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In this paper, a goodness-of-fit test for normality based on the comparison of the theoretical and empirical distributions is proposed. Critical values are obtained via Monte Carlo for several sample sizes and different significance levels.We study and compare the power of forty selected normality tests for a wide collection of alternative distributions. The new proposal is compared to some traditional test statistics, such as Kolmogorov-Smirnov, Kuiper, Cram´er-von Mises, Anderson-Darling, Pearson Chi-square, Shapiro-Wilk, Shapiro-Francia, Jarque-Bera, SJ, Robust Jarque-Bera, and also to entropy-based test statistics. From the simulation study results it is concluded that the best performance against asymmetric alternatives with support on the whole real line and alternative distributions with support on the positive real line is achieved by the new test. Other findings derived from the simulation study are that SJ and Robust Jarque-Bera tests are the most powerful ones for symmetric alternatives with support on the whole real line, whereas entropy-based tests are preferable for alternatives with support on the unit interval.
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Statistics & Operations Research Transactions
SORT 40 (1) January-June 2016, 55-88
Statistics &
Operations Research
Transactions
© Institut d’Estad´ıstica de Catalunya
sort@idescat.cat
ISSN: 1696-2281
eISSN: 2013-8830
www.idescat.cat/sort/
A test for normality based on the empirical
distribution function
Hamzeh Torabi1, Narges H. Montazeri1and Aurea Gran´e2
Abstract
In this paper, a goodness-of-fit test for normality based on the comparison of the theoretical and
empirical distributions is proposed. Critical values are obtained via Monte Carlo for several sample
sizes and different significance levels. We study and compare the power of forty selected normality
tests for a wide collection of alternative distributions. The new proposal is compared to some tradi-
tional test statistics, such as Kolmogorov-Smirnov, Kuiper, Cram ´
er-von Mises, Anderson-Darling,
Pearson Chi-square, Shapiro-Wilk, Shapiro-Francia, Jarque-Bera, SJ, Robust Jarque-Bera, and
also to entropy-based test statistics. From the simulation study results it is concluded that the best
performance against asymmetric alternatives with support on the whole real line and alternative
distributions with support on the positive real line is achieved by the new test. Other findings de-
rived from the simulation study are that SJ and Robust Jarque-Bera tests are the most powerful
ones for symmetric alternatives with support on the whole real line, whereas entropy-based tests
are preferable for alternatives with support on the unit interval.
MSC: 62F03, 62F10.
Keywords: Empirical distribution function, entropy estimator, goodness-of-fit tests, Monte Carlo
simulation, Robust Jarque-Bera test, Shapiro-Francia test, SJ test; test for normality.
1. Introduction
Let X1,...,Xnbe a nindependent an identically distributed (iid) random variables with
continuous cumulative distribution function (cdf) F(.)and probability density function
(pdf) f(.). All along the paper, we will denote the order statistic by (X(1),...,X(n)).
Based on the observed sample x1,...,xn, we are interested in the following goodness-
of-fit test for a location-scale family:
1Statistics Department, Yazd University, 89175-741, Yazd, Iran, htorabi@yazd.ac.ir, nmontazeri@stu.yazd.ac.ir
2Statistics Department, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe, Spain,
aurea.grane@uc3m.es
Received: April 2015
Accepted: February 2016
56 A test for normality based on the empirical distribution function
H0:FF
H1:F/F(1)
where F=F0(.;θ
θ
θ) = F0xµ
σ|θ
θ
θ= (µ,σ)Θ,Θ=R×(0,)and µand σare
unspecified. The family Fis called location-scale family, where F0(.)is the standard
case for F0(.;θ
θ
θ)for θ
θ
θ= (0,1). Suppose that f0(x;θ
θ
θ) = 1
σf0xµ
σis the corresponding
pdf of F0(x;θ
θ
θ).
The goodness-of-fit test problem for location-scale family described in (1) has been
discussed by many authors. For instance, Zhao and Xu (2014) considered a random
distance between the sample order statistic and the quasi sample order statistic derived
from the null distribution as a measure of discrepancy. On the other hand, Alizadeh
and Arghami (2012) used a test based on the minimum Kullback-Leibler distance. The
Kullback-Leibler divergence measure is a special case of a φ-divergence measure (2)
for φ(x) = xlog(x)x+1 (see p. 5 of Pardo, 2006 for details). Also φ-divergence is a
special case of the φ-disparity measure. The φ-disparity measure between two pdf’s f0
and fis defined by
Dφ(f0,f) = Zφf0(x;θ
θ
θ)
f(x)f(x)dx,(2)
where φ:(0,)[0,)is assumed to be continuous, decreasing on (0,1)and increas-
ing on (1,), with φ(1) = 0 (see p. 29 of Pardo, 2006 for details). In φ-divergence, φis
a convex function.
Inspired by this idea, in this paper we propose a goodness-of-fit statistic to test (1) by
considering a new proximity measure between two continuous cdf’s. The organization
of the paper is as follows. In Section 2 we define the new measure Hnand study its prop-
erties as a goodness-of-fit statistic. In Section 3 we propose a normality test based on
Hnand find its critical values for several sample sizes and different significance levels.
In Section 4 we review forty normality tests, including the most traditional ones such as
Kolmogorov-Smirnov, Cram´er-von Mises, Anderson-Darling, Shapiro-Wilk, Shapiro-
Francia, Pearson Chi-square, among others, and in Section 5 we compare their perfor-
mances to that of our proposal through a wide set of alternative distributions. We also
provide an application example where the Kolmogorov-Smirnov test fails to detect the
non normality of the sample.
2. A new discrepancy measure
In this section we define a discrepancy measure between two continuous cdf’s and study
its properties as a goodness-of-fit statistic.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran ´
e57
Definition 2.1 Let X and Y be two absolutely continuous random variables with cdf’s
F0and F , respectively. We define
D(F0,F) =
Z
h1+F0(x;θ
θ
θ)
1+F(x)dF (x) = EFh1+F0(X;θ
θ
θ)
1+F(X),(3)
where EF[.]is the expectation under F and h :(0,)R+is assumed to be continuous,
decreasing on (0,1)and increasing on (1,)with an absolute minimum at x =1such
that h(1) = 0.
Lemma 2.2 D(F0,F)0and equality holds if and only if F0=F, almost everywhere.
Proof. Using the non-negativity of function h, we have D(F0,F)0. It is clear that F0=
Fimplies D(F0,F) = 0. Conversely, if D(F0,F) = 0, since hhas an absolute minimum
at x=1, then F0=F.
Let us return to the goodness-of-fit test problem for a location-scale family described
in (1). Firstly, we estimate µand σby their maximum likelihood estimators (MLEs), i.e.,
ˆµand ˆσ, respectively, and we take zi= (xiˆµ)/ˆσ,i=1,...,n. Note that in this family,
F0(xi; ˆµ,ˆσ) = F0(zi). Secondly, consider the empirical distribution function (EDF) based
on data xi, that is
Fn(t) = 1
n
n
X
j=1
I[xjt],
where IAdenotes the indicator of an event A. Then, our proposal is based on the ratio
of the standard cdf under H0and the EDF based on the xi’s. Using (3) with F=Fn,
D(F0,Fn)can be written as
Hn:=D(F0,Fn) =
Z
h1+F0(x; ˆµ,ˆσ)
1+Fn(x)d Fn(x)
=1
n
n
X
i=1
h1+F0(x(i); ˆµ,ˆσ)
1+Fn(x(i))
=1
n
n
X
i=1
h1+F0(z(i))
1+i/n
Under H0, we expect that F0(t; ˆµ,ˆσ)Fn(t), for every tRand 1 +F0(t; ˆµ,ˆσ)
1+Fn(t). Note that, since h(1) = 0, we expect that h(1+F0(t))/(1+Fn(t))0 and
58 A test for normality based on the empirical distribution function
thus Hnwill take values close to zero when H0is true. Therefore, it seems justifiable
that H0must be rejected for large values of Hn. Some standard choices for hare: h(x) =
(x1)2/(x+1)2,xlog(x)x+1,(x1)log(x),|x1|or (x1)2(for more examples,
see p. 6 of Pardo, 2006 for details).
Proposition 2.3 The support of Hnis [0,max(h(1/2),h(2))].
Proof. Since F0(.)and Fnare cdf’s and take values in [0,1], we have that
1/21+F0(y)
1+Fn(y)2,yR.
Thus
0h1+F0(y)
1+Fn(y)max(h(1/2),h(2))
Finally, since Hnis the mean of h(.)over the transformed data, the result is obtained.
Proposition 2.4 The test statistic based on Hnis invariant under location-scale trans-
formations.
Proof. The location-scale family is invariant under the location-scale transformations of
the form gc,r(X1,...,Xn) = (rX1+c,...,rXn+c),cR,r>0, which induces similar
transformations on Θ:gc,r(θ
θ
θ) = (rµ+c,rσ)(See Shao, 2003). The estimator T0(X1,...,Xn)
for µis location-scale invariant if
T0(rX1+c,...,rXn+c) = rT0(X1,...,Xn) + c,r>0,cR,
and the estimator T1(X1,...,Xn)for σis location-scale invariant if
T1(rX1+c,...,rXn+c) = rT1(X1,...,Xn),r>0,cR.
