In this paper, we introduce a new approximation for the null
distribution of the likelihood ratio test for the general case. We
compare the the critical values obtained by the new approximation to
the values which are obtained by the exact distribution for the
cases k=1, 2 to test the accuracy of the new approximation. Also,
we compare the results to another approximation method (which is
known by Barnett and Lewis (1994)) for the cases k=3,4

Hassani, Hossein

Mahmoudvand, Rahim

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveA New Approximation for the NullDistribution of the Likelihood Ratio TestStatistics for k Upper Outliers in aNormal SampleRahim Mahmoudvand and Hossein HassaniPayame Noor University, IRAN, Cardiff University, UK1. September 2005Online at http://mpra.ub.uni-muenchen.de/5778/MPRA Paper No. 5778, posted 25. January 2008 19:32 UTCc Statistical Research and Training CenterJ. Statist. Res. Iran 2 (2005): 155-161 170164 XY '1384 u=DUtR w R}}=B '2 |xQ=tW '2 |xQwOA New Approximation for the Null Distributionof the Likelihood Ratio Test Statistics for kOutliers in a Normal SampleRahim Mahmoudvandy;and Hossein HassanizyPayam-e Noor University of ToyserkanzCentral Bank of the Islamic Republic of Iran, and Cardi University1 SummaryUsually when performing a statistical test or estimation procedure, we assumethe data are all observations of i.i.d. random variables, often from a normaldistribution. Sometimes, however, we notice in a sample one or more obser-vations that stand out from the crowd. These observation(s) are commonlycalled outlier(s). Outlier tests are more formal procedures which have been de-veloped for detecting outliers when a sample comes from a normal distribution(Thode, 2002). A lot of work has been done for testing outliers in a univariatesample, most of which corresponds to the normal and exponential distribution.Barnett and Lewis (1994) have presented a summary of tests for outliers andtheir critical values, many of which are specic to the detection of outliers innormal samples. The theoretical solution for the exact null distribution of thelikelihood ratio test for k = 1 is solved by Barnett and Lewis (1994) and Zhangand Yu (2006) for the case k = 2, but the problem is still open for k > 3.In this paper we introduce a new approximation for the null distribution ofthe likelihood ratio test for the general case. We compare the critical valuesobtained by the new approximation to the values, which are obtained by theexact distribution for the cases k = 1; 2 to test the accuracy of the new ap-proximation. Also, the results are compared to another approximation method(which is known by Barnett and Lewis (1994)) for the cases k = 3; 4.Corresponding author169 A New Approximation for the Null Distribution of . . .2 The Null Distribution of T(k)nLet x1; : : : ; xnbe the random sample taken from the normal distribution N(;2) (under the null hypothesis), where both  and 2are unknown, andx(1); : : : ; x(n)be the corresponding ordered statistics. Suppose the aim is toexamine that whether x(n k+1); : : : ; x(n)are the k upper outliers in this sam-ple. Under this assumption the alternative hypothesis can be formulated asx(1); : : : ; x(n k)belong to the N(; 2), and x(n k+1); : : : ; x(n)(the k upperdata) belong to the N(+ c; 2), where c > 0 is a constant. To answer whetherthe k upper data are outliers, we can build the hypothesis H0: c = 0 againstH1: c > 0.The likelihood ratio test statistic for testing k upper outliers (T(k)n) and klower outliers (Tn;(k)) are (Barnett and Lewis, 1994)T(k)n=x(n)+   + x(n k+1)  kxs; Tn;(k)=kx  x(1)       x(k)s; (1)where x and s are the sample mean and the sample standard deviation, re-spectively. Large values of T(k)nreject the null hypothesis H0, or recognizingthat the set of ordered observations