternative density function.

In this paper, we will consider the goodness-of-fit test for the generalized lambda distri-

bution (GLD) family. We will adopt the idea as that of Vexler and Gurevich [22] similarly to

construct a nonparametric goodness-of-fit test to test the null hypothesis of a GLD versus the

alternative hypothesis of some other unknown distribution. This paper is organized as fol-

lows. In Section 2, a brief introduction of the empirical likelihood method will be given. The

empirical likelihood ratio based goodness-of-fit test is proposed and its asymptotic properties

will be derived. In Section 3, an empirical procedure based on the simulations is provided

to approximate the p-value of the test statistic in Section 2 due to the dependency of the

test statistic on the estimated parameters. The proposed goodness-of-fit test is applied to the

roller data set and the pollen data set in Section 4 to illustrate the testing procedure and fitting

results are given. Discussion is provided in Section 5.

2. Statistical Method

2.1. The Empirical Likelihood (EL) Method

Consider the independently and identically distributed p-dimensional observations, say

x1, · · · , xn, from an unknown population distribution F . The main idea of empirical likelihood

methods, proposed and systematically developed by Owen [10] is to place a probability mass

at each observation. Therefore, let pi = P(X = x i) and the empirical likelihood function of F

be defined as

L(F) =

n∏

i=1

pi .

It is clear that L(F) subject to the constraints

pi ≥ 0 and
∑

i

pi = 1

is maximized at pi = 1/n, i.e., the likelihood L(F) attains its maximum n−n under the full

nonparametric model. When a population parameter θ identified by Em(X ,θ ) = 0 is of

interest where m(x ,θ ) is a real-valued function, the empirical log-likelihood maximum when

θ has the true value θ0 is obtained subject to the additional constraint
∑

pim(x i ,θ0) = 0.



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 25

The empirical log-likelihood ratio statistic to test θ = θ0 is given by

R(θ0) =max{
∑

i

log npi : pi ≥ 0,
∑

pi = 1,
∑

pim(x i ,θ0) = 0}.

Owen [10, 11] shows that similarly to the likelihood ratio test statistic in a parametric model

setup, with mild regular conditions θ0, −2 log R(θ0)→ χ
2
r in distribution under the null model

θ = θ0, where r is the dimension of m(x ,θ ). More details of the empirical likelihood and the

related work refer to Owen [12].

2.2. The EL Goodness-of-Fit Test

We will test the following hypothesis:

H0 : f = f0 ∼ GLD(λ1,λ2,λ3,λ4)

H1 : f = f1 � GLD(λ1,λ2,λ3,λ4).

The likelihood ratio test statistic for this hypothesis is defined as

LR=

∏n

i=1 fH1
(x i)∏n

i=1 fH0
(x i)

=

∏n

i=1 fH1
(x i)∏n

i=1 f (x i |λ)

where x1, x2, · · · , xn follows a GLD distribution with the parameter λ= (λ1,λ2,λ3,λ4) under

the null hypothesis. Neyman-Pearson lemma guarantees that such a test is the UMP test with

f0 and f1 both known. If they are both unknown, the maximum likelihood method will be

applied to estimate the parameters λ̂ = (λ̂1, λ̂2, λ̂3, λ̂4) of a GLD distribution under the null

hypothesis. The parameters can be estimated by using the R package GLDEX developed by Su

[18]. We then will apply maximum empirical likelihood method to estimate the numerator.

We rewrite

L f =

n∏

i=1

fH1
(x i) =

n∏

i=1

fH1
(x(i)) =

n∏

i=1

fi ,

where x(1) ≤ x(2) ≤ · · · ≤ x(n) are the order statistics of the observations x1, · · · , xn. We will

apply the empirical likelihood method introduced in Section 2.1 to derive the values of fi to

maximize L f with the constraint
∫

f (s)ds = 1 corresponding to the alternative hypothesis.

We first give the following lemma by Vexler and Gurevich [22] to express this constraint more

explicitly.

Lemma 1. Let X1, · · · , Xn be independent and identically distributed random variables with a

density function f (x). Then

n∑

j=1

∫ X( j+m)

X( j−m)

f (x)d x =2m

∫ X(n)

X(1)

f (x)d x −
m−1∑

k=1

(m− k)

∫ X(n−k+1)

X(n−k)

f (x)d x

−
m−1∑

k=1

(m− k)

∫ X(k+1)

X(k)

f (x)d x t 2m

∫ X(n)

X(1)

f (x)d x −
m(m− 1)

n
,

(3)



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 26

where X( j) = X(1) if j ≤ 1, and X( j) = X(n), if j ≥ n. X(1) < X(2) < · · · < X(n) are order statistics

of X1, · · · , Xn.

