Narinder Wolfgang Koessler

Wolfgang Kössler

N Kumar
email: nkumar67@yahoo.com

An adaptive test for the two-sample scale problem based on U-statistics

Keywords: Mood-test, Klotz-test, asymptotic efficacy, asymptotic power, simulation study, tailweight, skewness Mood-test, Klotz-test, asymptotic efficacy, asymptotic power, simulation study, tailweight, skewness. Subject classification: 62G10, 62G20

An adaptive test for the two-sample scale problem based on U-statistics

based on U-statistics. Our adaptive test is compared briefly with adaptive tests based on linear rank tests in Section 5. In Section 6 a simulation study is performed and the finite sample power is compared with the asymptotic power. A data example is provided in Section 7. We give some conclusions in Section 8.

Test statistics

We consider the class of U-statistics, which was proposed by [START_REF] Kumar | A New Class of Distribution-Free Tests for Location parameters[END_REF]. Let k, 1 ≤ k ≤ min(n 1 , n 2 ), and i, i ≤ k+1 2 , be fixed integers. Define Φ i (x 1 , . . . , x k , y 1 , . . . , y k ) =

     1 if x (i)k < y (i)k and x (k-i+1)k < y (k-i+1)k -1 if x (i)k > y (i)k and x (k-i+1)k > y (k-i+1)k 0 otherwise,
where x (i)k is the ith order statistic in a subsample of size k from the Xsample (and likewise for y's). Let U k:i be the U-statistic associated with kernel Φ i , i.e.

U k:i = n 1 n 2 n 1 k • n 2 k Φ i (X r 1 , . . . , X r k , Y s 1 , . . . , Y s k ),
where the summation extends over all possible combinations (r 1 , . . . , r k ) of k integers from {1, . . . , n 1 } and all possible combinations (s 1 , . . . , s k ) of k integers from {1, . . . , n 2 }. The null hypothesis H 0 is rejected in favour of H 1 for large values of U k:i .

Remark: For i = 1 or i = k we have the Kochar-Gupta test (cf. [START_REF] Kochar | .ca Communications in Statistics -Simulation and Computation two-sample scale problem[END_REF].

There is also a rank representation of the U k:i . Let R (s) be the rank of Y (s) in the joint ranking of the X-sample and Y -sample, where Y (s) is the sth order statistics of the Y -sample. Then 1,0 (x) = EΦ i (x, X 2 , . . . , X k , Y 1 , . . . , Y k ) ϕ

n 1 k n 2 k n 1 n 2 U k:i = n 2 s=i k j=i s -1 i -1 n 2 -s k -i R (s) -s j n 1 + s -R (s) k -j - n 2 s=k-i+1 k j=k-i+1 s -1 k -i n 2 -s i -1 R (s) -s j n 1 + s -R (s) k -j .
(i) 0,1 (y) = EΦ i (X 1 , . . . , X k , y, Y 2 , . . . , Y k ) ζ (i)
1,0 = Var(ϕ

(i) 1,0 (X)) ζ (i) 0,1 = Var(ϕ (i) 0,1 (Y )),
where E and Var denote the expectation and variance respectively. Moreover, let F (i)k (.) be the cumulative distribution function of the ith order statistics of a sample of size k.

Proposition 2.1 (cf. [START_REF] Kumar | A New Class of Distribution-Free Tests for Location parameters[END_REF] Under assumptions N → ∞, n 1 /N → λ, 0 < λ < 1 the limiting distribution of N 1/2 (U k:i -η k,i )/σ k,i is standard normal, where expectation η k,i = EU k:i and variance σ 2 k,i =Var (U k:i ) have the forms

η k,i = n 1 n 2 i k i 2 ∞ -∞ y(F (y)) 2i-2 (1 -F (y)) 2k-2i f 2 (y) dy - ∞ -∞
y(F (y)) 2k-2i (1 -F (y) 2i-2 f 2 (y) dy

σ 2 k,i = n 2 1 n 2 2 k 2 ζ (i) 10 λ + k 2 ζ (i) 01
1 -λ .

Remark: Under H 0 we have η k,i = 0 and

σ 2 k,i = n 2 1 n 2 2 k 2 ρ k,i λ(1 -λ) ,
where ρ k,i depends on k and i only. The expression for ρ k,i is rather long, that is why we do not write it out. It can be found in [START_REF] Kumar | A New Class of Distribution-Free Tests for Location parameters[END_REF].