fx(n k+1); : : : ; x(n)g as discordant obser-vations (which are named as k outliers). Please note that (x(n); : : : ; x(1))d=( x(1); : : : ; x(n)) then T(k)nd=Tn;(k); (k = 1; : : : ; n   1). In addition, one cansee that T(k)nd=Tn;(n k)and T(k)nd=T(n k)n. So, without loss of generality, weneed just to discuss the sample distribution of T(k)n.3 Some Approximations for the Null Distribu-tion of T(k)nIn this section we introduce a new approximation for the exact distribution ofT(k)n. Split the index set I = f1; : : : ; ng into[I ]k= fi16    6 ikg; 1 6 i1<    < ik6 n;and(I)k= In[I ]k:Set the notationT[I]k=Pi2[I]kxi  kxs: (2)156c 2005, SRTC IranRahim Mahmoudvand and Hossein Hassani 168By denition (2), we can rewrite T(k)n= max[I]kT[I]k, so the distribution ofthe max[I]kT[I]kis the upper bound for the null distribution of T(k)n. It can besimply proved that the nkrandom variables T[I]khave the distributionfT[I]k(x) =  n 12  12  n 22rnk(n  k)(n  1)1 nk(n  k)(n  1)x2n 22 1;(3)where jxj qk(n k)(n 1)n.3.1 Approximation ILet A1; : : : ; Ambe m arbitrary events. Based on the probability laws, one canwriteP m\k=1Ai!6 P (Ak); k = 1; 2; : : : ;mIt is easy to show thatPT(k)n6 t= Pmax[I]kT[I]k6 t= P0@\[I]kfT[I]k6 tg1A6 P T[I]k6 t: (4)Based on the Bonferroni general inequality we can write thatP m\k=1Ak!>mXk=1P (Ak) m+ 1:Barnet and Lewis (1994) introduced an approximation to calculate the criticalvalues of the null distribution of the likelihood ratio test for k > 2 based onthe above inequality, as below:PT(k)n6 t> mZtLfT[I]k(x) dx m+1; L =  rk(n  k)(n  1)n; (5)where m = nk.J. Statist. Res. Iran 2 (2005): 155-161 157167 A New Approximation for the Null Distribution of . . .3.2 Approximation IITo introduce a new approximation, we prove that T(k)nare asymptotically in-dependent. To do that, we just need to consider two conditions: the rst oneis to show that T(k)nare asymptotically normally distributed and the secondone is that T(k)nare asymptotically uncorrelated. The rst one (normality) iseasily obtained by taking the limitlimn!1  n 12  12  n 22rnk(n  k)(n  1)1 nk(n  k)(n  1)x2n 22 1=1p2kexp x22k:To consider the second one, dene T[I]k; T[J]k; (T[S]k= Tfs1;:::;skg, for S =I; J): By this denition, the coecient correlation between the pair T[I]k; T[J]kis Tfi1;:::;ikg; Tfj1;:::;jkg=zn  k2kn  k; z = 0; : : : ; k   1 (6)where z is the number of the equal indices in the pair T[I]k; T[J]k. Then,limn!1 Tfi1;:::;ikg; Tfj1;:::;jkg=zk; z = 0; : : : ; k   1 (7)On the other hand, the frequency distribution of the discrete variable Z isz 0 1 2 . . . k   1fi nk k0 n kk  nk k1 n kk 1  nk k2 n kk 2. . . nk kk 1 n k1By simple calculation, it can be shown that P (Z = 0) ! 1 as n ! 1. Wethen accept that T[I]kare asymptotically uncorrelated when n goes to innity.We can now assume that T[I]kare asymptotically independent. Based on thediscussion above, we introduced a new approximation for the exact distributionof the T[I]kfor the general case as below.PT(k)n6 t= Pmax[I]kT[I]k6 t= P0@\[I]k T[I]k6 t1AZtLfT[I]k(x) dx(nk); k = 1; 2; : : : ; n  1 (8)158c 2005, SRTC IranRahim Mahmoudvand and Hossein Hassani 166It must be mentioned that the identical distribution of the exact distribu-tion is kept for the third approximation formulae. It is easy to show thatT[I]kd= T[I]n k, and thenPT(n k)n6 t= Pmax[I]n kT[I]n k6 t= P0@\[I]n k T[I]n k6 t1AZtLfT[I]n k(x) dx(nn k)=ZtLfT[I]k(x) dx(nk); k = 1; : : : ; n  1 (9)We then conclude that the presented approximation distribution is the samefor T(k)nand T(n k)n.4 AccuracyIn this section we consider the accuracy of the new and previous approximation(approximation I and II). At the rst step we consider the discrepancy betweenthe obtained critical values by approximations I and II and the exact distribu-tion for the cases k = 1; 2. Tables 1 and 2 show some critical