We know that MLE of µand σare location-scale invariant for µand σ, respectively.
Therefore under H0, the distribution of Zi= (Xiˆµ)/ˆσdoes not depend on µand σ.
If Gnis the EDF based on data zi, then
Gn(zi) = 1
n
n
X
j=1
I[zjzi]=1
n
n
X
j=1
I[xjxi]=Fn(xi),
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e59
therefore
Hn=1
n
n
X
i=1
h1+F0(x(i); ˆµ,ˆσ)
1+Fn(x(i))=1
n
n
X
i=1
h1+F0(z(i))
1+Gn(z(i)).
Since the statistic Hnis a function of zi,i=1,...,n, is location-scale invariant. As a
consequence, the null distribution of Hndoes not depend on the parameters µand σ.
Proposition 2.5 Let F1be an arbitrary continuous cdf in H1. Then under the assumption
that the observed sample have cdf F1, the test based on Hnis consistent.
Proof. Based on Glivenko-Cantelli theorem, for nlarge enough, we have that Fn(x)
F1(x), for all xR. Also ˆµand ˆσare MLEs of µand σ, respectively, and hence are
consistent. Therefore
Hn=1
n
n
X
i=1
h1+F0(x(i); ˆµ,ˆσ)
1+Fn(x(i))=1
n
n
X
i=1
h1+F0(xi; ˆµ,ˆσ)
1+Fn(xi)
1
n
n
X
i=1
h1+F0(xi; ˆµ,ˆσ)
1+F1(xi)1
n
n
X
i=1
h1+F0(xi,µ,σ)
1+F1(xi)
EF1h1+F0(X,µ,σ)
1+F1(X)=:D(F0,F1),as n,
where EF1[.]is the expectation under F1, and µand σ2are, respectively, the expectation
and variance of F1. Note that the convergence holds by the law of large numbers and
D(F0,F1)is a divergence between F0and F1. So the test based on Hnis consistent.
3. A normality test based on Hn
Many statistical procedures are based on the assumption that the observed data are nor-
mally distributed. Consequently, a variety of tests have been developed to check the
validity of this assumption. In this section, we propose a new normality test based on
Hn.
Consider again the goodness-of-fit testing problem described in (1), where now
f0(x;µ,σ) = 1/2πσ2e(xµ)2/2σ2,xR, in which µRand σ>0 are both unknown,
and F0(.;µ,σ)is the corresponding cdf, where F0(.)is the standard case for F0(.;0,1).
First we estimate µand σby their maximum likelihood estimators (MLEs), i.e., ˆµ=
¯x=1/nPn
i=1xiand ˆσ2=s2=1/(n1)Pn
i=1(xi¯x)2, respectively. Let zi= (xi¯x)/s,
i=1,...,n. Then, the test statistic for normality is:
60 A test for normality based on the empirical distribution function
Hn=1
n
n
X
i=1
h1+F0(x(i),¯x,s)
1+Fn(x(i))=1
n
n
X
i=1
h1+F0(z(i))
1+i/n,(4)
where
h(x) = x1
x+12
.(5)
Note that h:(0,)R+is decreasing on (0,1)and increasing on (1,)with an ab-
solute minimum at x=1 such that h(1) = 0 (see Figure 1). We selected this function
h, because based on simulation study, it is more powerful than other functions h. For
example, we considered h2(x):=xlog(x)x+1 for comparison with h1(x):=x1
x+12
(see Tables 6 and 7).
Corollary 3.1 The support of Hnis [0,0.11].
Proof. From Proposition 2.3 and Figure 1, max(h(1/2),h(2)) = 0.11.
Table 1 contains the upper critical values of Hn, which have obtained by Monte Carlo
from 100000 simulated samples for different sample sizes nand significance levels
α=0.01,0.05,0.1.
0.5 1.0 1.5 2.0 2.5
0.0 0.1 0.2 0.3 0.4
x
h(x)=(x1)2
(x+1)2
h(x)=xlog(x)x+1
Figure 1: Plot of function h.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e61
Table 1: Critical values of Hnfor α=0.01,0.05,0.1.
n5 6 7 8 9 10 15 20 25 30 40 50
α
0.01 .0039 .0035 .0030 .0026 .0023 .0021 .0014 .0011 .0008 .0007 .0005 .0004
0.05 .0030 .0026 .0022 .0019 .0017 .0016 .0010 .0007 .0006 .0005 .0004 .0003
0.10 .0026 .0022 .0019 .0016 .0015 .0013 .0009 .0006 .0005 .0004 .0003 .0002
Remember that, Hnis expected to take values close to zero when H0is true. Hence,
H0will be rejected for large values of Hn. Also Hnis invariant under location-scale
transformations and consistent under the assumption H1, respectively, from Propositions
2.4 and 2.5.
4. Normality tests under evaluation
Comparison of the normality tests has received attention in the literature The goodness-
of-fit tests have been discussed by many authors including Shapiro et al. (1968), Poitras
(2006), Yazici and Yolacan (2007), Krauczi (2009), Romao et al. (2010), Yap and Sim
(2010) and Alizadeh and Arghami (2011).
In this section we consider a large number (forty) of recent and classical statistics that
have been used to test normality and in Section 5 we compare their performances with
that of Hn. In the following we prefer to keep the original notation for each statistic. Con-
cerning the notation, let x1,x2,...,xnbe a random sample of size nand x(1),x(2),...,x(n)
the corresponding order statistic. Also consider the sample mean, variance, skewness
and kurtosis, defined by
¯x=1
n
n
X
i=1
xi,s2=1
n
n
X
i=1
(xi¯x)2,pb1=m3
(m2)3/2,b2=m4
(m2)2,
respectively, where the j-th central moment mjis given by mj=1
nPn
i=1(xi¯x)jand
finally consider z(i)= (x(i)¯x)/s, for i=1,...,n.
1. Vasicek’s entropy estimator (Vasicek, 1976):
KLmn =exp{HVmn}
s
where
HVmn =1
n
n
X
i=1
lnnn
2mX(i+m)X(im)o,(6)
62 A test for normality based on the empirical distribution function
m<n/2 is a positive integer and X(i)=X(1)if i<1 and X(i)=X(n)if i>n.H0is
rejected for small values of KL. Vasicek (1976) showed that the maximum power
for KL was typically attained by choosing m=2 for n=10, m=3 for n=20 and
m=4 for n=50. The lower-tail 5%-significance values of KL for n=10,20 and
50 are 2.15, 2.77 and 3.34, respectively.
2. Ebrahimi’s entropy estimator (Ebrahimi, Pflughoeft and Soofi, 1994):
TEmn =exp{HEmn}
s,
where
HEmn =1
n
n
X
i=1
lnn
cimX(i+m)X(im),(7)
and ci= (1+i1
m)I[1,m](i) + 2I[m+1,nm](i) + (1+ni
m)I[nm+1,n](i). Ebrahimi et al.
(1994) proved the linear relationship between their estimator and (6). Thus for
fixed values of nand m, the tests based on (6) and (7) have the same power.
3. Nonparametric distribution function of Vasicek’s estimator:
TVmn =log q2πˆσ2
v+0.5HVmn,
where HVmn was defined in (6), ˆσ2
v=Vargv(X), and
gv(x) =
0x<ξ1or x>ξn+1,
2m
n(x(i+m)x(im))ξi<xξi+1i=1,. . ., n,
where ξi=x(im)+···+x(i+m1)/2m.H0is rejected for large values of TVmn .
(See Park, 2003).
4. Nonparametric distribution function of Ebrahimi estimator:
TEmn =logq2πˆσ2
e+0.5HEmn,
where HEmn was defined in (7), ˆσ2
e=Varge(X)and
ge(x) = (0x<η1or x>ηn+1
1
n(ηi+1ηi)ηi<xηi+1i=1,...,n,
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e63
with
ηi=
ξm+11
m+k1Pm
k=i(x(m+k)x(1))1im,
1
2mx(im)+···+x(i+m1)m+1inm+1,
ξnm+1+1
n+mk+1Pi
k=nm+2(x(n)x(km1))nm+2in+1,
and ξi=x(im)+···+x(i+m1)/2m.H0is rejected for large values of TEmn. (See
Park, 2003).