The asymptotic efficacies

The asymptotic (Pitman-) efficacies AE of the statistics U k:i under the alternative θ N = N -1/2 • θ are given by where f (•) denotes the probability density function belonging to the c.d.f. F (•) and

AE(U k:i |f ) = λ(1 -λ) • C 2 k,i (f ),
C k,i (f ) = ( k i i) 2 (k 2 ρ k,i ) 1/2 • ∞ -∞ x(F (x)) 2i-2 (1 -F (x)) 2k-2i f 2 (x) dx- ∞ -∞ x(F (x)) 2k-2i (1 -F (x)) 2i-2 f 2 (x) dx .
Note that the asymptotic efficacy is defined by the limit of η 2 k,i /σ 2 k,i , cf. [START_REF] Noether | On a Theorem of Pitman[END_REF]. Since only the factor C k,i (f ) is of interest here we refer to this factor if we say AE.

Remark: It may be shown that the Mood test is asymptotically equivalent to the test U 2:1 .

Before we continue with the investigation of our class of U-statistics, we will compare the asymptotic efficacies of our tests with that of other tests based on U-statistics. First, let us note that the statistics SJ(2k-1, r, 1) with r = 1 of Joshi and Shetty are asymptotically equivalent to our statistics U k:1 . (Joshi and Shetty consider only the case r = 1.) In Table 1 we find the AE of the tests of [START_REF] Sukhatme | A two-sample distribution-free test for comparing variances[END_REF], [START_REF] Kusum | A new distribution-free test for the two-sample scale problem[END_REF], three of the best variants of [START_REF] Joshi | A note on Rosenbaum's test for twosample scale problem[END_REF] (abbreviated by SJ(5, 1), SJ(9, 1), SJ(5, 2)) as well as of the tests U 8:1 , U 2:1 and U 5:2 for five densities, the uniform, logistic, Cauchy, DE, and normal. The AE of the best of these eight tests is written in bold style, respectively. Missing entries are not relevant or not computed. Most computations are done with the Mathematica package, few could be done analytically. From Table 1 we see that Sukhatme, Kusum, SJ(5,1) and SJ(5,2) are in no cases the best tests. Recall that SJ(9, 1) is asymptotically equivalent to U 5:1 .

Table 1 about here

For further investigations we have to make some restrictions. We restricted the choice of k at first to k ≤ 6 ourselves just for illustrative purpose. Of course, computations can be made for values of k beyond that.

We compute the asymptotic Pitman efficacies for all tests the U-L by Gastwirth (1965), the RST is named after Ramberg, Schmeiser and Tukey, cf. Ramberg andSchmeiser (1972, 1974), CN( , σ) is the scale contaminated normal with contaminating proportion , and the skew-logistic is proposed by [START_REF] Nadarajah | The skew-logistic distribution[END_REF].

U k:i with 1 ≤ i ≤ k 2 , k ≤ 6. Obviously, for i = k+1 2 we have C k,i = 0.
The bold entries denote, for the given density, the asymptotically best test among the considered tests. (If the Klotz-test is the asymptotically best, it is denoted in italics.) If no test is bold, then one of the tests U 7:1 , U 8:1 , or U 9:1 has larger AE-values (see below), and the corresponding entry is bold in Table 3.

On the first view we see that the columns for U 8:1 , U 5:1 and U 2:1 have the most bold entries. This observation gives rise to the idea to use these few statistics in our adaptive test.

To get a closer idea how to classify symmetric densities we apply the method of Hall and Joiner (1982). The content of information in the asymptotic efficacy matrix is analysed by a principal component analysis where the densities are the observations and the efficacies of the U k:i are the variables. The first principal component explains already 98% of the variability. For better visibility we display in Figure 1 the values of the first two principal components (Factors one and two). Nearly symmetric densities with short tails are denoted by a green plus, those with short-medium and long-medium tails by a cyan X and a blue star, respectively, and that with very long tails with a red dot.

On the left side we have densities with long tails, in the centre that with medium tails, and on the right that with short tails. For an exact definition what we understand by very long, (long-and short-) medium and short tails see below. On the first view we see that the AE(U k:i ) classify the densities according to their tailweight.

Since very often the tests U 6:1 and U 5:1 are asymptotically the best it might be desirable to investigate also the tests U 7:1 , U 8:1 , and U 9:1 . From this investigation we found that it might be a good idea to choose the test U 8:1 for short tail densities. AE values for the test U 8:1 are, togehter with that of U 5:1 given in Table 3.

When we consider skew densities we have to differ between densities defined on the whole real line (such as the Gumbel or the skew-logistic) and that starting at some point, e.g. at zero (as it is, e.g. for time data). Skew densities of the former group may be included in the scheme designed for symmetric densities. For the latter the considered U-tests are bad (cf. [START_REF] Kössler | An adaptive test for the two-sample location problem based on U-statistics[END_REF]. In Table 4 we included the Mann-Whitney test, denoted by U 1,1 . The location U-test U 2,1 of [START_REF] Kössler | An adaptive test for the two-sample location problem based on U-statistics[END_REF] seems to be asymptotically slightly better than the Mann-Whitney test, but for convenience of the reader we preferred the latter test here.