See Vexler and Gurevich [22] for the detailed proof.

Since
∫ X(n)

X(1)
f (x)d x ≤

∫∞
−∞

f (x)d x = 1, from Lemma 1 we have

Λm ≤ 1,Λm =
1

2m

n∑

j=1

∫ X( j+m)

X( j−m)

f (x)d x (4)

Therefore, Λm → 1 as m

n
→ ∞. The integration on the right side of the equation (4) can be

approximated as,

n∑

j=1

∫ X( j+m)

X( j−m)

f (x)d x t (X( j+m)− X( j−m)) f (x( j)) = (X( j+m)− X( j−m)) f j .

Thus,

Λm t
1

2m

n∑

j=1

(X( j+m)− X( j−m)) f j ¬
eΛm,

therefore, eΛm ≤ 1. To maximize
∏

f j with this constraint, we apply the Lagrange multiplier

method and have

l( f1, · · · , fn,η) =

n∑

j=1

log f j +η(
1

2m

n∑

j=1

(X( j+m)− X( j−m)) f j − 1), (5)

where η is a lagrange multiplier. By taking the derivative of the equation (5) respect to each

f j and η, we obtain

∂ l

∂ fi

=0⇒
1

f j

+
η

2m
(X( j+m)− X( j−m)) = 0 (6)

∂ l

∂ η
=0⇒

1

2m

n∑

j=1

(X( j+m)− X( j−m)) f j − 1= 0. (7)

from the equation (6) and (7), we have,

∑
f j ·

1

f j

+η
1

2m

∑
f j(X( j+m)− X( j−m)) = 0⇒ η= −n. (8)

Hence, we will obtain the estimate of f j to maximize
∏

f j as

f j =
2m

n(X( j+m)− X( j−m))
, (9)



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 27

where X( j) = X(1), if j ≤ 1, and X( j) = X(n), if j ≥ n. We then construct the likelihood ratio test

statistic for the goodness-of-fit test for the GLD based on the maximum empirical likelihood

method as

GLDmn =

∏n

j=1
2m

n(X( j+m)−X( j−m))

max
λ

∏n

j=1 fH0
(X j|λ)

, (10)

where λ = (λ1,λ2,λ3,λ4) is the parameter vector of a GLD. We notice that the test statistic

GLDmn strongly depends on the integer m. To make the test more efficient, Vexler and Gure-

vich [22] and Vexler et al. [23] reconstructed the test statistic according to the properties

of the empirical likelihood method. We follow their suggestions here to reconstruct the test

statistic in (10) as

GLDn =

min
1≤m<nδ

∏n

j=1
2m

n(X( j+m)−X( j−m))

max
λ

∏n

j=1 fH0
(X j|λ)

, (11)

and 0 < δ < 1. Here, we choose δ = 1/3 for the convenience of computations. Then the

equation (11) will be changed to

GLDn =

min
1≤m<n1/3

∏n

j=1
2m

n(X( j+m)−X( j−m))

∏n

j=1 fH0
(X j |λ̂)

, (12)

2.3. Asymptotic Results

In this section, we will derive some asymptotic properties of the test statistic proposed in

(12). First we denote

hi(x ,λ) =
∂ log fH0

(x;λ)

∂ λi

, i = 1,2,3,4

and λ= (λ1,λ2,λ3,λ4). We assume the following conditions hold:

C1. E(log f (X1))
2 <∞.

C2. Under the null hypothesis, |λ̂−λ|= max
1≤i≤4
|λ̂i −λi | → 0 in probability.

C3. Under alternative hypothesis, λ̂ → λ0 in probability where λ0 is a constant vector with

finite components.

C4. There are open intervals Θ0 ⊆ R
4 and Θ1 ⊆ R

4 containing λ and λ0 respectively.

There also exists a function s(x) such that |h(x ,ξ)| ≤ s(x) for all x ∈ R and ξ ∈ Θ0 ∪Θ1.

Theorem 1. Assume that the conditions C1-C4 hold. Then under H0,

1

n
log(GLDn)→ 0 (13)

in probability as n→∞.



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 28

Theorem 2. Assume that the conditions C1-C4 hold. Then under H1,

1

n
log(GLDn)→ E log

�
fH1
(X1)

fH0
(X1,λ0)

�
(14)

in probability as n→∞. That is, the test is consistent.