values for T(k)nat = 0:01 and  = 0:05, respectively, which are obtained by dierent methods.The bold font indicates the better approximation method which produces theclosest values to those which are obtained from the exact distribution. Thereare some interesting results, as is evident from Tables 1 and 2: (a) The valuesof the new and previous approximation methods are very close to the values ofthe exact distribution; (b) The new approximation method is better than theprevious method. It must be emphasized also, that there is no scienticTable 1. Critical values for T(k)nat level  = 0:01 for exact distributionn5 10 20 30 50 100k = 1Exact 1.749 2.410 2.884 3.103 3.337 3.600Approx I 1.751 2.413 2.886 3.106 3.340 3.650Approx II 1.749 2.410 2.883 3.102 3.335 3.600k = 2Exact 2.160 3.402 4.437 4.946 5.497 6.118Approx I 2.164 3.406 4.465 4.890 5.552 6.193Approx II 2.160 3.402 4.435 4.951 5.516 6.136J. Statist. Res. Iran 2 (2005): 155-161 159165 A New Approximation for the Null Distribution of . . .Table 2. Critical values for T(k)nat level  = 0:05 for exact distributionn5 10 20 30 50 100k = 1Exact 1.671 2.176 2.557 2.745 2.945 3.207Approx I 1.672 2.179 2.559 2.749 2.967 3.225Approx II 1.670 2.173 2.553 2.743 2.995 3.207k = 2Exact 2.010 3.197 4.110 4.651 5.058 5.638Approx I 2.103 3.198 4.123 4.600 5.123 5.745Approx II 2.100 3.193 4.113 4.584 5.094 5.657Table 3. Critical values for T(k)nat level  = 0:01 by using simulationn10 20 30 50 100k = 3Simulation 3.997 5.612 6.431 7.329 8.388Approx I 4.004 5.630 6.467 7.399 8.493Approx II 3.998 5.614 6.451 7.388 8.474k = 4Simulation 4.323 6.530 7.660 8.935 10.309Approx I 4.331 6.544 7.717 9.052 10.621Approx II 4.323 6.529 7.700 9.035 10.599Table 4. Critical values for T(k)nat level  = 0:05 by using simulationn10 20 30 50 100k = 3Simulation 3.813 5.311 6.051 6.871 7.855Approx I 3.818 5.321 6.111 6.999 8.056Approx II 3.814 5.314 6.100 6.992 8.044k = 4Simulation 4.155 6.249 7.235 8.408 9.772Approx I 4.161 6.261 7.379 8.661 10.182Approx II 4.155 6.253 7.370 8.651 10.172discrepancy between the two approximations (I and II) for the small value of n,but for the large value of n the new approximation is better than the previousone; (c) The accuracy for the case k = 1 is more than for the case k = 2;because, for the case k = 1 the coecient correlation is more closer to zero160c 2005, SRTC IranRahim Mahmoudvand and Hossein Hassani 164than for k = 2, and thus converges to zero faster; (d) The accuracy of theapproximation methods reduce when the  increases, and the approximation Idoes not work for some value of the .At the second step, we consider the accuracy of the approximation meth-ods by the critical values which are obtained for the cases k = 3; 4. Tables3 and 4 show the results. For simulation part, we used S-Plus software andthe number 10,000 for simulation. As it appears from Tables 3 and 4, the newapproximation is again better than the previous approximation. Also it mustbe mentioned that conditions (a)-(d) hold for these tables as well.Keywords. outlier; normal sample; likelihood ratio test; approximation.ReferencesBarnett, V.; Lewis, T. (1994). Outliers in Statistical Data, 3rd ed. Wiley, Chichester.Thode, H.C. (2002). Testing for Normality. Marcel Dekker, New York.Zhang, J.; Yu, k. (2006). The null distribution of the likelihood-ratio test for one or twooutliers in a normal sample. Test 15, 141-150.Rahim MahmoudvandStatistics Department,Payam-e Noor University of Toyserkan,Toyserkan,Iran.e-mail: r{mahmodvand@yahoo.comHossein HassaniCenteral Bank of the Islamic Republic of Iran,207.1, Pasdaran Avenue,Tehran,Iran.School of Mathematics,Statistics Department,Cardi University,Cardi, UK.e-mail: hassanih@cf.ac.ukThe full version of the paper, in Persian, appears on pages 167-184.J. Statist. Res. Iran 2 (2005): 155-161 161