Insert Figure 1 about here 4 Adaptive test At first we have to decide whether we have a density with some starting point (which is generally zero) or not. In the first case we apply a location test on the data, such as the Mann-Whitney-Wilcoxon test. Alternatively we may perform a logarithmic transformation and then apply an adaptive location test, e.g. that of [START_REF] Kössler | An adaptive test for the two-sample location problem based on U-statistics[END_REF] which is also based on U-statistics.

Assume now that we have a symmetric density or a skew density that is defined on the whole real line. We apply the concept of [START_REF] Hogg | Adaptive robust procedures: partial review and some suggestions for future applications and theory[END_REF], that is, to classify at first the type of the underlying density with respect to one measure of tailweight t, which is defined by

t = Q(0.95) -Q(0.05) Q(0.85) -Q(0.15) (1)
where Q(u) is the so-called classical quantile estimate of F -1 (u),

Q(u) =      X (1) -(1 -δ) • (X (2) -X (1) ) if u < 1/(2 • N ) (1 -δ) • X (j) + δ • X (j+1) if 1 2•N ≤ u ≤ 2•N -1 2•N X (N ) + δ(X (N ) -X (N -1) ) if u > (2 • N -1)/(2 • N ), (2)
where δ = N • u + 1/2 -j and j = N • u + 1/2 . In Table 5, together with the AE of the adaptive tests, the values of the corresponding theoretical measures t for various selected densities are presented.

Comparing Tables 2 and3 with the tailweight measures roughly we see that the tests U 4:1 , U 5:1 and U 6:1 are asymptotically good tests for symmetric densities with small to medium tailweight, U 3:1 and U 2:1 for symmetric densities with longer tailweight, U 4:2 , U 5:2 , U 6:2 and U 6:3 for symmetric densities with very large tailweight.

From Table 3 we see that for medium tails the tests U 5:1 or U 6:1 are the best choices whereas for short tails the tests U 8:1 and U 9:1 are the best.

The reasoning of the last two sections gives rise to the following adaptive test.

Define regions E 1 , . . . , E 4 which are based on the selector statistic t which is given by ( 1),

E 1 = { t < 1.61} "short tails" E 2 = {1.61 ≤ t < 1.93} "light medium tails" E 3 = {1.93 ≤ t ≤ 2.5} "long tails" E 4 = { t > 2.5}
"very long tails" The cutoff values of the regions are determined in such a way that the vast majority of densities is classified correctly, i.e. they fall in the class that has the highest asymptotic power (cf. Tables 2 and3 with Table 5). For example, the normal (tailweight t=1.59, cf. Table 5) is classified to E 1 , and the test U 8:1 , which is the best among the considered tests (cf. Tables 2 and3), is performed. The logistic is mapped to region E 2 and the test U 5:1 is performed. Similar observations for the other densities lead to the given cutoff values. In few cases, if the classification doesn't be correct, then the efficacy loss is very small in almost all cases. In Tables 2 and3 the chosen test is underlined if it is not already the (bold) best. Now, we propose the Adaptive test A which is based on the four Ustatistics U 8:1 , U 5:1 , U 2:1 , and U 5,2 . We denote the tests by (8:1), (5:1), (2:1) and (5:2), respectively.

A = A( t) =          (8 : 1) if Ŝ ∈ E 1 (5 : 1) if Ŝ ∈ E 2 (2 : 1) if Ŝ ∈ E 3 (5 : 2) if Ŝ ∈ E 4 (3)
In Figure 2 the corresponding adaptive scheme is given.

Insert Figure 2 The two-stage procedure defined above is asymptotically distribution-free since the selector statistic t is based on the order statistic only and the Ustatistics are based on the ranks only.

The Adaptive test A is only asymptotically distribution-free because asymptotic critical values are used in the adaptive scheme.

Proposition 4.1 Let {θ N } be a sequence of 'near' alternatives with √ N θ N → θ. The asymptotic power function of the Adaptive test A equals

β(θ) =          1 -Φ(z 1-α -AE(U 8:1 |f ) • θ) if f ∈ E 1 1 -Φ(z 1-α -AE(U 5:1 |f ) • θ) if f ∈ E 2 1 -Φ(z 1-α -AE(U 2:1 |f ) • θ) if f ∈ E 3 1 -Φ(z 1-α -AE(U 5:2 |f ) • θ) if f ∈ E 4 Proof. Let be h = 1 if (k : i) = (8 : 1), h = 2 if (k, i) = (5 : 1), h = 3 if (k, i) = (2 : 1), h = 4 if (k, i) = (5 : 2) Let be T 1 = U 8:1 , T 2 = U 5:1 , T 3 = U 2:1 , T 4 = U 5:2 .
The proposition follows from the total probability theorem and from the consistency of the selector statistics, i.e.