Please see both proofs in the Appendix.

3. Approximations to the p-value of GLDn

From (12) in section 2.3, we observe that the values of the test statistic depend on the

estimated parameters based on the data, and the asymptotic null distribution is not available.

Therefore, we will provide an empirical procedure through the simulations to approximate

the p-value asymptotically as follows.

1. Fit the original data x1, x2, . . . , xn with a GLD and obtain the estimated values

λ̂= (λ̂1, λ̂2, λ̂3, λ̂4).

2. Simulate a GLD distributed data y1, y2, · · · , yn with the parameter λ̂.

3. Calculate the test statistic (12) for the original data and denote by GLD1
n. Calculate the

test statistic (12) for the simulated sample y1, · · · , yn and denote as GLD1B
n .

4. Repeat the above simulation procedure M times and obtain M test statistics GLD1B
n , · · · ,

GLDMB
n .

5. The p-value then will be approximated by

p̂ =
1

M

M∑

i=1

I(GLDiB
n ≥ GLD1

n),

where I(·) is an indicator function taking value 1 when GLDiB
n ≥ GLD1

n, and taking value

0 when GLDiB
n < GLD1

n.

4. Application

In this section, we apply the proposed nonparametric goodness-of-fit test to two real data

sets to show that GLD does offer sufficient fittings for both data sets.

4.1. Roller Data

The data set consists of 1150 heights measured at 1 micron intervals along the drum of a

roller for the purpose of the study of surface roughness of the rollers. The data is available at

from Carnegie Mellon University Statistics Department∗. We obtain the estimated parameters

∗http://lib.stat.cmu.edu/jasadata/laslett



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 29

of an assumed GLD distribution of the data as λ̂1 = 3.642, λ̂2 = 2.921 and λ̂3 = −0.145,

λ̂4 = 0.166 with the R package GLDEX by Su [18]. Following the procedure proposed in

Section 3, we generate 1000 samples with GLD(3.642,2.921,−0.145,0.166) with the sample

size n = 1150. The test statistic is calculated from (12) with the approximated p-value as

0.068, which leads to fail to reject the null hypothesis at the significance level α = 0.05, that

is, the data can be fitted by a GLD model sufficiently.

We also apply the resample Kolmogorov-Smirnov test [Su 17] for the goodness-of-fit pur-

pose. There are 960 times out of 1000 times that the p-value does not reject the null hypoth-

esis, which indicates the sufficiency of the GLD fitting. Left graph in Figure 1 shows the fitted

GLD density function with the histogram of the roller data. Three different estimated GLD

density functions with different colors provided by GLDEX package [Su 17]. The dark blue

one with the label RPRS corresponds to the GLD introduced in Section 2 due to Ramberg and

Schmeiser [13], which is estimated by the maximum likelihood method. The light blue one

with the label RMFMKL corresponds to a slight different version of GLD due to Freimer et al.

[3], which is estimated by the maximum likelihood method. The pink one with the label STAR

corresponds to the Freimer et al. [3] version of the GLD, which is estimated by the starship

method [King and MacGillivray 6]). The right graph in Figure 1 is the quantile plot of the

Ramberg and Schmeiser [13] version of the GLD fits. In Table 1, the estimated values of GLDs

are all for Ramberg and Schmeiser [13] version of the GLD. From the graphs and tables, we

can see the GLD fits the data pretty well. In Table 1, we also compare the first four moments

of the real data with those of the fitted GLD distribution. From comparison we can observe

that the moments of the fitted GLD are close to true moments of the data.

Roller

D
en

si
ty

1 2 3 4 5

0.
0

0.
2

0.
4

0.
6

0.
8

RPRS
RMFMKL
STAR

(a) Histogram of 1150 roller with GLD fits.

0.0 0.2 0.4 0.6 0.8 1.0

−
2

0
2

4
6

D
at

a 
va

lu
es

Data
Fitted Distribution

(b) Quantile plot for GLD fits of the data

using maximum likelihood estimation.

Figure 1: Results of fitting GLD to the roller data.



W. Ning / Eur. J. Pure Appl. Math, 7 (2014), 22-36 30

Table 1: Moments comparison of roller data and the fitted GLD.

DATA GLD

Mean 3.53474 3.53544

Variance 0.42212 0.42135

Skewness −0.98659 −0.98786

Kurtosis 4.86310 5.53253

4.2. Pollen Data

The second data is pollen data which also is available from Carnegie Mellon University†.

