International Journal of Scientific & Engineering Research, Volume 6, Issue 5, May-2015 1889

ISSN 2229-5518

MULTI-SAMPLE ADAPTIVE TEST AND OTHER COMPETITATORS IN LOCATION PROBLEM

1.1 Introduction:

If the parametric assumptions are fulfilled, classical F test is the appropriate test for the multi sample location problem. On the other hand, if assumptions not satisfies ,F-test is not a suitable tests. So we have to search for some other tests. Test based on ranks or scores are found to be more powerful and robust in many situations. It is seen that many of the practicing statisticians have no idea regarding the data type. In this situations, the adaptive tests based on Hogg’s concept , may help the statistician to identify data type with respect to some measures like skewness and tail weight and then to select an appropriate rank test or test based on scores for classified type of distribution. In this chapter, we have first discussed the adaptive test procedure that are used to select an appropriate test and then compare an adaptive test and some of the tests procedure s with the help of Monte Carlo simulation technique. Both empirical level and power of these tests are calculated and comparison are made with F-test and different other adaptive tests. We have observed that adaptive tests behave well in broad class of distributions.

1.2 Selection Statistics:

Here we will use a selection statistics S =(Q1 ,Q2 ), where Q1 and Q2 are Hoggs measure of skewness and tailweight defined by –
Q1 =

𝑈�5%−𝑀� 50%

𝑀� 50% −𝐿� 5%

and Q2 =

𝑈�5% −𝐿�5%

𝑈�50% −𝐿�50%

Where , 𝑈�5% , 𝑀�50% and 𝐿�5% are the averages of the upper 5% , middle 50% and lower 5%
of the order statistics of the combined sample. 𝑈�50% and 𝐿�50% are the averages of the upper
50% and lower 50% of the order statistics of the combined sample
Table 1.1: Theoretical values of Q1 and Q2 for some selected distributions-

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Now let us define four categories of S- D1 = {S/0≤Q1≤ R2, 1≤Q2 ≤2}
D2 = {S/0≤Q1 ≤2; 2≤Q2 ≤3}
D 3 = {S/ Q 1 ≥0; Q2 >3}
D 4 = {S/ Q1 >2; 1≤Q2 ≤3}
This means that the distribution is short or medium tails if S falls in the
Category D1 or D2 respectively; long tail if S falls in the category D3 and right skewed tail if it falls in the category D4
Buning (1996) proposed the following adaptive test A :
𝐺 𝑖𝑓 𝑆 ∈ 𝐷1
𝐴 = � 𝐾𝑊 𝑖𝑓 𝑆 ∈ 𝐷2
𝐿𝑇 𝑖𝑓 𝑆 ∈ 𝐷3
𝐻𝐹𝑅 𝑖𝑓 𝑆 ∈ 𝐷4

1.3 Test Procedures :

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Let 𝑋𝑖1 , 𝑋𝑖2 , … , 𝑋𝑖𝑛𝑖 , i =1,2,…,c be independent random variables with absolutely
continuous distribution function F(x-𝜃R i).
Here the null hypothesis H0 :𝜃1 = 𝜃2 = ⋯ = 𝜃𝑐
Against the alternative hypothesis H1 :𝜃𝑟 ≠ 𝜃𝑠 for at least one pair (r,s), r≠s.

1.3.1 F-Test:

For normally distributed random samples with equal variance, in testing equality of means the likelihood ratio F test is the best one. The test statistics defined as
F = (𝑁−𝑐) ∑𝑖=1

𝑛𝑖 (�𝑋�𝚤−𝑋� )2

∑ ∑𝑛

−𝑋 )

(𝐶−1)

𝐶

𝑖=1

𝑖 (𝑋𝑖𝑗

� �𝚤 2

where N=∑𝑐
𝑛𝑖 , 𝑋�
= 1
∑𝑛𝑖 𝑋
, 𝑋� =

1 ∑𝑐

𝑛𝑖 𝑋�𝑖

𝑖=1

𝑖 𝑛𝑖

𝑗=1

𝑖𝑗

𝑁 𝑖=1

Under H0 the test statistics follows F distribution with c-1 and N-c degrees of freedom.

1.3.2 Kruskal-Wallis(KW) test:

Let Rij be the rank of the observation xij in the pooled sample. The Kruskal- Wallis test for two-sided alternative which based on the statistic

KW =

12 ∑ 1 R

− ni ( N + 1) 

N ( N + 1) i =1 ni  2 

KW = 12
∑𝑐

1 [𝑅

- 𝑛𝑖 (𝑁+1) ]2

𝑁(𝑁+!)