β(θ) = 4 h=1 P θ (T h > c αh |T h chosen) • P θ (T h chosen) = 4 h=1 1 -Φ(z 1-α -AE(T h |f ) • θ) + o(1) • 1 + o(1) if f ∈ E h o(1) else ∼ 1 -Φ(z 1-α -AE(T h |f ) • θ) if f ∈ E h ,
where c αh is the (1 -α)-quantile of the asymptotic null distribution of T h .

The Adaptive test A( t) is based on selector statistics computed from the pooled sample. However, location differences may effect the estimates of tailweight and skewness. That is why we also consider a modification A( t * ) of the adaptive test, where tailweights are estimated from the single samples. Let ti , i = 1, 2 be statistics of the form (1) for tailweight. Applying the A( t * )-test the selector statistic t * ) with

t * = n 1 N t1 + n 2 N t2
is used instead of t. This procedure is also asymptotically distribution-free. However, it is not distribution-free also if the exact critical values are used. This property is since the selector statistic is no longer based on the pure order statistic. For various densities asymptotic power functions (together with finite power functions) are given in Figures 3 and4.

The blue dotted line is for U 8:1 , the violet short-dashed line for U 5:1 , the green long-dashed line for U 2:1 , the red dashed-dotted line for U 5:2 (and the black continuous line for the adaptive test).

Comparison to adaptive tests based on linear rank statistics

Restrictive adaptive tests for the two-sample scale problem based on linear rank tests are proposed by Rünstler (1987) and [START_REF] Kössler | Restrictive adaptive tests for the treatment of the twosample scale problem[END_REF], as well as by [START_REF] Hall | Adaptive inference for the two-sample scale problem[END_REF], cf. also [START_REF] Büning | Robuste und adaptive Tests[END_REF]. All of them are based on the concept of [START_REF] Hogg | Adaptive robust procedures: partial review and some suggestions for future applications and theory[END_REF], and they use few linear rank statistics, with some scores. Rünstler (1987) considered only symmetric densities, and he used the Gastwirth scores (for short tails), Mood scores (for medium tails), and Cauchy scores (for long tails). [START_REF] Kössler | Restrictive adaptive tests for the treatment of the twosample scale problem[END_REF] used the Klotz scores (for symmetric densities with short tails), logistic scores (for symmetric densities with medium tails), Ansari-Bradley scores (for long-tail densities), the Savage and Wilcoxon scores for skew densities with short or medium tails, respectively. [START_REF] Hall | Adaptive inference for the two-sample scale problem[END_REF] performed a transformation to handle unequal locations and applied the Klotz-test and the Wilcoxon test to the transformed data (their Adaptive procedure II). For the estimation of tailweight Rünstler (1987) and [START_REF] Hall | Adaptive inference for the two-sample scale problem[END_REF] used integral measures whereas [START_REF] Kössler | Restrictive adaptive tests for the treatment of the twosample scale problem[END_REF] used quantile measures for his tailweight and skewness estimates.

Let us denote the adaptive test of [START_REF] Kössler | Restrictive adaptive tests for the treatment of the twosample scale problem[END_REF] by B( Ŝ), where Ŝ = ( t, ŝ) and ŝ is a suitable skewness estimator. In Table 5 we compare, for each density considered, the AE of the test (asymptotically) chosen by the Adaptive tests A( t) and B( Ŝ). In that table the respective larger AE-value is underlined.

For most of the classical densities considered the Adaptive test B( Ŝ) based on linear rank statistics has slightly higher asymptotic power than the new test A( t). The same is true for the densities of the L-DE family (not included in Table 5). For densities with larger tails and for the majority of the contaminated normal densities the U-statistics based test A( t) is asymptotically better. However, the differences in asymptotic efficacies are small in most cases. For the contaminated normal and for the very skew densities defined on the whole real line the new adaptive test is clearly better.

Simulation study

In order to assess whether the asymptotic theory can also be applied for medium to small sample sizes a simulation study (10,000 replications each for the null case, 1,000 replications each for the alternative cases) is performed. We choose the following six distributions:

-Uniform distribution (density with small tailweight), -Normal distribution (density with medium tailweight), -Logistic distribution (density with medium tailweight), -Doubleexponential distribution (density with large tailweight), -Cauchy distribution (density with very large tailweight), -Gumbel distribution (skew density)

We consider the four single U-tests U 8,1 , U 5,1 , U 2,1 , U 5,2 , and the Adaptive test A( t). (Results for the Klotz-test may be obtained, e.g. from [START_REF] Kössler | Restrictive adaptive tests for the treatment of the twosample scale problem[END_REF], they are in accordance with the asymptotic results.) The sample sizes n 1 = n 2 = 10, 20, 40, 100 and the alternatives θ N = N -1/2 θ with various θ are considered. Estimated levels of significance are summarized in Table 6 for the uniform density. For the other densities we get very similar values. The test U 8:1 has the largest levels. That is why we have, for short-tail densities, for the adaptive tests also relatively large values. For densities with longer tails these values are slightly smaller (about 0.02 for n 1 = n 2 = 10 and 0.01 for n 1 = n 2 = 40).