5. Nonparametric distribution function of Alizadeh and Arghami estimator (Alizadeh Noughabi
and Arghami, 2010, 2013):
TAmn =logq2πˆσ2
a+0.5HAmn,
where
HAmn =1
n
n
X
i=1
lnn
aimX(i+m)X(im),
with ai=I[1,m](i) + 2I[m+1,nm](i) + I[nm+1,n](i), ˆσ2
a=Varga(X)and
ga(x) = (0x<η1or x>ηn+1,
1
n(ηi+1ηi)ηi<xηi+1i=1,...,n,
with
ηi=
ξm+11
mPm
k=i(x(m+k)x(1))1im,
1
2mx(im)+···+x(i+m1)m+1inm+1,
ξnm+1+1
mPi
k=nm+2(x(n)x(km1))nm+2in+1,
and ξi=x(im)+···+x(i+m1)/2m. Also m= [n+1].H0is rejected for large
values of TAmn. The upper-tail 5%-significance values of TA for n=10,20 and 50
are 0.4422, 0.2805 and 0.1805, respectively.
6. Dimitriev and Tarasenko’s entropy estimator (Dimitriev and Tarasenko, 1973):
TDmn =exp{HDmn}
s
64 A test for normality based on the empirical distribution function
where
HDmn =
Z
ln(ˆ
f(x)) ˆ
f(x)dx,
where ˆ
f(x)is the kernel density estimation of f(x)given by
ˆ
f(Xi) = 1
nh
n
X
j=1
kXiXj
h,(8)
where kis a kernel function satisfying R
k(x)dx =1 and his a bandwidth. The
kernel function kbeing the standard normal density function and the bandwidth
h=1.06 ˆσn1/5.H0is rejected for small values of TDmn.
7. Corea’s entropy estimator (Corea, 1995):
TCmn =exp{HCmn}
s,
where
HCmn =1
n
n
X
i=1
ln(Pi+m
j=imX(j)˜
X(i)(ji)
nPi+m
j=imX(j)˜
X(i)2)
and ˜
X(i)=Pi+m
j=imX(j)/(2m+1).H0is rejected for small values of TCmn.
8. Van Es’s entropy estimator (Van Es, 1992):
TEsmn =exp{HEsmn}
s,
where
HEsmn =1
nm
nm
X
i=1lnn+1
m(X(i+m)X(i))+
n
X
k=m
1
k+ln(m)ln(n+1).
H0is rejected for small values of TEsmn.
9. Zamanzade and Arghami’s entropy estimator (Zamanzade and Arghami, 2012):
TZ1mn =exp{HZ1mn}
s,
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e65
where HZ1mn =1
nPn
i=1ln(bi), with
bi=X(i+m)X(im)
Pk2(i)1
j=k1(i)(ˆ
f(X(j+1)) + ˆ
f(X(j)))(X(j+1)X(j))/2
(9)
where ˆ
fis defined as in (8) with the kernel function kbeing the standard normal
density function and the bandwidth h=1.06 ˆσn1/5.H0is rejected for small values
of TZ1. For n=10,20 and 50, the lower-tail 5%-significance critical values are
3.403, 3.648 and 3.867.
10. Zamanzade and Arghami’s entropy estimator (Zamanzade and Arghami, 2012):
TZ2mn =exp{HZ2mn}
s,
where HZ2mn =Pn
i=1wiln(bi), being coefficients bi’s were defined in (9) and
wi=
(m+i1)/Pn
i=1wi1im,
2m/Pn
i=1wim+1inm,
(ni+m)/Pn
i=1winm+1in,
i=1,...,n,
are weights proportional to the number of points used in computation of bis. H0
is rejected for small values of TZ2. For n=10,20 and 50, the lower-tail 5%-
significance critical values are 3.321, 3.520 and 3.721.
11. Zhang and Wu’s statistics (Zhang and Wu, 2005):
ZK=max
1in(i0.5)ln i0.5
nF0(Z(i))+ (ni+0.5)ln ni+0.5
n(1F0(Z(i))),
ZC=
n
X
i=1log (1/F0(Z(i))1)
(n0.5)/(i0.75)12
,
and
ZA=
n
X
i=1logF0(Z(i))
ni+0.5+log(1F0(Z(i))
i0.5,
The null hypothesis H0is rejected for large values of the three test statistics.
66 A test for normality based on the empirical distribution function
12. Classical test statistics for normality based skewness and kurtosis from D’Agostino
and Pearson (D’Agostino and Pearson, 1973):
pb1=m3
(m2)3/2,b2=m4
(m2)2,
The null hypothesis H0is rejected for both small and large values of the two test
statistics.
13. Transformed skewness and kurtosis statistic from D’Agostino et al. (1990):
K2=hZ(pb1)i2+ [Z(b2)]2,
where
Z(pb1) = log(Y/c+p(Y/c)2+1)
plog(w),
Z(b2) = "12
9A3
s12/A
1+yp2/(A4)#r9A
2,
where
c1=6+8/c2(2/c2+q1+4/c2
2),
c2= (6(n25n+2)/(n+7)(n+9))p6(n+3)(n+5)/n(n2)(n3),
c3= (b23(n1)/(n+1))/q24n(n2)(n3)/(n+1)2(n+3)(n+5).
and
Y=pb1s(n+1)(n+3)
6(n2),w2=p2β211,
β2=3(n2+27n70)(n+1)(n+3)
(n2)(n+5)(n+7)(n+9);c=s2
(w21).
Transformed skewness Z(b1)and transformed kurtosis Z(b2)is obtained by
D’Agostino (1970) and Anscombe and Glynn (1983), respectively. The null hy-
pothesis H0is rejected for large values of K2.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e67
14. Transformed skewness and kurtosis statistic by Doornik and Hansen (1994):
DH =hZ(pb1)i2+z2
2,
where
z2="ξ
2a1/3
1+1
9a#9a,
and
ξ= (b21b1)2k,
k=(n+5)(n+7)(n3+37n2+11n313)
12(n3)(n+1)(n2+15n4),
a=(n+5)(n+7)(n2)(n2+27n70) + b1(n7)(n2+2n5)
6(n3)(n+1)(n2+15n4),
Transformed kurtosis z2is obtained by Shenton and Bowman (1977). The null
hypothesis H0is rejected for large values of DH.
15. Bonett and Seier’s statistic (Bonett and Seier, 2002):
Zw=n+2(ˆw3)
3.54 ,
where ˆw=13.29ln m2log n1Pn
i=1|xi¯x|.H0is rejected for both small
and large values of Zw.
16. D’Agostino’s statistic (D’Agostino, 1971):
D=Pn
i=1(i(n+1)/2)X(i)
n2qPn
i=1x(i)¯
X2,
H0is rejected for both small and large values of D.
17. Chen and Shapiro’s statistic (Chen and Shapiro, 1995):
QH =1
(n1)s
n1
X
i=1
X(i+1)X(i)
M(i+1)M(i)
,
68 A test for normality based on the empirical distribution function
where Mi=Φ1((i0.375)/(n+0.25)), where Φis the cdf of a standard normal
random variable. H0is rejected for small values of QH.
18. Filliben’s statistic (Filliben, 1975):
r=Pn
i=1x(i)M(i)
qPn
i=1M2
(i)p(n1)s2,
where M(i)=Φ1(m(i))and m(1)=10.51/n,m(n)=0.51/nand m(i)= (i0.3175)/(n+
0.365)for i=2,...,n1. H0is rejected for small values of r.
19. del Barrio et al.’s statistic (del Barrio et al., 1999):
Rn=1Pn
k=1X(k)Rk/n
(k1)/nF1
0(t)dt2
m2
,
where m2is the sample standardized second moment. H0is rejected for large val-
ues of Rn.
20. Epps and Pulley statistic (Epps and Pulley, 1983):
TEP =1
3+1
n2
n
X
k=1
n
X
j=1
exp(XjXk)2
2m22
n
n
X
j=1
exp(Xj¯
X)2
4m2,
where m2is the sample standardized second moment. H0is rejected for large val-
ues of TEP.
21. Martinez and Iglewicz’s statistic (Martinez and Iglewicz, 1981):
In=Pn
i=1(XiM)2
(n1)S2
b
,
where Mis is the sample median and
S2
b=nP|˜
Zi|<1(XiM)2(1˜
Z2
i)4
P|˜
Zi|<1(1˜
Z2
i)(15˜
Z2
i)2,
with ˜
Zi= (XiM)/(9A)for |˜
Zi|<1 and ˜
Zi=0 otherwise, and Ais the median of
|XiM|.H0is rejected for large values of In.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
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22. deWet and Venter statistic (de Wet and Venter, 1972):
En=
n
X
i=1X(i)¯
XsΦ1i
n+12s2.
H0is rejected for large values of En.
23. Optimal test (Cs ¨orgo and R´ev´esz, 1971):
Mn=
n
X
i=1X(i)¯
XsΦ1i
n+12
φΦ1i
n+1Φ1i
n+1λ1
.
H0is rejected for large values of Mn.
24. Pettitt statistic (Pettitt, 1977):
Qn=
n
X
i=1ΦX(i)¯
X
si
n+12φΦ1i
n+12
.
H0is rejected for large values of Qn.
25. Three test statistics from LaRiccia (1986):
T1n=C2
1n/(s2B1n),T2n =C2
2n/(s2B2n),T3n =T1n +T2n,
where
C1n=1
n
n
X
i=1W1i
n+1A1nX(i),
C2n=1
n
n
X
i=1W2i
n+1A2nΦ1i
n+1X(i),
Also W1(u) = [Φ1(u)]21 and W2(u) = [Φ1(u)]33Φ1(u). The constants A1n,
A2n,B1nand B2nare given in Table 1 from LaRiccia (1986). For all three statistics
H0is rejected for large value.