𝑖=1 𝑛𝑖

𝑖 2 P

= 12
∑𝑐

2

𝑖 − 3(𝑁 + 1)

𝑁(𝑁+1)

𝑖=1 𝑛𝑖

where

ni

Ri = ∑ Rij

and N =∑𝑖=1 𝑛𝑖 .

j =1

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For sample size ni and sample number k large , H o is rejected if KW> χ 2
. When k is
small and ni are small then exact distribution table of KW can be used.
Now let us define linear rank statistics. Let us consider a the combined ordered sample 𝑋(1),𝑋(2),…. 𝑋(𝑁) of 𝑋11, … 𝑋1𝑛1,….𝑋𝑐1, … 𝑋𝑐𝑛𝑐 and indicator variables Vik given by
𝑉 = 1 𝑖𝑓 𝑋(𝑘) 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝑡ℎ𝑒 𝑖𝑡ℎ 𝑠𝑎𝑚𝑝𝑙𝑒 , 𝑖 = 1, … , 𝑐, 𝑘 = 1, … , 𝑁
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
where N=∑𝑐
𝑛𝑖 ,
Let a(k) , k =1,2,…,N be real valued score with mean 𝑎� = 1 ∑𝑁
𝑎(𝑘). now we define for

𝑁 𝑘=1

each sample a statistics Ai in the following way:
Ai =

1

𝑛𝑖

𝑁

𝑘=1

𝑎(𝑘)𝑉𝑖𝑘, 1≤i≤c
Then the linear rank statistics LN is given by
LN =

(𝑁−1) ∑𝑐

𝑛𝑖 (𝐴𝑖−𝑎�)2

∑𝑁

(𝑎(𝑘)−𝑎�)2

𝑘=1

Under H0 ,LN is distribution free and follows asymptotically chi- square distribution with c-1 degrees of freedom .
Some of the scores to obtain more powerful test for types of distribution according to
Buning(1991,1994) are as follows:

1.3.3 Gastwrith test G (short tails ):

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⎧ 𝑘 −
⎪

(𝑁 + 1)
4
𝑖𝑓 𝑘 ≤

(𝑁 + 1)
4
𝑎𝐺 (𝑘) =
0 𝑖𝑓
⎨

(𝑁 + 1)
4
≤ 𝑘 ≤

3(𝑁 + 1)
4
⎪
⎩𝑘 −

3(𝑁 + 1)
4 𝑖𝑓 𝑘 ≥

3(𝑁 + 1)
4

1.3.4 Kruskal wallis test KW (medium tails):

If 𝑎𝐾𝑊 (𝑘) = 𝑘, test transform to above KW test

1.3.5 Test LT (long tails ):

𝑁
⎧ −(�
⎪ 4

𝑁
� + 1 𝑖𝑓 𝑘 < � 4
� + 1
𝑎𝐿𝑇 =
𝑘 −
⎨

(𝑁 + 1)
2

𝑁
𝑖𝑓 � 4
� + 1 ≤ 𝑘 ≤ [

3(𝑁 + 1)
4 ]
⎪ 𝑁
3(𝑁 + 1)


⎩ � 4 � + 1 𝑖𝑓 𝑘 > [ 4 ]

3.3.6 Hogg Fisher Randles test HFR(right skewed ):

𝑎𝐻𝐹𝑅 = �
𝑘 −

(𝑁 + 1)
2
𝑖𝑓 𝑘 ≤

(𝑁 + 1)
2
(𝑁 + 1)

0 𝑖𝑓 𝑘 > 2
For left-skewed distributions we change the terms k- (N+1)/2 and 0 in the definition of the scores above.

1.4 Monte Carlo simulation:

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We investigate the power of the tests via Monte Carlo simulation. For this purpose we have repeated 10000 times. The criteria of the test comparisons are the level 𝛼 and the
power 𝛽 of the tests. The concept of 𝛼 robustness can be defined as follows. For a nominal level 𝛼 and underlying distribution function F, the critical region 𝐶𝛼 of a statistic T may be uniquely determined by 𝑃𝐻0 (𝑇 ∈ 𝐶𝛼 ! 𝐹) = 𝛼. We now assume a distribution function for the data and determine the actual level 𝛼 ∗ of the test, i.e. 𝛼 ∗ = 𝑃𝐻0 (𝑇 ∈ 𝐶𝛼 ! 𝐺), T is then called
‘𝛼 − 𝑟𝑜𝑏𝑢𝑠𝑡’ if ! 𝛼 − 𝛼 ∗ ! is small. In case of 𝛼 ∗ ≤ 𝛼, we call the test conservative;
otherwise , it is anticonservative.
The selected distributions for the robustness and power study are Normal, Logistic, Cauchy, Lognormal, Double Exponential, Exponential and Uniform. We consider cases of three samples and Four samples with sample size combinations (10,10,10),(10,15,20), (10,10,10,10) and (10,15,20,25). Various combinations of location parameters are considered which are shown in respective Tables. For generating the samples from the normal distribution formula given by Hammersley and Hanscomb(1964 ) is used and for other distributions we have used method inverse integration. Necessary modification are made in generated sample to represent the location shift.

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Table 1.2 Empirical Level and power of tests under Normal distribution:

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Fig-1.1 Empirical power of tests under Normal distribution for n1 =n2 =n 3 =10

at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Normal Dist.

Variable

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig-1.2 Empirical power of tests under Normal distribution for n1 =10, n2 =15,

n3 =20, n4 =25 at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Normal Dist.