Table 6 about here

As we see from Table 6 all the tests are, for n 1 = n 2 ≤ 100 slightly anticonservative. For n 1 = n 2 = 100 the level is almost always less than 0.053, and therefore = 0.1-robust in almost all cases. (For the notion of -robustness see e.g. [START_REF] Rasch | How robust are tests for two independent samples[END_REF] For n 1 = n 2 = 10, 20, 40 almost all tests are more or less anticonservative, for n 1 = n 2 = 10 the estimated attained level of significance is always in (0.085,0.125), for n 1 = n 2 = 20 it is always in (0.063,0.085), and for n 1 = n 2 = 40 it is always in (0.053,0.068), For n 1 = n 2 = 100 the results of the simulation study are summarized in Figures 3 and4. Again, the blue dotted line is for U 8:1 , the violet short-dashed line for U 5:1 , the green long-dashed line for U 2:1 , the red dashed-dotted line for U 5:2 and the continuous line is for the Adaptive test A( Ŝ). At first we see that, for n 1 = n 2 = 100, the finite power is well approximated by the asymptotic power. Moreover, it can be seen that, for a given density, there is always, sometimes together with another test, a single test which is the best. The test U 8:1 is the best for the uniform and for the normal (together with the Adaptive test A( t)) the test U 5:1 is the best for the logistic density (together with the Adaptive test A( t)), and the test U 5:2 is the best for the double exponential and for the Cauchy (again together with the Adaptive test A( t)). All these facts are not surprising. Also, not surprisingly, that all the tests, except the adaptive test, may be bad for some densities. The Adaptive test A( t) is, over all densities, the best.

Although the tests are slightly anticonservative for smaller sample sizes the order of the tests is almost the same as in the asymptotic case. For small sample sizes, n 1 = n 2 ≤ 20 the adaptive test has slightly lower power as expected because of the higher misclassification probabilities.

Insert Figures 3 and 4 about here 7 Data example

In this section we give a data example to illustrate our adaptive procedure.

Example (cf. Example 9 of [START_REF] Büning | Robuste und adaptive Tests[END_REF]. In a statistics lesson variances of median and mean are estimated for the Doubleexponential by two independent groups of students of size 24 each. Figure 7 gives a box plot of these data.

Insert Figure 5 At first, we estimate the tailweight, cf. (1), and we obtain t = 1.4916. According to our adaptive procedure the test U 8:1 is chosen. We obtain U 8:1 = -3.401 an one-sided p-value of 0.0003, a result which is, of course, not surprising. Because of the very low p-value it is no problem that the (adaptive) procedure is slightly anticonservative for the given sample size.

Conclusions

What are the results of our study? At first, we see that the finite power of the considered tests based on U-statistics can be well approximated by their asymptotic power. However, the 'convergence' of the finite power to the asymptotic power is slower than in the location case. Second, there are modifications of the "classical" Klotz test that may have (considerably) higher power than Klotz for symmetric as well asymmetric densities. Third, the Adaptive test A( Ŝ) is a serious alternative to the Klotz test or to the Mood test U 2:1 for moderate to large sample sizes, if the densities have very short tails or very long tails or if they are skew. Also, for short-tail or for long-tail densities or for skew densities the adaptive test considered here may be better than the adaptive test B( Ŝ) based on linear rank tests.

Many tests based on U-Statistics have good power properties. They can be applied to a broader class of underlying densities than a linear rank test with fixed scores if the sample sizes are moderate to large.

For moderate to large sample sizes (about n 1 , n 2 ≥ 100) the recommendation for the practising statistician is as follows: If the density is known to be exactly normal take the F -test. If it is only known that it has medium tails then take the Klotz test or the test U 5:1 . For densities with short tails take the test U 8:1 , and for those with long tails the test U 2:1 . If the density is known to be defined on a half interval only and is skew, then take a location test. For a completely unknown density take the adaptive test A( Ŝ).

For relatively small sample sizes (about n 1 , n 2 ≤ 10) take the Klotz test.

Note that we considered the scale problem for continuous data only. How the procedures behave for ordinal data may be a topic for further studies. Further investigations are also desirable for more general alternatives, especially for the case of unknown and unequal locations or for the case of F = G.

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Figure 1 :

 1 Figure 1: The first two principal components

Figure 3 :Figure 4 :

 34 Figure 3: The asymptotic and finite (n 1 = n 2 = 100) power functions of the tests U 2:1 , U 5:1 , U 8:1 , U 5:2 and A( t); densities: uniform, normal and logistic. asymptotic finite

ca Communications in Statistics -Simulation and Computation

	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	Values of the factors
	C 2 k,i (f ) for various densities are, together with that of the well established
	Klotz test presented in Table 2.