26. Kolmogorov-Smirnov’s (Lilliefors) statistic (Kolmogorov, 1933):
KS =maxmax
1jnj
nF0(Z(j)),max
1jnF0(Z(j))j1
n.
70 A test for normality based on the empirical distribution function
Lilliefors (1967) computed estimated critical points for the Kolmogorov-Smirnov’s
test statistic for testing normality when mean and variance estimated.
27. Kuiper’s statistic (Kuiper, 1962):
V=max
1jnj
nF0(Z(j))+max
1jnF0(Z(j))j1
n.
Louter and Kort (1970) computed estimated critical points for the Kuiper test
statistic for testing normality when mean and variance estimated.
28. Cram ´er-von Mises’ statistic (Cram´er, 1928 and von Mises, 1931):
W2=1
12n+
n
X
j=1F0(Z(j))2j1
2n2
.
29. Watson’s statistic (Watson, 1961):
U2=W2n
1
n
n
X
j=1
F0(Z(j))1
2
2
.
30. Anderson-Darling’s statistic (Anderson, 1954):
A2=n1
n
n
X
i=1
(2i1)log(F0(Z(i))) + log 1F0(Z(ni+1)).
These classical tests are based on the empirical distribution function and H0is
rejected for large values of KS, V, W2, U2and A2.
31. Pearson’s chi-square statistic (D’Agostino and Stephens, 1986):
P=X
i
(CiEi)2/Ei,
where Ciis the number of counted and Eiis the number of expected observations
(under H0) in class i. The classes are build is such a way that they are equiprobable
under the null hypothesis of normality. The number of classes used for the test is
2n2/5where .is ceiling function.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
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32. Shapiro-Wilk’s statistic (Shapiro and Wilk, 1965):
SW =P[n/2]
i=1a(ni+1)X(ni+1)X(i)2
Pn
i=1X(i)¯
X2,
where coefficients ai’s are given by
(a1,...,an) = mTV1
(mTV1V1m)1/2,(10)
and mT= (m1,...,mn)and Vare, respectively, the vector of expected values and
the covariance matrix of the order statistic of niid random variables sampled from
the standard normal distribution. H0is rejected for small values of SW.
33. Shapiro-Francia’s statistic (Shapiro and Francia, 1972) is a modification of SW. It
is defined as
SF =Pn
i=1biX(i)2
Pn
i=1(X(i)¯
X)2,
where
(b1,...,bn) = mT
(mTm)1/2
and mis defined as in (10). H0is rejected for small values of SF.
34. SJ statistic discussed in Gel, Miao and Gastwirth (2007). It is based on the ratio
of the classical standard deviation ˆσand the robust standard deviation Jn(average
absolute deviation from the median (MAAD)) of the sample data
SJ =s
Jn
,(11)
where Jn=pπ
2
1
nPn
i=1|XiM|and Mis the sample median. H0is rejected for
large values of SJ.
35. Jarque-Bera’s statistic (Jarque and Bera, 1980, 1987):
JB =n
6b1+n
24 (b23)2,
72 A test for normality based on the empirical distribution function
where b1and b2are the sample skewness and sample kurtosis, respectively. H0
is rejected for large values of JB.
36. Robust Jarque-Bera’s statistic (Gel and Gastwirth, 2008):
RJB =n
C1m3
J3
n2
+n
C2m4
J4
n32
,
where Jnis defined as in (11), C1and C2are positive constants. For a 5%-significance
level, C1=6 and C2=64 according to Monte Carlo simulations. H0is rejected for
large values of RJB.
5. Simulation study
In this section we study the power of the normality test based on Hnand compare it
with a large number of recent and classical normality tests. To facilitate comparisons of
the power of the present test with the powers of the mentioned tests, we select two sets
of alternative distributions:
Set 1. Alternatives listed in in Esteban et al. (2001).
Set 2. Alternatives listed in Gan and Koehler (1990) and Krauczi (2009).
Set 1 of alternative distributions
Following Esteban et al. (2001) we consider the following alternative distributions, that
can be classified in four groups:
Group I: Symmetric distributions with support on (,):
Standard Normal (N);
Student’s t(t) with 1 and 3 degrees of freedoms;
Double Exponential (DE) with parameters µ=0 (location) and σ=1 (scale);
Logistic (L) with parameters µ=0 (location) and σ=1 (scale);
Group II: Asymmetric distributions with support on (,):
Gumbel (Gu) with parameters α=0 (location) and β=1 (scale);
Skew Normal (SN) with with parameters µ=0 (location), σ=1 (scale) and
α=2 (shape);
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
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Group III: Distributions with support on (0,):
Exponential (Exp) with mean 1;
Gamma (G) with parameters β=1 (scale) and α=.5,2 (shape);
Lognormal (LN) with parameters µ=0 and σ=.5,1,2;
Weibull (W) with parameters β=1 (scale) and α=.5,2 (shape);
Group IV: Distributions with support on (0,1):
Uniform (Unif);
Beta (B) with parameters (2,2), (.5,.5), (3,1.5) and (2,1).
Set 2 of alternative distributions
Gan and Koehler (1990) and Krauczi (2009) considered a battery of “difficult alterna-
tives” for comparing normality tests. We also consider them in order to evaluate the
sensitivity of the proposed test. Let Uand Zdenote a [0,1]-Uniform and a Standard
Normal random variable, respectively.
Contaminated Normal distribution (CN) with parameters (λ,µ1,µ2,σ)given by
the cdf F(x) = (1λ)F0(x,µ1,1) + λF0(x,µ2,σ);
Half Normal (HN) distribution, that is, the distribution of |Z|.
Bounded Johnson’s distribution (SB) with parameters (γ,δ)of the random variable
e(Zγ)/δ/(1+e(Zγ)/δ);
Unbounded Johnson’s distribution (UB) with parameters (γ,δ)of the random vari-
able sinh((Zγ)/δ);
Triangle type I (Tri) with density function f(x) = 1|t|,1<t<1;
Truncated Standard Normal distribution at aand b(TN);
Tukey’s distribution (Tu) with parameter λof the random variable Uλ(1U)λ.
Cauchy distribution with parameters µ=0 (location), σ=1 (scale).
Chi-squared distribution χ2with kdegrees of freedom.
Tables 2-3 contain the skewness (β1)and kurtosis (β2)of the previous sets of alter-
native distributions. Alternatives in Set 2 are roughly ordered and grouped in five groups
according to their skewness and kurtosis values in Table 3. These groups correspond
to: symmetric short tailed, symmetric closed to normal, asymmetric short tailed, asym-
metric long tailed. Figure 2 illustrates some of the possible shapes of the pdf’s of the
alternatives in Set 1 and Set 2.
74 A test for normality based on the empirical distribution function
−3 −1 0 1 2 3
0.0 0.4 0.8
Symmetric short−tailed
TN(−3,3)
TN(−1,1)
Tri
−0.5 0.0 0.5 1.0 1.5
0.0 0.4 0.8
SB(0,.5)
SB(0,.707)
0.0 0.4 0.8
0.0 1.0 2.0
Beta(.5,.5)
Beta(1,1)
Beta(2,2)
−1.0 0.0 0.5 1.0
0.0 0.2 0.4 0.6
Tu(3)
Tu(1.5)
Tu(.7)
−1.0 0.0 0.5 1.0
0 5 10 15
Symmetric long−tailed
Tu(10)
−10 −5 0 5 10
0.0 0.2 0.4
DE
t(3)
cauchy
SU(0,1)
−0.5 0.0 0.5
0.0 1.0 2.0
Symmetric close to normal
Tu(.1)
−4 −2 0 2 4
0.0 0.2 0.4 0.6
SU(0,3)
L
t(10)
0 1 2 3 4
0.0 0.2 0.4 0.6 0.8
Asymmetric short−tailed
W(2)
HN
0.0 0.4 0.8
0.0 0.5 1.0 1.5 2.0
Beta(2,1)
Beta(3,2)
−2 0 2 4
0.0 0.2 0.4 0.6 0.8
SN(2)
TN(−3,1)
−0.5 0.0 0.5 1.0 1.5
0.0 0.5 1.0 1.5 2.0
SB(.533,.5)
SB(1,1)
SB(1,2)
0 1 2 3 4 5 6 7
0.0 0.4 0.8 1.2
Asymmetric long−tailed
LN(2)
LN(.5)
LN(1)
−20 −10 0 5 10
0.0 0.1 0.2 0.3
SU(1,1)
Gu
0 5 10 15
0.0 0.1 0.2 0.3 0.4
chisq(1)
chisq(4)
0 1 2 3 4 5 6 7
0.0 0.4 0.8
Exp(1)
W(.5)
G(.5)
G(2)
Figure 2: Plots of alternative distributions in Set 1 and Set 2.