Albanian

A New Approximation for the Null Distribution of the Likelihood Ratio Test Statistics for k Upper Outliers in a Normal Sample

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
A New Approximation for the Null
Distribution of the Likelihood Ratio Test
Statistics for k Upper Outliers in a
Normal Sample
Rahim Mahmoudvand and Hossein Hassani
Payame Noor University, IRAN, Cardiff University, UK
1. September 2005
Online at http://mpra.ub.uni-muenchen.de/5778/
MPRA Paper No. 5778, posted 25. January 2008 19:32 UTC
c Statistical Research and Training Center
J. Statist. Res. Iran 2 (2005): 155-161 170164 XY '1384 u=DUtR w R}}=B '2 |xQ=tW '2 |xQwO
A New Approximation for the Null Distribution
of the Likelihood Ratio Test Statistics for k
Outliers in a Normal Sample
Rahim Mahmoudvand
y;
and Hossein Hassani
z
y
Payam-e Noor University of Toyserkan
z
Central Bank of the Islamic Republic of Iran, and Cardi University
1 Summary
Usually when performing a statistical test or estimation procedure, we assume
the data are all observations of i.i.d. random variables, often from a normal
distribution. Sometimes, however, we notice in a sample one or more obser-
vations that stand out from the crowd. These observation(s) are commonly
called outlier(s). Outlier tests are more formal procedures which have been de-
veloped for detecting outliers when a sample comes from a normal distribution
(Thode, 2002). A lot of work has been done for testing outliers in a univariate
sample, most of which corresponds to the normal and exponential distribution.
Barnett and Lewis (1994) have presented a summary of tests for outliers and
their critical values, many of which are specic to the detection of outliers in
normal samples. The theoretical solution for the exact null distribution of the
likelihood ratio test for k = 1 is solved by Barnett and Lewis (1994) and Zhang
and Yu (2006) for the case k = 2, but the problem is still open for k > 3.
In this paper we introduce a new approximation for the null distribution of
the likelihood ratio test for the general case. We compare the critical values
obtained by the new approximation to the values, which are obtained by the
exact distribution for the cases k = 1; 2 to test the accuracy of the new ap-
proximation. Also, the results are compared to another approximation method
(which is known by Barnett and Lewis (1994)) for the cases k = 3; 4.