The L-DE density was proposed by

[START_REF] Policello | Adaptive robust procedures for the one-sample location problem[END_REF]

, 5 Page 6 of 26 URL: http://mc.manuscriptcentral.com/lssp E-mail: comstat@univmail.cis.mcmaster.

Table 4 )

 4 . Much better in this case are tests that are designed for testing locations,

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  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	(8:1)	(5:1)	(2:1)	(5:2)
	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.61	1.93	2.5
			Tailweight t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1 :

 1 The asymptotic efficacies for various U-tests and various densities

	density	Sukhatme Kusum SJ(5,1) SJ(5,2) * SJ(9,1)
				U 3:1		U 5:1	U 8:1	U 2:1	U 5:2
	Uniform	12.00	12.44	6.222	2.333	9.800 15.66 5.000 1.802
	Logistic	1.040	1.333	1.307	1.037	1.372		1.250 0.947
	Cauchy	0.493	0.407	0.435	0.496	0.350		0.462 0.500
	Doubleex	0.75	0.913	0.902	0.676	0.936		0.868 0.676
	Normal	1.216	1.681	1.624	1.187	1.795 1.862 1.520 1.061
	Gumbel	1.224		1.316	0.950	1.467 1.537 1.228 0.847

* : computed from

Joshi and Pandit (2008)

.

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Table 2 :

 2 The asymptotic efficacies of the U k:i -tests and the corresponding factor for the Klotz-test for the scale problem

	density	Klotz U 2:1	U 3:1	U 4:1	U 4:2	U 5:1	U 5:2	U 6:1	U 6:2	U 6:3
	uniform	∞	5.000 6.222 7.936 1.583 9.800 1.802 11.735 2.160 1.020
	logistic	1.396 1.250 1.307 1.353 0.890 1.372 0.947 1.371 1.026 0.708
	Cauchy	0.277 0.462 0.435 0.394 0.497 0.350 0.500 0.311 0.496 0.468
	DE	0.997 0.868 0.902 0.927 0.641 0.936 0.676 0.933 0.722 0.526
	normal	2.000 1.520 1.624 1.727 0.987 1.795 1.061 1.837 1.169 0.759
	Gumbel	1.670 1.228 1.316 1.406 0.786 1.467 0.847 1.506 0.935 0.603
	skew-log 1 0.804 1.089 1.149 1.204 0.745 1.233 0.797 1.245 0.870 0.584
	skew-log 2		0.788 0.841 0.892 0.216 0.925 0.206 0.943
	t 1.5	0.454 0.659 0.641 0.605 0.626 0.560 0.642 0.515 0.658 0.555
	t 2	0.610 0.800 0.794 0.768 0.700 0.729 0.730 0.686 0.761 0.604
	t 3	0.853 0.983 0.997 0.993 0.790 0.968 0.829 0.934 0.880 0.654
	t 4	1.030 1.093 1.122 1.135 0.837 1.124 0.884 1.100 0.947 0.681
	t 5	1.163 1.166 1.207 1.232 0.866 1.232 0.918 1.215 0.989 0.696
	t 10	1.509 1.330 1.398 1.457 0.925 1.485 0.988 1.491
	RST -1.0	0.249 0.414 0.389 0.351 0.450 0.313 0.451 0.278 0.445 0.429
	RST -0.5	0.508 0.684 0.672 0.643 0.628 0.604 0.646 0.563 0.665 0.553
	RST -0.4	0.603 0.765 0.760 0.737 0.673 0.701 0.697 0.662 0.724 0.582
	RST -0.3	0.726 0.859 0.863 0.850 0.722 0.820 0.752 0.785 0.789 0.612
	RST -0.2	0.886 0.969 0.986 0.986 0.774 0.966 0.812 0.937 0.861 0.643
	RST -0.1	1.102 1.098 1.132 1.151 0.830 1.147 0.877 1.129 0.940 0.675
	RST 0.05 1.588 1.336 1.407 1.471 0.922 1.504 0.984 1.516 1.073 0.725
	RST 0.14 2.035 1.509 1.613 1.716 0.981 1.784 1.054 1.826 1.161 0.756
	RST 0.2	2.435 1.641 1.771 1.906 1.022 2.006 1.103 2.075 1.223 0.777
	RST 0.4	4.948 2.188 2.446 2.750 1.166 3.015 1.278 3.238 1.449 0.846
	L-DE 0.55 0.977 0.872 0.905 0.931 0.645 0.938 0.680 0.936 0.726 0.530
	L-DE 0.61 0.986 0.888 0.920 0.944 0.663 0.950 0.697 0.946 0.743 0.548
	L-DE 0.7 1.025 0.939 0.969 0.989 0.714 0.991 0.749 0.984 0.795 0.595
	L-DE 0.75 1.061 0.983 1.013 1.031 0.752 1.031 0.789 1.020 0.837 0.626
	L-DE 0.8 1.106 1.037 1.069 1.087 0.793 1.085 0.833 1.071 0.887 0.655
	L-DE 0.9 1.241 1.166 1.208 1.236 0.863 1.239 0.914 1.226 0.984 0.695
	L-DE 0.95 1.328 1.223 1.274 1.313 0.884 1.324 0.939 1.318 1.015 0.705
	L-DE 0.97 1.361 1.239 1.294 1.337 0.888 1.352 0.944 1.348 1.022 0.707
	L-DE 0.99 1.388 1.249 1.305 1.351 0.890 1.369 0.947 1.368 1.026 0.708
								continuation on the following page
					22