Tables 4-5 contain the estimated value of Hn(for h(x) = (x1)2/(x+1)2and h(x) =
xlog(x)x+1, respectively), for each alternative distribution, computed as the average
value from 10000 simulated samples of sizes n=10,20,50,100,1000. In the last row of
these tables (n=)), we show the value of D(F0,F1)computed with the the command
integrate in R Software, with (µ)and (σ2)being the expectation and variance of F1,
respectively. These tables show consistency of the test statistic Hn.
Tables 6-7 report the power of the 5% significance level of forty normality tests based
on the statistics considered in Section 4 for the Set 1 of alternatives.
Tables 8-9 contain the power of the 5% significance level test of normality based on
the most powerful statistics and the alternatives listed in Set 2.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e75
Table 2: Skewness and kurtosis of alternative distributions in Set 1.
Group I Group II Group III Group IV
t(1) t(3) L DE Gu SN(2) Exp G(2) G(.5) LN(1) LN(2) LN(.5) W(.5) W(2) Unif B(2,2) B(.5,.5) B(3,.5) B(2,1)
β10 0 0 0 1.30 .45 2 1.41 2.83 6.18 414.36 1.75 6.62 .63 0 0 0 1.575 .57
β2 4.2 6 5.4 .31 9 6 15 113.94 9220560 8.90 87.72 3.25 1.8 2.14 1.5 5.22 2.4
Table 3: Skewness and kurtosis of alternative distributions in Set 2.
Symmetric Asymmetric
Short tailed Close to Normal Long tailed Short tailed Long tailed
Tu Tu Tu SB Tri TN TN Tu SU t Tu SU caushy TN SB SB SB HN SU χ2χ2
(.7) (1.5) (3) (0,.5) (1,1) (3,3) (.1) (0,3) (10) (10) (0,1) (3,1) (1,1) (1,2) (.533,.5) (1,1) (1) (4)
β10 0 0 0 0 0 0 0 0 0 0 0 0 .55 .73 .28 .65 .97 5.37 2.83 1.41
β21.92 1.75 2.06 1.63 2.4 1.94 2.84 3.21 3.53 4 5.38 36.2 2.78 2.91 2.77 2.13 3.78 93.4 15 6
76 A test for normality based on the empirical distribution function
Table 4: Estimated value of Hnwith h1(x) = (x1)2/(x+1)2under H1, based on 10000 simulations for several values of n.
Group I Group II Group III Group IV
t(1) t(3) L DE Gu SN(2) Exp G(2) G(.5) LN(1) LN(2) LN(.5) W(.5) W(2) Unif B(2,2) B(.5,.5) B(3,.5) B(2,1)
n
10 .0011 .00086 .0010 .0011 .00092 .0017 .0013 .0025 .00226 .0040 .0013 .0035 .00097 .0009 .00082 .0012 .0013 .0008
20 .0007 .00043 .0006 .0007 .00047 .0014 .0009 .0023 .00213 .0045 .0009 .0036 .00054 .0005 .00041 .0008 .0011 .0005
50 .0005 .00018 .0004 .0004 .00022 .0012 .0006 .0022 .00211 .0052 .0007 .0037 .00028 .0003 .00019 .0006 .0011 .0003
100 .0004 .00011 .0003 .0003 .00013 .0011 .0006 .0022 .00215 .0056 .0006 .0039 .00019 .0003 .00012 .0006 .0011 .0003
1000 .0004 .00004 .0002 .0002 .00006 .0010 .0005 .0021 .00226 .0066 .0005 .0040 .00012 .0002 .00006 .0005 .0011 .0002
.0004 .00003 .0002 .0002 .00005 .0010 .0005 .0021 .00228 .0074 .0005 .0040 .00011 .0002 .00006 .0005 .0011 .0002
Table 5: Estimated value of Hnwith h2(x) = xlog(x)x+1under H1, based on 10000 simulations for several values of n.
Group I Group II Group III Group IV
t(1) t(3) L DE Gu SN(2) Exp G(2) G(.5) LN(1) LN(2) LN(.5) W(.5) W(2) Unif B(2,2) B(.5,.5) B(3,.5) B(2,1)
n
10 .0021 .00167 .0019 .0022 .0018 .0034 .0027 .0048 .0044 .0077 .0026 .0065 .0020 .0019 .0017 .0025 .0027 .0017
20 .0014 .00086 .0012 .0013 .0009 .0028 .0017 .0045 .0042 .0088 .0018 .0070 .0010 .0011 .0009 .0017 .0024 .0010
50 - .0010 .00037 .0007 .0008 .0004 .0023 .0013 .0044 .0042 .0106 .0013 .0075 .0005 .0006 .0004 .0013 .0023 .0006
100 - .0009 .00021 .0006 .0006 .0003 .0022 .001 .0043 .0043 .0113 .0012 .0079 .0004 .0005 .0003 .0012 .0023 .0005
1000 .0009 .00007 .0004 .0005 .0001 .0021 .0009 .0043 .0046 .0139 .0010 .0084 .0002 .0004 .0001 .0011 .0023 .0004
.0009 .00006 .0004 .0005 .0001 .0021 .0009 .0043 .0047 .0163 .0010 .0084 .0002 .0004 .0001 .0010 .0023 .0004
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e77
Table 6: Power comparisons for the normality test for Set 1 of alternative distributions, α=0.05, n =10.
Group I II III IV
altern. N t(1) t(3) L DE Gu SN Exp G(2) G(.5) LN(1) LN(2) LN(.5) W(.5) W(2) Unif B(2,2) B(.5,.5) B(3,.5) B(2,1)
1 KL .048 .442 .091 .051 .091 .101 .058 .416 .179 .782 .552 .938 .181 .931 .075 .167 .082 .512 .108 .173
2 TV .048 .375 .082 .048 .053 .092 .055 .397 .151 .762 .519 .933 .144 .923 .073 .181 .084 .514 .656 .170
3 TE .052 .460 .112 .058 .077 .111 .059 .454 .185 .794 .581 .945 .181 .935 .074 .158 .071 .481 .686 .164
4 TA .053 .507 .134 .065 .094 .124 .062 .477 .213 .810 .616 .951 .208 .940 .080 .129 .064 .451 .704 .162
5 TD .051 .583 .201 .087 .163 .154 .071 .394 .222 .631 .565 .869 .249 .813 .076 .028 .025 .080 .065 .093
6 TC .054 .409 .083 .047 .057 .097 .053 .404 .173 .786 .542 .936 .171 .926 .071 .170 .086 .489 .110 .182
7 TEs .049 .591 .167 .074 .140 .113 .062 .330 .158 .679 .485 .892 .176 .876 .064 .061 .037 .238 .064 .092
8 TZ1 .053 .632 .212 .089 .177 .145 .068 .359 .209 .581 .524 .846 .229 .784 .074 .030 .025 .078 .061 .081
9 TZ2 .051 .638 .216 .091 .181 .144 .066 .353 .205 .572 .516 .840 .228 .776 .073 .026 .023 .060 .058 .076
10 ZK.055 .587 .174 .075 .154 .126 .071 .352 .180 .636 .509 .885 .192 .842 .079 .078 .053 .221 .510 .109
11 ZC.053 .580 .183 .079 .154 .157 .074 .450 .245 .740 .606 .926 .248 .898 .089 .094 .044 .336 .621 .130
12 ZA.053 .608 .199 .083 .167 .162 .071 .457 .246 .744 .612 .928 .255 .901 .086 .050 .032 .204 .621 .115
13 b1.057 .587 .219 .096 .184 .165 .073 .372 .226 .557 .532 .928 .247 .751 .088 .019 .024 .035 .437 .083
14 b2.053 .536 .170 .073 .136 .113 .060 .227 .148 .340 .353 .907 .159 .508 .072 .115 .057 .270 .235 .092
15 K2.058 .592 .220 .096 .190 .154 .073 .314 .197 .467 .464 .754 .221 .662 .082 .020 .021 .065 .336 .067
16 DH .055 .625 .207 .084 .183 .130 .067 .344 .183 .590 .507 .860 .195 .797 .069 .071 .037 .238 .467 .093
17 Zw.055 .501 .150 .068 .130 .075 .088 .125 .091 .181 .210 .416 .097 .311 .055 .100 .056 .215 .123 .073
18 D .051 .584 .175 .071 .142 .111 .060 .270 .146 .478 .434 .799 .168 .717 .064 .042 .044 .039 .335 .061
19 QH .053 .598 .189 .081 .159 .160 .075 .455 .245 .742 .609 .928 .250 .901 .090 .094 .046 .321 .625 .135
20 r .054 .635 .214 .088 .187 .160 .074 .421 .231 .692 .578 .907 .245 .868 .089 .042 .031 .164 .561 .099
21 Rn.054 .609 .196 .083 .167 .162 .075 .448 .244 .733 .604 .924 .251 .894 .090 .077 .042 .276 .613 .125
22 TEP .053 .602 .200 .088 .170 .167 .077 .427 .244 .663 .587 .891 .256 .842 .070 .054 .040 .152 .538 .115
23 In.055 .157 .151 .084 .151 .120 .066 .209 .149 .199 .215 .100 .151 .134 .070 .024 .025 .043 .207 .065
24 En.055 .638 .218 .089 .193 .158 .073 .407 .226 .670 .567 .898 .240 .852 .082 .035 .028 .126 .536 .091
25 Mn.054 .631 .226 .095 .198 .147 .071 .326 .189 .524 .484 .808 .214 .733 .073 .014 .020 .029 .385 .061
26 Qn.053 .604 .175 .074 .152 .141 .071 .426 .220 .728 .585 .923 .222 .894 .081 .094 .051 .285 .610 .130