Corresponding author
169 A New Approximation for the Null Distribution of . . .
2 The Null Distribution of T
(k)
n
Let x
1
; : : : ; x
n
be the random sample taken from the normal distribution N(;

2
) (under the null hypothesis), where both  and 
2
are unknown, and
x
(1)
; : : : ; x
(n)
be the corresponding ordered statistics. Suppose the aim is to
examine that whether x
(n k+1)
; : : : ; x
(n)
are the k upper outliers in this sam-
ple. Under this assumption the alternative hypothesis can be formulated as
x
(1)
; : : : ; x
(n k)
belong to the N(; 
2
), and x
(n k+1)
; : : : ; x
(n)
(the k upper
data) belong to the N(+ c; 
2
), where c > 0 is a constant. To answer whether
the k upper data are outliers, we can build the hypothesis H
0
: c = 0 against
H
1
: c > 0.
The likelihood ratio test statistic for testing k upper outliers (T
(k)
n
) and k
lower outliers (T
n;(k)
) are (Barnett and Lewis, 1994)
T
(k)
n
=
x
(n)
+   + x
(n k+1)
  kx
s
; T
n;(k)
=
kx  x
(1)
       x
(k)
s
; (1)
where x and s are the sample mean and the sample standard deviation, re-
spectively. Large values of T
(k)
n
reject the null hypothesis H
0
, or recognizing
that the set of ordered observations fx
(n k+1)
; : : : ; x
(n)
g as discordant obser-
vations (which are named as k outliers). Please note that (x
(n)
; : : : ; x
(1)
)
d
=
( x
(1)
; : : : ; x
(n)
) then T
(k)
n
d
=T
n;(k)
; (k = 1; : : : ; n   1). In addition, one can
see that T
(k)
n
d
=T
n;(n k)
and T
(k)
n
d
=T
(n k)
n
. So, without loss of generality, we
need just to discuss the sample distribution of T
(k)
n
.
3 Some Approximations for the Null Distribu-
tion of T
(k)
n
In this section we introduce a new approximation for the exact distribution of
T
(k)
n
. Split the index set I = f1; : : : ; ng into
[I ]
k
= fi
1
6    6 i
k
g; 1 6 i
1
<    < i
k
6 n;
and
(I)
k
= In[I ]
k
:
Set the notation
T
[I]
k
=
P
i2[I]
k
x
i
  kx
s
: (2)
156
c
 2005, SRTC Iran
Rahim Mahmoudvand and Hossein Hassani 168
By denition (2), we can rewrite T
(k)
n
= max
[I]
k
T
[I]
k
, so the distribution of
the max
[I]
k
T
[I]
k
is the upper bound for the null distribution of T
(k)
n
. It can be
simply proved that the
 
n
k

random variables T
[I]
k
have the distribution
f
T
[I]
k
(x) =
 
 
n 1
2

 
 
1
2

 
 
n 2
2

r
n
k(n  k)(n  1)

1 
n
k(n  k)(n  1)
x
2

n 2
2
 1
;
(3)
where jxj 
q
k(n k)(n 1)
n
.
3.1 Approximation I
Let A
1
; : : : ; A
m
be m arbitrary events. Based on the probability laws, one can
write
P
 
m
\
k=1
A
i
!
6 P (A
k
); k = 1; 2; : : : ;m
It is easy to show that
P

T
(k)
n
6 t

= P

max
[I]
k
T
[I]
k
6 t

= P
0
@
\
[I]
k
fT
[I]
k
6 tg
1
A
6 P
 
T
[I]
k
6 t

: (4)
Based on the Bonferroni general inequality we can write that
P
 
m
\
k=1
A
k
!
>
m
X
k=1
P (A
k
) m+ 1:
Barnet and Lewis (1994) introduced an approximation to calculate the critical
values of the null distribution of the likelihood ratio test for k > 2 based on
the above inequality, as below:
P

T
(k)
n
6 t

> m
Z
t
L
f
T
[I]
k
(x) dx m+1; L =  
r
k(n  k)(n  1)
n
; (5)
where m =
 
n
k

.
J. Statist. Res. Iran 2 (2005): 155-161 157
167 A New Approximation for the Null Distribution of . . .
3.2 Approximation II
To introduce a new approximation, we prove that T
(k)
n
are asymptotically in-
dependent. To do that, we just need to consider two conditions: the rst one
is to show that T
(k)
n
are asymptotically normally distributed and the second
one is that T
(k)
n
are asymptotically uncorrelated. The rst one (normality) is
easily obtained by taking the limit
lim
n!1
 
 
n 1
2

 
 