Table 3 :

 3 The asymptotic efficacies of the tests U 8:1 and U 5:1 for the scale

	problem, short tail densities only
	density	tailweight U 8:1	U 5:1 density	tailweight U 8:1	U 5:1
	uniform	1.286	15.66 9.800 U-L 0.55	1.668	1.425 1.463
	normal	1.587	1.862 1.795 U-L 0.61	1.623	1.580 1.623
	Gumbel	1.655	1.537 1.467 U-L 0.7	1.534	1.993 2.032
	RST 0.05	1.657	1.499 1.504 U-L 0.75	1.474	2.393 2.404
	RST 0.14	1.591	1.854 1.784 U-L 0.8	1.409	3.027 2.947
	RST 0.2	1.552	2.146 2.006 U-L 0.9	1.300	5.888 4.945
	RST 0.4	1.446	3.576 3.015 U-L 0.95		9.214 6.854
				U-L 0.97		11.28 7.874
				U-L 0.99		11.68 8.958
	CN 2,0.01	1.592	1.812 1.762 CN 2,0.02	1.597	1.765 1.730
	CN 2,0.05	1.611	1.639 1.641 CN 2,0.10	1.636	1.470 1.519
	CN 3,0.01	1.596	1.755 1.725 CN 3,0.02	1.605	1.655 1.659
	CN 3,0.05	1.635	1.400 1.483 CN 3,0.10	1.697	1.086 1.257
	CN 5,0.01	1.600	1.690 1.684 CN 5,0.02	1.614	1.535 1.582
	CN 5,0.05	1.665	1.154 1.321 CN 5,0.10	1.974	0.728 1.009
	CN 10,0.01	1.603	1.633 1.649 CN 10,0.02	1.622	1.433 1.516
	CN 10,0.05	1.694	0.966 1.189 CN 10,0.10	1.944	0.498 0.825
	skew-log 1	1.672	1.233 1.233
	skew-log 2	1.684		0.925
	RST 0.2,0.4	1.546	2.852 2.500
	RST 0.2,0.49	1.507	3.374 2.868
	RST 0.4,0.49	1.456	4.070 3.342

Table 4 :

 4 The AE of the tests U k:i , for the scale problem, together with that for the Mann-Whitney test U 1,1 , skew densities with F (0) = 0.

	Page 25 of 26
	density	U 1,1	U 2:1	U 3:1	U 4:1	U 4:2	U 5:1	U 5:2	U 6:1	U 6:2	U 6:3
	exponential 0.750 0.139 0.148 0.157 0.092 0.162 0.098 0.165 0.108 0.071
	Gamma1.5 1.216 0.150 0.159 0.169 0.098 0.175 0.105 0.179 0.116 0.076
	Gamma2	1.688 0.155 0.165 0.175 0.101 0.182 0.109 0.185 0.120 0.078
	Gamma2.5 2.162 0.158 0.168 0.179 0.103 0.185 0.111 0.189 0.122 0.080
	Gamma3	2.637 0.160 0.170 0.181 0.104 0.188 0.112 0.192 0.123 0.080
	Gamma4	3.589 0.162 0.173 0.184 0.106 0.191 0.114 0.195 0.125 0.081
	Gamma5	4.542 0.164 0.175 0.186 0.107 0.193 0.115 0.197 0.126 0.082
	LogNormal 0.955 0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
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Table 5 :