27 T1n.054 .516 .179 .083 .145 .173 .072 .475 .264 .726 .626 .918 .274 .884 .095 .036 .030 .093 .605 .114
28 T2n.053 .555 .168 .072 .155 .075 .055 .106 .075 .167 .206 .453 .088 .326 .049 .090 .046 .284 .105 .060
29 T3n.057 .647 .225 .093 .204 .146 .070 .360 .199 .625 .518 .882 .216 .831 .074 .039 .026 .203 .487 .076
30 KS .053 .581 .164 .073 .148 .124 .072 .312 .170 .545 .469 .828 .182 .761 .078 .066 .051 .163 .424 .103
31 V .050 .593 .163 .071 .143 .119 .065 .365 .180 .662 .530 .894 .188 .856 .074 .087 .054 .240 .540 .108
32 W2.052 .624 .186 .080 .164 .143 .073 .396 .210 .674 .562 .898 .220 .860 .082 .083 .050 .236 .552 .116
33 U2.052 .618 .178 .076 .159 .135 .071 .382 .200 .661 .547 .893 .211 .853 .081 .091 .056 .260 .540 .120
34 A2.051 .619 .190 .083 .165 .147 .073 .417 .225 .670 .578 .911 .233 .877 .085 .086 .048 .268 .580 .126
35 P .042 .531 .148 .083 .136 .127 .080 .397 .200 .704 .545 .903 .199 .878 .087 .086 .061 .229 .594 .136
36 SW .052 .597 .187 .082 .159 .159 .075 .451 .245 .740 .608 .927 .248 .899 .088 .090 .045 .312 .622 .133
37 SF .054 .631 .214 .088 .185 .161 .074 .426 .234 .701 .584 .912 .248 .872 .085 .047 .033 .183 .571 .104
38 SJ .055 .655 .217 .096 .211 .121 .068 .253 .147 .429 .416 .756 .176 .660 .060 .012 .021 .022 .285 .046
39 JB .059 .600 .223 .096 .192 .149 .075 .352 .219 .532 .511 .804 .242 .731 .087 .016 .021 .029 .396 .073
40 RJB .056 .644 .228 .097 .205 .165 .072 .485 .189 .504 .470 .784 .214 .700 .076 .015 .021 .025 .345 .061
h2Hn.051 .596 .173 .074 .150 .190 .091 .504 .285 .780 .659 .940 .290 .918 .114 .074 .046 .218 .331 .054
h1Hn.051 .587 .169 .073 .144 .199 .095 .516 .296 .784 .665 .942 .301 .920 .119 .079 .049 .220 .300 .047
78 A test for normality based on the empirical distribution function
Table 7: Power comparisons for the normality test for Set 1 of alternative distributions, α=0.05, n =20.
Group I II III IV
altern. N t(1) t(3) L DE Gu SN Exp G(2) G(.5) LN(1) LN(2) LN(.5) W(.5) W(2) Unif B(2,2) B(.5,.5) B(3,.5) B(2,1)
1 KL .045 .737 .165 .051 .091 .198 .073 .846 .457 .992 .927 .999 .404 1.00 .132 .442 .131 .914 .224 .438
2 TV .047 .684 .121 .046 .062 .176 .067 .830 .429 .992 .910 1.00 .364 1.00 .126 .443 .136 .910 .980 .428
3 TE .047 .786 .205 .064 .129 .237 .079 .865 .508 .993 .934 1.00 .445 1.00 .143 .391 .112 .891 .984 .423
4 TA .048 .858 .301 .095 .229 .279 .101 .870 .533 .993 .937 1.00 .485 1.00 .145 .258 .064 .824 .983 .358
5 TD .049 .872 .371 .134 .304 .310 .102 .790 .507 .959 .909 .997 .517 .995 .148 .084 .028 .408 .129 .221
6 TC .047 .687 .138 .043 .070 .185 .076 .836 .443 .991 .919 .999 .386 .999 .133 .438 .135 .902 .225 .432
7 TEs .054 .871 .330 .114 .271 .195 .073 .646 .322 .955 .825 .997 .360 .997 .089 .076 .027 .460 .069 .131
8 TZ1 .056 .885 .377 .133 .309 .294 .099 .745 .459 .947 .895 .996 .470 .994 .123 .099 .028 .442 .114 .200
9 TZ2 .062 .900 .402 .147 .344 .282 .096 .688 .416 .915 .865 .994 .445 .987 .110 .028 .013 .145 .079 .130
10 ZK.055 .861 .308 .109 .252 .251 .088 .797 .438 .983 .906 .992 .423 .999 .118 .132 .054 .512 .952 .253
11 ZC.050 .844 .333 .121 .249 .313 .104 .838 .529 .983 .931 .999 .520 .999 .159 .231 .052 .782 .953 .307
12 ZA.052 .864 .347 .124 .268 .323 .108 .866 .559 .989 .943 .999 .541 .999 .166 .142 .032 .674 .967 .318
13 b1.052 .775 .345 .135 .286 .324 .114 .708 .471 .891 .869 .990 .508 .979 .151 .006 .008 .013 .762 .125
14 b2.049 .832 .333 .111 .239 .181 .076 .365 .230 .544 .600 .877 .279 .787 .093 .324 .109 .683 .316 .122
15 K2.048 .849 .370 .139 .282 .267 .100 .570 .371 .777 .781 .967 .418 .936 .119 .133 .030 .491 .587 .093
16 DH .050 .871 .382 .141 .316 .258 .089 .730 .429 .941 .888 .997 .444 .994 .110 .101 .024 .494 .855 .186
17 Zw.049 .853 .326 .108 .280 .120 .062 .203 .135 .340 .427 .756 .173 .602 .059 .225 .089 .539 .160 .111
18 D .051 .882 .347 .119 .276 .202 .075 .517 .280 .805 .758 .984 .330 .963 .086 .094 .075 .031 .607 .067
19 QH .053 .862 .327 .115 .251 .313 .103 .841 .533 .983 .933 .999 .520 .999 .157 .229 .059 .761 .957 .326
20 r .053 .895 .389 .145 .325 .311 .108 .794 .492 .970 .911 .998 .504 .999 .145 .073 .019 .460 .916 .207
21 Rn.054 .875 .353 .128 .281 .320 .108 .833 .528 .981 .931 .999 .524 .999 .158 .176 .045 .683 .946 .292
22 TEP .054 .868 .332 .115 .257 .309 .104 .778 .502 .954 .912 .998 .507 .995 .147 .130 .043 .478 .888 .266
23 In.053 .144 .268 .145 .286 .216 .091 .387 .289 .286 .310 .038 .313 .084 .095 .004 .006 .013 .382 .070
24 En.053 .901 .398 .150 .337 .302 .105 .763 .467 .959 .899 .998 .488 .997 .135 .038 .013 .326 ..891 .169
25 Mn.050 .894 .409 .153 .339 .274 .099 .661 .395 .897 .841 .992 .431 .984 .112 .005 .004 .025 .771 .087
26 Qn.053 .874 .311 .105 .257 .277 .092 .847 .508 .988 .932 .999 .486 .999 .142 .176 .051 .663 .963 .333
27 T1n.050 .656 .255 .106 .179 .345 .111 .838 .569 .972 .938 .999 .565 .999 .178 .029 .018 .082 .924 .246
27 T2n.049 .866 .343 .116 .296 .100 .059 .150 .101 .269 .362 .734 .141 .554 .049 .311 .079 .773 .109 .119
27 T3n.050 .897 .387 .143 .330 .278 .096 .779 .453 .973 .905 .999 .466 .999 .121 .174 .032 .732 .926 .225
30 KS .056 .847 .268 .089 .227 .214 .084 .595 .338 .884 .799 .992 .349 .985 .109 .102 .056 .377 .761 .192
31 V .052 .863 .273 .090 .236 .199 .073 .697 .352 .955 .859 .998 .348 .997 .093 .148 .063 .495 .885 .205
32 W2.056 .880 .308 .105 .265 .254 .091 .732 .420 .954 .883 .998 .429 .996 .123 .149 .056 .517 .882 .237
33 U2.055 .878 .297 .099 .261 .225 .083 .694 .381 .942 .862 .997 .391 .995 .113 .167 .064 .554 .863 .230
34 A2.054 .880 .324 .110 .268 .279 .094 .780 .463 .968 .906 .999 .467 .998 .133 .179 .056 .624 .917 .269
35 P .049 .777 .182 .067 .144 .141 .063 .656 .282 .956 .827 .998 .267 .994 .074 .082 .053 .272 .880 .162
36 SW .054 .867 .337 .119 .266 .317 .105 .840 .534 .982 .933 .999 .526 .999 .160 .208 .053 .738 .954 .314
37 SF .053 .893 .383 .143 .318 .313 .107 .802 .498 .973 .915 .998 .507 .999 .148 .086 .022 .499 .922 .220
38 SJ .053 .915 .404 .147 .377 .188 .078 .416 .234 .676 .695 .959 .301 .911 .064 .003 .006 .002 .435 .033
39 JB .050 .864 .384 .146 .300 .285 .104 .630 .404 .840 .825 .984 .448 .964 .125 .003 .004 .006 .677 .080
40 RJB .054 .906 .410 .159 .354 .266 .099 .563 .357 .790 .787 .977 .410 .949 .106 .002 .004 .004 .594 .061
h2Hn.055 .874 .302 .100 .254 .322 .116 .832 .540 .982 .933 .999 .525 .999 .176 .154 .061 .524 .734 .140
h1Hn.053 .869 .293 .097 .244 .330 .120 .835 .546 .983 .934 .999 .532 .999 .181 .156 .062 .525 .709 .126
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e79
Table 8: Power comparisons for the normality test for Set 2 of alternative distributions, α=0.05, n =10.