We fit the variable “nub”, which consists of 481 values from measuring geometric charac-

teristics of certain type of pollen with a GLD model. We obtain the estimated parameters

of an assumed GLD distribution of the data as λ̂1 = −0.425, λ̂2 = 0.240 and λ̂3 = 0.328,

λ̂4 = 0.186 with the R package GLDEX. Following the procedure proposed in Section 3, we

generate 1000 samples with GLD(−0.425,0.240,0.328,0.186) with the sample size n= 481.

The test statistic is calculated from (12) with the approximated p-value as 0.156, which leads

to fail to reject the null hypothesis at the significance level α = 0.05, that is, the data can be

fitted by a GLD model sufficiently. The results are listed in Table 2 provides the comparison of

the moments of the data and the fitted GLD. We observe that the moments of the GLD model

are close to true values of the moments of the data.

The resample Kolmogorov-Smirnov test [Su 18] shows that 959 times among 1000 times

the p-value does not reject the null hypothesis, which indicates the sufficiency of the GLD fits.

The histogram and quantile plots listed below also show the good fit of the GLD on the data.

The left graph in Figure 2 shows the estimated density function of the GLDs by the maximum

likelihood method and the starship method with the histogram of the pollen data. The colors

have the same meaning as in the Figure 1. The right graph shows the quantile plot of the

estimated Ramberg and Schmeiser [13] version of the GLD. We observe that the GLD fits the

data very well.

Table 2: Moments comparison of pollen data and the fitted GLD

DATA GLD

Mean −0.04826 −0.04813

Variance 26.94185 27.02739

Skewness 0.23295 0.25234

Kurtosis 2.59438 2.64018

†http://lib.stat.cmu.edu/datasets/pollen.data



REFERENCES 31

nub

D
en

si
ty

−10 −5 0 5 10 15

0.
00

0.
02

0.
04

0.
06

RPRS
RMFMKL
STAR

(a) Histogram of 481 observations with

GLD fits.

0.0 0.2 0.4 0.6 0.8 1.0

−
10

−
5

0
5

10
15

20

D
at

a 
va

lu
es

Data
Fitted Distribution

(b) Quantile plot for GLD fits of the data

using maximum likelihood estimation.

Figure 2: Results of fitting GLD to the pollen data.

5. Discussion

In this paper, we investigate the problems of the goodness-of-fit for the generalized lambda

distribution (GLD) family due to its high flexibility in the data fitting, especially for the heavy-

tailed data. We propose an empirical likelihood based goodness-of-fit test for this distribution

family. The motivation of the proposed test is based on the UMP likelihood ratio test guar-

anteed by Neyman-Pearson lemma with the null and alternative distributions both known.

However, in practice, the alternative distribution is usually not known. Therefore, the pro-

posed nonparametric goodness of fit test is constructed to approximate the optimal test for

the scenario with an unknown alternative distribution. Asymptotic results have been derived

for the proposed test. Since the explicit form of the asymptotic null distribution is not avail-

able and the test statistic is data dependent, we propose an empirical procedure through the

simulations to approximate p-value of the test statistic for given data sets. The results show

that the generalized lambda distribution offers sufficient fittings for both roller and pollen

data sets, which match the conclusions obtained from the other goodness of fit tests such as

the resample Kolmogorov-Smirnov test.

ACKNOWLEDGEMENTS The author would like to thank Professor Arjun Gupta for his valu-

able discussions and suggestions.

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Group, Princeton University, 1960.

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ratio based goodness-of-fit test for inverse Gaussian distributions. Journal of Statistical

Planning and Inference 141:2128-2140, 2011.



REFERENCES 34

Appendix

Proof. [Theorem 1] We consider the following statistic,

Tn =
1

n
log min

1≤m<n1/3

n∏

j=1

2m

n(X( j+m)− X( j−m))

=− max
1≤m<n1/3

tmn

where tmn =
1

n

∑n

i=1 log( n

2m
(X( j+m) − X( j−m)). It is a part of n−1 log(GLDn), where GLDn is

the test statistic defined in (12). Similar to the work by Vasicek [21], we rewrite

tmn = −
1

n

n∑

i=1

log f (x i) + Vmn+ Umn, (A1)

where

Vmn =−
1

n

n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

f (x(i))(x(i+m)− x(i−m))

�
,

Umn =
1

n

n∑

i=1

log

�
n

2m
(F(X(i+m))− F(X(i−m)))

�
.

where F is the distribution function of X ′s. Denote Fn the empirical distribution function of