 5 The test chosen by the Adaptive tests A( t) and B( Ŝ) and their

						Page 26 of 26
	factors C 2 k,i (f )
			Adaptive test A( t)	Adaptive test B( Ŝ)
	density	tailweight chosen test AE	chosen test	AE
	uniform	1.286	U 8:1	15.66 Klotz	∞
	logistic	1.697	U 5:1	1.372 logistic scores	1.428
	Cauchy	3.217	U 5:2	0.500 Ansari-Bradley 0.498
	DE	1.912	U 5:1	0.936 logistic scores	0.996
	Normal	1.587	U 8:1	1.862 Klotz	2.000
	t 1.5	2.433	U 2:1	0.659 Ansari-Bradley 0.657
	t 2	2.107	U 2:1	0.800 Ansari-Bradley 0.750
	t 3	1.798	U 5:1	0.968 logistic scores	0.942
	t 4	1.786	U 5:1	1.124 logistic scores	1.111
	t 5	1.737	U 5:1	1.232 logistic scores	1.232
	t 10	1.672	U 5:1	1.485 logistic scores	1.537
	RST -1	3.451	U 5:2	0.451 Ansari-Bradley 0.448
	RST -0.5 2.302	U 2:1	0.684 Ansari-Bradley 0.661
	RST -0.4 2.146	U 2:1	0.765 Ansari-Bradley 0.720
	RST -0.3 2.010	U 2:1	0.859 logistic scores	0.808
	RST -0.2 1.891	U 5:1	0.966 logistic scores	0.963
	RST -0.1 1.788	U 5:1	1.147 logistic scores	1.164
	RST 0.05 1.657	U 5:1	1.516 logistic scores	1.592
	RST 0.14 1.591	U 8:1	1.854 Klotz	2.015
	RST 0.2 1.552	U 8:1	2.146 Klotz	2.394
	RST 0.4 1.446	U 8:1	3.576 Klotz	4.531
	U-L 0.55 1.668	U 5:1	1.463 logistic scores	1.517
	U-L 0.61 1.623	U 5:1	1.623 Klotz	1.602
	U-L 0.7	1.534	U 8:1	1.993 Klotz	2.016
	U-L 0.75 1.474	U 8:1	2.393 Klotz	2.377
	U-L 0.8	1.409	U 8:1	3.027 Klotz	2.925
	U-L 0.9	1.300	U 8:1	5.888 Klotz	5.595
	U-L 0.95		U 8:1	9.214 Klotz	9.585
	U-L 0.99		U 8:1	11.28 Klotz	13.62
				continuation on the following page
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	continuation from previuos page
			Adaptive test A( t)	Adaptive test B( Ŝ)
	density	tailw. skewn. chosen test AE	chosen test	AE
	CN 2,0.01	1.592	U 8:1	1.812 Klotz	1.861
	CN 2,0.02	1.597	U 8:1	1.765 Klotz	1.782
	CN 2,0.05 F CN 2,0.1 CN 3,0.01 o CN 3,0.02 r 1.611 1.636 1.596 1.605 CN 3,0.05 1.635	U 5:1 U 5:1 U 8:1 U 5:1 U 5:1	1.641 Klotz 1.519 Klotz 1.755 Klotz 1.655 Klotz 1.483 Klotz	1.610 1.449 1.711 1.569 1.292
	CN 3,0.1 CN 5,0.01 CN 5,0.02 CN 5,0.05 CN 5,0.1 CN 10,0.01	1.697 P 1.600 1.614 e 1.665 e 1.794 r 1.603	U 5:1 U 8:1 U 5:1 U 5:1 U 5:1 U 8:1	1.257 logistic scores 1.690 Klotz 1.582 Klotz 1.321 logistic scores 1.009 logistic scores 1.634 Klotz	1.198 1.560 1.403 1.202 0.956 1.387
	CN 10,0.02 CN 10,0.05 CN 10,0.1 Gumbel skew-log1 skew-log2 RST-0.4,-0.49 2.226 0.375 1.622 1.694 R U 5:1 U 5:1 1.944 U 2:1 e 1.655 U 5:1 1.672 0.1313 U 5:1 v i 1.516 Klotz 1.189 logistic scores 0.961 logistic scores 1.467 logistic scores 1.233 logistic scores 1.684 0.2279 U 5:1 0.925 logistic scores U 2:1 0.718 Wilcoxon e RST-0.2,-0.49 2.195 0.490 U 2:1 0.646 Wilcoxon RST-0.2,-0.4 2.073 U 2:1 0.751 Ansari-Bradley 0.665 1.318 1.044 0.798 ... 0.955 0.047 0.002 w RST 0.2,0.4 1.546 U 8:1 2.852 Klotz 3.415
	RST 0.2,0.49 1.507 RST 0.4,0.49 1.456	U 8:1 U 8:1	n 3.374 Klotz O 4.070 Klotz	4.222 3.962
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Table 6 :

 6 Estimated levels of significance for various sample sizes and for various U-tests and for the adaptive test.

	Page 27 of 26
		U 2:1	U 5:1	U 8:1	U 5:2 Adaptive
	n 1 = n 2 = 10	0.085 0.100 0.125 0.118	0.123
	n 1 = n 2 = 20	0.065 0.071 0.085 0.072	0.085
	n 1 = n 2 = 40	0.058 0.060 0.067 0.060	0.067
	n 1 = n 2 = 100 0.052 0.051 0.053 0.053	0.053
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