Symmetric Asymmetric
Short tailed Close to Normal Long tailed Short tailed Long tailed
Tu Tu Tu SB Tri TN TN Tu SU t Tu SU caushy TN SB SB SB HN SU χ2χ2
(.7) (1.5) (3) (0,.5) (-1,1) (-3,3) (.1) (0,3) (10) (10) (0,1) (-3,1) (1,1) (1,2) (.533,.5) (1,1) (1) (4)
TV .122 .205 .095 .313 .057 .125 .053 .044 .045 .057 .281 .086 .369 .027 .130 .055 .442 .206 .260 .766 .171
TA .086 .155 .068 .244 .043 .089 .047 .050 .051 .048 .460 .162 .496 .095 .143 .051 .431 .177 .357 .814 .225
ZA.040 .060 .033 .101 .031 .037 .043 .062 .067 .080 .523 .258 .608 .071 .133 .050 .296 .196 .438 .755 .254
b1.018 .019 .017 .024 .025 .017 .040 .061 .069 .084 .372 .255 .574 .061 .101 .042 .130 .156 .412 .566 .221
r .035 .050 .021 .086 .031 .034 .044 .064 .068 .087 .595 .276 .634 .070 .123 .050 .253 .177 .436 .704 .239
Rn.057 .094 .042 .149 .035 .054 .046 .061 .066 .075 .552 .251 .609 .073 .142 .054 .323 .194 .436 .743 .251
TEP .046 .067 .037 .101 .035 .047 .050 .063 .068 .085 .480 .180 .602 .072 .143 .057 .271 .193 .448 .684 .256
En.029 .038 .026 .067 .030 .028 .044 .063 .068 .090 .602 .278 .639 .067 .114 .048 .224 .168 .433 .682 .232
Mn.015 .017 .017 .020 .043 .016 .045 .064 .075 .100 .521 .287 .634 .061 .087 .044 .115 .137 .397 .550 .200
T1n.032 .041 .026 .060 .031 .030 .044 .061 .063 .069 .338 .220 .517 .077 .140 .051 .253 .204 .441 .739 .266
T3n.029 .047 .026 .088 .030 .030 .045 .064 .072 .038 .579 .288 .645 .083 .093 .046 .200 .139 .405 .644 .198
A2.063 .102 .049 .154 .037 .063 .050 .060 .064 .068 .630 .245 .620 .075 .137 .054 .319 .182 .422 .715 .234
SW .064 .109 .049 .170 .036 .064 .046 .060 .064 .074 .532 .242 .598 .078 .144 .054 .345 .199 .433 .751 .253
SF .037 .055 .031 .093 .030 .036 .044 .063 .067 .082 .588 .270 .630 .070 .124 .050 .261 .179 .435 .709 .238
SJ .014 .016 .019 .015 .032 .016 .047 .067 .072 .093 .678 .290 .660 .049 .070 .046 .086 .101 .360 .442 .157
RJB .015 .016 .016 .018 .026 .016 .045 .064 .073 .093 .569 .290 .645 .057 .088 .044 .105 .132 .394 .504 .195
Hn.066 .094 .052 .141 .047 .061 .053 .060 .064 .060 .625 .222 .592 .026 .208 .073 .416 .262 .264 .807 .315
80 A test for normality based on the empirical distribution function
Table 9: Power comparisons for the normality test for Set 2 of alternative distributions, α=0.05, n =20.
Symmetric Asymmetric
Short tailed Close to Normal Long tailed Short tailed Long tailed
Tu Tu Tu SB Tri TN TN Tu SU t Tu SU caushy TN SB SB SB HN SU χ2χ2
(.7) (1.5) (3) (0,.5) (-1,1) (-3,3) (.1) (0,3) (10) (10) (0,1) (-3,1) (1,1) (1,2) (.533,.5) (1,1) (1) (4)
TV .291 .515 .188 .729 .075 .268 .051 .042 .047 .048 .724 .159 .683 .180 .314 .070 .877 .458 .547 .993 .433
TA .131 .310 .083 .531 .036 .122 .040 .051 .065 .091 .909 .376 .853 .171 .307 .057 .807 .477 .678 .992 .515
ZA.064 .168 .040 .343 .020 .057 .037 .060 .077 .103 .721 .421 .859 .154 .305 .058 .709 .462 .714 .989 .541
b1.005 .007 .008 .009 .011 .006 .035 .065 .084 .113 .354 .401 .771 .111 .190 .050 .174 .307 .708 .882 .446
r .034 .084 .021 .193 .017 .030 .037 .066 .085 .109 .851 .480 .890 .105 .230 .052 .534 .360 .720 .966 .472
Rn.085 .198 .050 .385 .028 .078 .038 .059 .077 .102 .817 .440 .872 .135 .282 .058 .681 .414 .721 .980 .509
TEP .073 .149 .045 .267 .034 .065 .042 .058 .071 .090 .807 .417 .866 .129 .284 .062 .580 .368 .722 .952 .488
En.019 .047 .014 .114 .015 .019 .036 .067 .087 .112 .859 .494 .897 .094 .205 .049 .444 .329 .712 .957 .450
Mn.003 .005 .006 .010 .011 .004 .034 .067 .090 .116 .774 .501 .894 .076 .140 .041 .191 .253 .675 .895 .381
T1n.018 .029 .017 .046 .020 .019 .040 .057 .070 .089 .261 .292 .645 .152 .301 .058 .448 .436 .723 .971 .547
T3n.074 .212 .043 .409 .037 .070 .039 .061 .083 .110 .775 .482 .896 .098 .205 .045 .646 .333 .695 .971 .433
A2.105 .206 .060 .374 .040 .092 .048 .057 .070 .084 .906 .423 .878 .117 .266 .064 .651 .359 .704 .970 .459
SW .108 .250 .067 .452 .034 .100 .040 .058 .077 .097 .805 .424 .866 .143 .305 .063 .723 .435 .719 .982 .522
SF .041 .102 .025 .222 .018 .036 .038 .065 .084 .106 .848 .477 .888 .109 .242 .054 .561 .372 .722 .970 .482
SJ .002 .001 .005 .004 .018 .003 .037 .066 .086 .115 .930 .509 .917 .039 .065 .044 .054 .109 .594 .669 .227
RJB .002 .002 .004 .003 .011 .003 .036 .068 .092 .121 .819 .507 .902 .065 .119 .041 .091 .206 .666 .784 .348
Hn.095 .176 .058 .308 .041 .082 .049 .056 .065 .078 .914 .384 .867 .056 .345 .083 .719 .441 .574 .981 . 527
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e81
Table 10: Ranking from first to the fifth of average powers computed from values in Tables 6-7 for
Set 1 of alternative distributions.
Group I Group II Group III Group IV
Symmetric (,)Asymmetric (,)Asymmetric(0,)(0,1)
Rank n=10 n=20 n=10 n=20 n=10 n=20 n=10 n=20
1 SJ SJ HnT1nHnZATV TV
2 RJB RJB T1nHnTV T1nTE TE
3 T3nMnTEP ZAA HnTV TA
4 MnTZ2 b1RnT1nSW ZCQH
5 EnEnRnSW ZAQH QH ZC
Table 11: Ranking from first to the fifth of average powers computed from values in Tables 8-9 for
Set 2 of alternative distributions.