Variable

F H G LT

HF R

0.0

(-.1,1,-.2,.2)

(-.5,.5,-1,1)

(-1,1,-1.5,1.5)

means

(-1.5,1.5,-2,2)

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Table 1.3 Empirical Level and power of tests under Cauchy distribution:

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Fig 1.3 Empirical power of tests under Cauchy distribution for

n1 =n2 =n 3 =10 at 5% level:

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Empirical power of tests under Cauchy dist.

Variable

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig.1.4Empirical power of tests under Cauchy distribution for n1 =10, n 2 =15,

n3 =20, n4 =25 at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical powerof tests under Cauchy dist.

Variable

F H G LT

HF R

0.0

(-.1,.1,-.2,.2)

(-.5,.5,-1,1)

(-1,1,-1.5,1.5)

means

(-1.5,1.5,-2,2)

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Table 1.4 Empirical Level and power of tests under Logistic distribution:

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Fig. 1.5 Empirical power of tests under Logistic distribution for n1 =n2 =n3 =10

at 5% level:

Empirical power of tests under Logistic dist.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Variable

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig. 1.6 Empirical power of tests under Logistic distribution for n1 =10, n2 =15,

n3 =20, n4 =25 at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Logistic distribution

Variable

F H G LT

HF R

0.0

(-.1,.1,-.2,.2)

(-.5,.5,-1,1)

(-1,1,-1.5,1.5)

means

(-1.5,1.5,-2,2)

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Table 1.5 Empirical Level and power of tests under Lognormal distribution:

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Fig. 1.7 Empirical power of tests under Lognormal distribution for n1 =n2 =n 3 =10

at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Lognormal distribution

Variable

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig. 1.8 Empirical power of tests under Lognormal test for n1 =10, n2 =15, n3 =20,

n4 =25 at 5% level:

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Empirical power of tests under Lognormal distribution

Variable

F H G LT

HF R

0.1

1

2 3 4

means

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Table 1.6 Empirical Level and power of tests under Exponential distribution:

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Fig. 1.9 Empirical power of tests under Exponential distribution for n1 =n 2 =n3 =10

at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Exponential dist

Variable

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig. 1.10 Empirical power of tests under Exponential test for n1 =10, n2 =15, n3 =20,

n4 =25 at 5% level:

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

Empirical power of tests under Exponenetial dist

Variable

F H G LT

HF R

0.1

(-.1,.1,-.2,.2)

(-.5,.5,-1,1)

(-1,1,-1.5,1.5)

means

(-1.5,1.5,-2,2)

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Table 1.7 Empirical Level and power of tests under Double Exponential

Distribution:

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Fig.1.11 Empirical power of tests under Double Exponential distribution for

n1 =n2 =n3 =10 at 5% level:

Empirical power of tests under Double Expo. dist

1.0

0.8

0.6

0.4

0.2

Variab le

F H G LT

HF R

0.0

(0,-.2,.2)

(0,-.5,.5)

means

(0,-1,1)

(0,-1.5,1.5)

Fig.1.12 Empirical power of tests under Double Exponential distribution for n1 =10,

n2 =15, n3 =20, n4 =25 at 5% level:

1.0

0.8

0.6

0.4

0.2

Empirical power of tests under Double Expo. dist

Variable

F H G LT

HF R

0.0

(-.1,.1,-.2,.2)

(-.5,.5,-1,1)

(-1,1,-1.5,1.5)

means

(-1.5,1.5,-2,2)

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1.5 Discussion:

From Table 3.2 it is seen that empirical level of almost all tests satisfies the nominal level. In case of power , F- test seems to be more powerful than other tests followed by Kruskal –Wallis test. But as the location shift , sample size increases power of all the tests going to be almost equal.
In case Cauchy distribution, only Kruskal-Wallis and Long-tail test satisfies the nominal levels under the null situations. F-test not at all satisfies the nominal level. However,G, HFR slightly better than F-test. Power of Long –tail test (LT) seems to be the highest of all the tests discussed here.
Table 3.4 shows the empirical level and power of six tests under logistic distribution. Here we have observe that all the test satisfies the nominal levels. It is seen that power of F-test and Kruskal –Wallis tests are almost similar. However, KW tests are slightly higher in some situations. Out of three score tests, power LT test is higher than other two tests.
Table 3.5 shows the empirical level and power of tests under lognormal distribution. It is seen that except F –test all other tests satisfy the nominal level approximately. Here we have seen that power of HFR test is more than other tests. Power of F and G test are found to be less than other tests.
Table 3.6 shows empirical levels and powers of tests under exponential distribution . Here we have found similar results as lognormal distribution. Since both are right-skewed distribution that why we get similar results.
From 3.7 , we have found the empirical levels and power of the six tests. It is
observe that except G test, empirical level of other test are closed to nominal levels. It is also

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clear that power of LT test is the highest of all , followed by KW and F test and HFR
respectively.

3.6 Conclusions:

From the above results we can conclude that ,F test is suitable for the normal distribution. For log tailed distribution LT test and H test is more preferable than other test. G test is suitable for short tailed distribution and HFR test is preferable for the right-skewed distribution. From these results it is clear that prior information regarding the observation distribution help in choosing the appropriate test. So, adaptive procedure certainly help the practioner for appropriate test selection and help to arrive at right conclusion.

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