1
2

 
 
n 2
2

r
n
k(n  k)(n  1)

1 
n
k(n  k)(n  1)
x
2

n 2
2
 1
=
1
p
2k
exp

 
x
2
2k

:
To consider the second one, dene
 
T
[I]
k
; T
[J]
k

; (T
[S]
k
= T
fs
1
;:::;s
k
g
, for S =
I; J): By this denition, the coecient correlation between the pair
 
T
[I]
k
; T
[J]
k

is

 
T
fi
1
;:::;i
k
g
; T
fj
1
;:::;j
k
g

=
zn  k
2
kn  k
; z = 0; : : : ; k   1 (6)
where z is the number of the equal indices in the pair
 
T
[I]
k
; T
[J]
k

. Then,
lim
n!1

 
T
fi
1
;:::;i
k
g
; T
fj
1
;:::;j
k
g

=
z
k
; z = 0; : : : ; k   1 (7)
On the other hand, the frequency distribution of the discrete variable Z is
z 0 1 2 . . . k   1
f
i
 
n
k
 
k
0
 
n k
k
  
n
k
 
k
1
 
n k
k 1
  
n
k
 
k
2
 
n k
k 2

. . .
 
n
k
 
k
k 1
 
n k
1

By simple calculation, it can be shown that P (Z = 0) ! 1 as n ! 1. We
then accept that T
[I]
k
are asymptotically uncorrelated when n goes to innity.
We can now assume that T
[I]
k
are asymptotically independent. Based on the
discussion above, we introduced a new approximation for the exact distribution
of the T
[I]
k
for the general case as below.
P

T
(k)
n
6 t

= P

max
[I]
k
T
[I]
k
6 t

= P
0
@
\
[I]
k
 
T
[I]
k
6 t

1
A


Z
t
L
f
T
[I]
k
(x) dx

(
n
k
)
; k = 1; 2; : : : ; n  1 (8)
158
c
 2005, SRTC Iran
Rahim Mahmoudvand and Hossein Hassani 166
It must be mentioned that the identical distribution of the exact distribu-
tion is kept for the third approximation formulae. It is easy to show that
T
[I]
k
d
= T
[I]
n k
, and then
P