X ′s and combine the first two terms in (A-1), we obtain

−
1

n

n∑

i=1

log f (x i) + Vmn =−
1

n

n∑

i=1

log f (x i)−
1

n

n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

f (x(i))(x(i+m)− x(i−m))

�

=−
1

n

n∑

i=1

log f (x(i))−
1

n

n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

f (x(i))(x(i+m)− x(i−m))

�

=−
1

n

n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

(x(i+m)− x(i−m))

�

=−
1

2m

2m∑

j=1

n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

(x(i+m)− x(i−m))

�
(Fn(x(i+m))− Fn(x(i−m))),

where i ≡ j (mod 2m). Let

S j = −
n∑

i=1

log

�
F(x(i+m))− F(x(i−m))

(x(i+m)− x(i−m))

�
(Fn(x(i+m))− Fn(x(i−m))), i ≡ j (mod 2m),

then

tmn =
1

2m

2m∑

j=1

S j + Umn.



REFERENCES 35

With the argument in Theorem 1. by Vasicek [21],

1

2m

2m∑

j=1

S j → H( f ), a.s,

as m/n → 0 uniformly for all 1 ≤ m ≤ n1/3, where H( f ) = E(− log f (x i)) = E(− log f (x1).

Since the statistics Umn is a non-positive variable by the definition and is independent of F ,

Umn→ 0 in probability as m→∞ and n→∞ by Lemma 1 in Vasicek [21]. Therefore,

Tn ≤− tn1/3,n

p
→ E f (log(X1)),

Tn ≥− max
1≤m<n1/3

(2m)−1
2m∑

j=1

S j

p
→ E f (log(X1))

as n→∞. It implies that

Tn→ E f (log( f (X1))), as n→∞. (A2)

Now, we rewrite the test statistic GLDn as

1

n
log(GLDn) =Tn−

1

n

n∑

i=1

log( fH0
(X i |λ))

+
1

n

 
n∑

i=1

log( fH0
(X i |λ))−

n∑

i=1

log( fH0
(X i |λ̂n))

!
,

(A3)

where λ̂n = (λ̂1, λ̂2, λ̂3, λ̂4). Under the null hypothesis, Tn→ E fH0
(log( fH0

(X1))) in probability

since (A2). With the condition (C1), we have

1

n

n∑

i=1

log( fH0
(X i|λ))

p
→ E fH0

(log( fH0
(X1|λ))). (A4)

With the condition (C2) holds and applying one-term Taylor Series expansion to the third part

in the equation (A3), we obtain

1

n




n∑

i=1

log( fH0
(X i|λ))−

n∑

i=1

log( fH0
(X i |λ̂n))


∼=

1

n

n∑

i=1

4∑

j=1

hi(X i; λ̂n
)(θ j − λ̂n j),

where hi(·) is defined in Section 2.3. Since (C4), we obtain

1

n
{

n∑

i=1

log( fH0
(X i |λ))−

n∑

i=1

log( fH0
(X i |λ̂n))} ∼=

1

n

n∑

i=1

4∑

j=1

hi(X i; λ̂n
)(λ j − λ̂n j)

≥
1

n

n∑

i=1

4∑

j=1

|hi(X i;ξi)|(λ j − λ̂n j)
p
→ 0

(A5)



REFERENCES 36

where |ξi −λ| ≤ |λ− λ̂n|. Thus under the null hypothesis, the equations (A3), (A4) and (A5)

provide
1

n
log(GLDn)

p
→ 0, as n→∞. (A6)

This completes the proof of Theorem 1.

Proof. [Theorem 2] Under H1, we have

1

n
log(GLDn) =Tn−

1

n

n∑

i=1

log( fH1
(X i)) +

1

n

n∑

i=1

log

�
fH1
(X i)

fH0
(X i |λ0)

�

+
1

n

n∑

i=1

log

 
fH0
(X i |λ0)

fH0
(X i |λ̂n)

! (A7)

Since (A2), similarly under H1

Tn→ E fH1
(log( fH1

(X1))), as n→∞, (A8)

and the condition (C1) leads to

1

n

n∑

i=1

log( fH1
(X i|λ))

p
→ E fH1

(log( fH1
(X1|λ))). (A9)

With the condition (C3),

1

n

n∑

i=1

log

 
fH0
(X i|λ0)

fH0
(X i |λ̂n)

!
P
−→ 0 (A10)

as n→ 0. With the equations (A8),(A9) and (A10), we obtain,

1

n
log(GLDn)

P
−→

1

n

n∑

i=1

log

�
fH1
(X i)

fH0
(X i|λ0)

�
> 0 as n→∞. (A11)