Symmetric Asymmetric
Rank Short tailed Close to Normal Long tailed Short tailed Long tailed
n=10 n=20 n=10 n=20 n=10 n=20 n=10 n=20 n=10 n=20
1 TV TV MnRJB SJ SJ HnHnT1nT1n
2 TA TA SJ MnRJB RJB TA TV SW SW
3 SW RnRJB SJ A2SF TV TA RnRn
4 HnSW SF SF SF A2SW SW HnTA
5 A2A2SW T3nT3nMnRnRnTA Hn
Tables 10-11 contain the ranking from first to the fifth of the average powers com-
puted from the values in Tables 6-7 and 8-9, respectively. By average powers we can
select the tests that are, on average, most powerful against the alternatives from the
given groups.
Power against an alternative distribution has been estimated by the relative frequency
of values of the corresponding statistic in the critical region for 10000 simulated sam-
ples of size n = 10, 20. The maximum reached power is indicated in bold. For computing
the estimated powers of the new test, R software is used. We also use R software for
computing Pearson chi-square and Shapiro-Francia tests by the package (nortest), com-
mand pearson.test and sf.test, respectively, and also the package (lawstat), com-
mand sj.test and rjb.test for SJ and Robast Jarque-Bera tests, respectively. For the
entropy-based test statistics, powers are taken from Zamanzadeh and Arghami (2012)
and Alizadeh and Arghami (2011, 2013). In the case of the test based on Hn, we also
consider h2(x):=xlog(x)x+1 for comparison with h1(x):=x1
x+12.
82 A test for normality based on the empirical distribution function
Results and recommendations
Based on these comparisons, the following recommendations can be formulated for the
application of the evaluated statistics for testing normality in practice.
Set 1 of alternative distributions (Tables 6-7 and 10): In Group I, for n=10 and
20, it is seen that the tests based on SJ, RJB, T3n, TZ2, Mnand Enare the most powerful
whereas the tests based on In, TV, TC and KL are the least powerful. The difference
of powers between KL and the others is substantial. In Group II, for n=10 and 20,
it is seen that the tests based on Hn, T1n, TEP, Rn, ZAand b1are the most powerful
whereas those based on T2n, TV, TC, Kl and Zware the least powerful. In Group III,
the most powerful tests for n=10 are those based on Hn, TV, TA and T1n, and for
n=20, those based on ZA, T1n, Hnand SW are the most powerful. On the other hand,
the least powerful tests are those based on Inand Zware the least powerful. Finally, in
group IV, the results are not in favour of the proposed tests. In this group, for n=10 and
20, the most powerful tests are those based on TV, TE, TA, ZC, ZAand r, whereas the
tests based on TZ2, SJ and RJB are the least powerful. The SJ and RJB show very poor
sensitivity against symmetric distributions in [0,1]such as Unif, B(2,2)or B(.5, .5). For
example, for n=20, in the case of the [0,1]-Unif alternative, the SJ test has a power
of .002 while even the Hntest has a power of .156. From Tables 6-7 one can see that
the proportion of times that the SJ and RJB statistics lie below the 5% point of the null
distribution are greater than those of the Hnstatistic.
Note that for the proposed test, the maximum power in Group II and III was typically
attained by choosing h1.
From the simulation study implemented for Set 1 of alternative distributions we can
lead to different conclusions from that existing in the literature. New and existing results
are reported in Table 12.
Table 12: Comparison of most powerful tests in Groups I–IV, according to
Alizadeh and Arghami (2011, 2013) and Zamanzade and Arghami (2012) with new simulation results.
Alizadeh and Arghami (2011) JB SW KLaor SW KL
Alizadeh and Arghami (2013) A2SW TA TVb
Zamanzadeh and Arghami (2012) TZ2 TZ2 or TD TZ1, KL or TD KL or TC
New simulation study SJ or RJB Hnor T1nHnor ZATV or TE
aStatistic based on Vasicek’s estimator
bStatistic using nonparametric distribution of Vasicek’s estimato
Set 2 of alternative distributions (Tables 8-9 and 11): For symmetric short-tailed
distributions, it is seen that the tests based on TV, TA and SW are the most powerful.
For symmetric close to normal and symmetric long tailed distributions, RJB, JB and Mn
are the most powerful. For asymmetric short tailed distributions, Hn, TV and TA are the
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e83
−5 0 5 10
0.00 0.10 0.20 0.30
contaminated normal density
f(x)=(1− λ)φ(x, µ1, 1)+ λφ(x, µ2, σ)
µ1 =−3, µ2 =3, σ =2
λ=0.2
λ=0.5
λ=0.8
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
power of 5% for n=20
λ
Hn
KS
A2
Rn
Figure 3: Left panel: Probability density functions of Contaminated Normal distribution for several values
of the parameter λ. Right panel: Power of the tests based on Hn,KS,A2and Rnas a function of λagainst
alternative CN(λ,µ1=3,µ2=3,σ=2).
most powerful. Finally, for asymmetric long tailed distributions, T1n, SW and Rnare the
most powerful. It is also worth mentioning that the differences between the power of
tests based on TV and Hnin T N (3,3)alternative are not considerable.
In Figure 3 we compare the power of the tests based on Hn, KS, A2and Rnagainst
a family of Contaminated Normal alternatives CN(λ,µ1=3,µ2=3,σ=2). The left
panel of Figure 3 contains the probability density functions of Contaminated Normal
alternatives CN(λ,µ1=3,µ2=3,σ=1), for λ=.2,.5,.8, whereas the right panel
contains the power comparisons for n=20 and α=0.05. We can see the good power
results of Hnfor 0.2<λ<0.6.
In general, we can conclude that the proposed test Hnhas good performance and
therefore can be used in practice.
Numerical example
Finally, we illustrate the performance of the new proposal through the analysis of a
real data set. One of the most famous tests of normality among practitioners is the
Kolmogorov-Smirnov test, mostly because it is available in any statistical software.
However, one of its drawbacks is the low power against several alternatives (see also
Gran´e and Fortiana, 2003; Gran´e, 2012; Gran´e and Tchirina, 2013).We would like to
emphasize this fact through a numerical example.
Armitage and Berry (1987) provided the weights in ounces of 32 newborn babies(see
also data set 3 of Henry, 2002, p. 342). The approximate ML estimators of ˆµ=111.75
and ˆσ=331.03 =18.19. Also sample skewness and kurtosis are b1=.64 and
84 A test for normality based on the empirical distribution function
Histogram and theoretical densities
data
Density
80 100 120 140
0.000 0.005 0.010 0.015 0.020 0.025 0.030
Normal
emp.
Figure 4: Histogram and theoretical (normal) distribution for ounces of 32 newborn babies data.
b2=2.33, respectively. From the histogram of these data it can be observed that the
birth weights are skewed to the left and may be bimodal (see Figure 4).
When fitting the normal distribution to these data, we find that the KS (Kolmogorov-
Smirnov) test does not reject the null hypothesis providing a p-value of 0.093. How-
ever with the Hnstatistic we are able to reject the null hypothesis of normality at a
5% significance level, since we obtain Hn=.0006 and the corresponding critical value
for n=32 is .00047. Also associated p-values of the Hn, SW (Shapiro-Wilk) and SF
(Shapiro-Francia) tests are .015, .024 and .036, respectively. Thus, the non-normality is
more pronounced by the new test at 5% level. In Appendix, we provide an Rsoftware
program, to calculate the Hnstatistics, the critical points and corresponding p-value.
6. Conclusions
In this paper we propose a statistic to test normality and compare its performance with
40 recent and classical tests for normality and a wide collection of alternative distribu-
tions. As expected (Janssen, 2000), the simulation study shows that none of the statistics
under evaluation can be considered to be the best one for all the alternative distributions
studied. However, the tests based on RJB or SJ have the best performance for symmetric
distributions with the support on (,)and the same happens to TV or TA for distri-
butions with the support on (0,1). Regarding our proposal, Hnand also T1nare the most
powerful for asymmetric distributions with the support on (,)and distributions
with the support on (0,), mainly for small sample sizes.
Hamzeh Torabi, Narges H. Montazeri and Aurea Gran´
e85
Acknowledgements
This work has been partially supported by research grant project MTM2014-56535-
R (Spanish Ministry of Economy and Competitiveness). The authors are thankful to
two Referees and the Editor, whose helpful comments and suggestions contributed to
improve the quality of the paper.
Appendix
h=function(x) (x-1)ˆ2/(x+1)ˆ2
Hn=function(x) {x=sort(x);n=length(x);
F=pnorm(x, mean(x), sd(x)*sqrt(n/(n-1)))+1;
Fn=1:n/n+1; mean(h(F/Fn))}
##weights in ounces of 32 newborn babies,
data=c(72,80,81,84,86,87,92,94,103,106,107,111,112,115,116,118,
119,122,123,123,114,125,126,126,126,127,118,128,128,132,133,142)
Hn(data) ## statistics
n=length(data); B=10000; x=matrix(rnorm(n*B, 0, 1), nrow=B, ncol=n)
H0=apply(x, 1, Hn); Q=quantile(H0, .95); Q ## critical point
length(H0[H0>Hn(data)])/B ##p-value
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