T
(n k)
n
6 t

= P

max
[I]
n k
T
[I]
n k
6 t

= P
0
@
\
[I]
n k
 
T
[I]
n k
6 t

1
A


Z
t
L
f
T
[I]
n k
(x) dx

(
n
n k
)
=

Z
t
L
f
T
[I]
k
(x) dx

(
n
k
)
; k = 1; : : : ; n  1 (9)
We then conclude that the presented approximation distribution is the same
for T
(k)
n
and T
(n k)
n
.
4 Accuracy
In this section we consider the accuracy of the new and previous approximation
(approximation I and II). At the rst step we consider the discrepancy between
the obtained critical values by approximations I and II and the exact distribu-
tion for the cases k = 1; 2. Tables 1 and 2 show some critical values for T
(k)
n
at
 = 0:01 and  = 0:05, respectively, which are obtained by dierent methods.
The bold font indicates the better approximation method which produces the
closest values to those which are obtained from the exact distribution. There
are some interesting results, as is evident from Tables 1 and 2: (a) The values
of the new and previous approximation methods are very close to the values of
the exact distribution; (b) The new approximation method is better than the
previous method. It must be emphasized also, that there is no scientic
Table 1. Critical values for T
(k)
n
at level  = 0:01 for exact distribution
n
5 10 20 30 50 100
k = 1
Exact 1.749 2.410 2.884 3.103 3.337 3.600
Approx I 1.751 2.413 2.886 3.106 3.340 3.650
Approx II 1.749 2.410 2.883 3.102 3.335 3.600
k = 2
Exact 2.160 3.402 4.437 4.946 5.497 6.118
Approx I 2.164 3.406 4.465 4.890 5.552 6.193
Approx II 2.160 3.402 4.435 4.951 5.516 6.136
J. Statist. Res. Iran 2 (2005): 155-161 159
165 A New Approximation for the Null Distribution of . . .
Table 2. Critical values for T
(k)
n
at level  = 0:05 for exact distribution
n
5 10 20 30 50 100
k = 1
Exact 1.671 2.176 2.557 2.745 2.945 3.207
Approx I 1.672 2.179 2.559 2.749 2.967 3.225
Approx II 1.670 2.173 2.553 2.743 2.995 3.207
k = 2
Exact 2.010 3.197 4.110 4.651 5.058 5.638
Approx I 2.103 3.198 4.123 4.600 5.123 5.745
Approx II 2.100 3.193 4.113 4.584 5.094 5.657
Table 3. Critical values for T
(k)
n
at level  = 0:01 by using simulation
n
10 20 30 50 100
k = 3
Simulation 3.997 5.612 6.431 7.329 8.388
Approx I 4.004 5.630 6.467 7.399 8.493
Approx II 3.998 5.614 6.451 7.388 8.474
k = 4
Simulation 4.323 6.530 7.660 8.935 10.309
Approx I 4.331 6.544 7.717 9.052 10.621
Approx II 4.323 6.529 7.700 9.035 10.599
Table 4. Critical values for T
(k)
n
at level  = 0:05 by using simulation
n
10 20 30 50 100
k = 3
Simulation 3.813 5.311 6.051 6.871 7.855
Approx I 3.818 5.321 6.111 6.999 8.056
Approx II 3.814 5.314 6.100 6.992 8.044
k = 4
Simulation 4.155 6.249 7.235 8.408 9.772
Approx I 4.161 6.261 7.379 8.661 10.182
Approx II 4.155 6.253 7.370 8.651 10.172
discrepancy between the two approximations (I and II) for the small value of n,
but for the large value of n the new approximation is better than the previous
one; (c) The accuracy for the case k = 1 is more than for the case k = 2;
because, for the case k = 1 the coecient correlation is more closer to zero
160
c
 2005, SRTC Iran
Rahim Mahmoudvand and Hossein Hassani 164
than for k = 2, and thus converges to zero faster; (d) The accuracy of the
approximation methods reduce when the  increases, and the approximation I
does not work for some value of the .
At the second step, we consider the accuracy of the approximation meth-
ods by the critical values which are obtained for the cases k = 3; 4. Tables
3 and 4 show the results. For simulation part, we used S-Plus software and
the number 10,000 for simulation. As it appears from Tables 3 and 4, the new
approximation is again better than the previous approximation. Also it must
be mentioned that conditions (a)-(d) hold for these tables as well.
Keywords. outlier; normal sample; likelihood ratio test; approximation.
References
Barnett, V.; Lewis, T. (1994). Outliers in Statistical Data, 3rd ed. Wiley, Chichester.
Thode, H.C. (2002). Testing for Normality. Marcel Dekker, New York.
Zhang, J.; Yu, k. (2006). The null distribution of the likelihood-ratio test for one or two
outliers in a normal sample. Test 15, 141-150.
Rahim Mahmoudvand
Statistics Department,
Payam-e Noor University of Toyserkan,
Toyserkan,
Iran.
e-mail: r
{
mahmodvand@yahoo.com
Hossein Hassani
Centeral Bank of the Islamic Republic of Iran,
207.1, Pasdaran Avenue,
Tehran,
Iran.
School of Mathematics,
Statistics Department,
Cardi University,
Cardi, UK.
e-mail: hassanih@cf.ac.uk
The full version of the paper, in Persian, appears on pages 167-184.
J. Statist. Res. Iran 2 (2005): 155-161 161


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A New Approximation for the Null Distribution of the Likelihood Ratio Test Statistics for k Upper Outliers in a Normal Sample

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