A CLASS OF TWO-SAMPLE SCALE TESTS BASED ON U-STATISTICS

A class of tests based on U-statistic is proposed for two-sample scale problem. The U-statistic is function of extremes of subsamples taken from random samples of two absolutely continuous distributions. The asymptotic distribution, null distribution and efficacy of the class of tests are studied. A comparison of its performance with respect to other tests is studied in terms of Pitman ARE and small sample performance through its empirical power. Application of the class of tests is illustrated through an example.


Introduction
Variability is a fundamental component of almost all the phenomena observed in Nature. In some cases it is the outcome to be studied while in some other cases it is pre-existent. For instance, in a clinical trial studying variances in the effects of a treatment when subjects are provided with different dosages of a single drug would be an attempt to study the variability in the outcomes. On the other hand, studying variances in the effects of the same drug when subjects are provided with same dosages would be studying individual differences among the subjects. In the first case variance is induced while in the second, variance is a pre-existent nuisance. Whatever the case be, testing for variances among two populations is a fundamental problem in statistical inference.
Under parametric setup, Snedecor's F-test (given in Rohatgi and Saleh (2001)) is used to test 0 against 1 . Tests due to Mood (1954) and Siegel and Tukey (1960) are some earlier nonparametric tests based on ranks. Many distribution-free tests based on U-statistics for testing 0 against 1 are available in the literature. Some of them are due to, Sukhatme (1957Sukhatme ( , 1958, Deshpande and Kusum (1984). Kusum (1985), Kochar and Gupta (1986), Mehra and Rao (1992), Shetty and Bhat (1993) Lehmann (1951) established asymptotic distribution of the two-sample U-statistics. An extensive review of tests for two-sample scale problem is given by Duran (1976).
In this paper, we propose a class of distribution-free tests, ( 1 , 2 ) based on U-statistics which is a function of extremes of subsamples of sizes 1 and 2 respectively from and samples. The class of tests with its alternative form is given in section 2. Section 3 contains the distributional properties of ( 1 , 2 ), section 4 deals with its efficacy, Pitman ARE and empirical power. In sections 5, we furnish conclusions along with an illustration of the application of ( 1 , 2 ).

A Class of Distribution-Free Tests
In this section, we propose a class of tests based on two-sample U-statistic being a function of subsample extremes since the information contained in the tails of the distribution plays an important role in detecting difference in scales. The proposed class of tests is given by where, denotes summation over all possible respectively denote minimum order statistics of positive and negative ( ) observations and 1 ( 2 ) is the size of the subsample from ( ) sample satisfying The newly proposed class of tests ( 1 , 2 ) can also be obtained from tests ℎ ( 1 , 2 ) and The class of tests ( 1 , 2 ) is distribution-free for 1 ≤ 1 ≤ 1 and 1 ≤ 2 ≤ 2 and large values of the test statistic are significant for testing 0 against 1 .
Assuming that there are no ties, following Bhat (1995) an alternative form of Where, is the rank of ( 1 ) + in the joint rankings of  ( 2 ) − is the rank of ( 2 ) − in the joint respectively are ordered negative and observations.

Distribution of ( , )
In this section, we derive the null mean and asymptotic distribution of the proposed class of tests. Also, we obtain the null distribution of * ( 1 , 2 ) using Monte-Carlo simulation. The mean of ( 1 , 2 ) is given by, and the null mean is given by According to Lehmann (1951) as → ∞ such that 0 < = lim follows asymptotic normal distribution with mean zero and variance 2 = 1 2 10 + 2 2 01 Where, On similar lines by defining We get From (7) and (8) The null distribution of * ( 1 , 2 ) is obtained by generating 10000 random samples from uniform distribution for different values of 1 , 2 , 1 + , 2 + , 1 and 2 using (3) and is presented in figure  1.  We observe from the figure that the distribution of * ( 1 , 2 ) is symmetric and is asymptotically normal.

Performance of The Proposed Class Of Tests
We assess the performance of the proposed class of tests in terms of large and small samples. The large sample performance is assessed using Pitman ARE while the small sample performance in terms of empirical power.
It is observed from table 2 that, under uniform distribution ( 1 , 2 ) outperforms and it outperforms for ≥ 5. Under normal distribution it outperforms , , 1 and 2 . The ARE of ( 1 , 2 ) wrt , , 1 and 2 are increasing for increasing values of . Table 3 shows that, ( 1 , 2 ) is better than 3 ( 1 , 2 ) under normal distribution. The ARE is increasing with increasing , but decreases for a given value of as increases. It is better than The ARE of ( 1 , 2 ) wrt 5 ( 1 , 2 ) is increasing with increasing values of and . The class of tests ( 1 , 2 ) outperforms 5 ( 1 , 2 ) under normal distribution and under uniform distribution when ≥ 4, ≥ 10.
As far as small sample performance is concerned, from table 4, we see that the empirical power of * ( 1 , 2 ) is higher for smaller values of 1 , 2 , 1 + , 2 + , 1 , 2 and it decreases as they increase. It is high for normal, logistic, Laplace and Cauchy distributions as compared to uniform distribution when ≤ 1.2 whereas, empirical power under uniform distribution is higher as increases for larger values of , , 1 + and 2 + .

Conclusions
To illustrate the application of the proposed class of tests, following example of two samples of gasket diameters from two brands given in Deshpande et. al. (1995) is considered. The measurements of diameters are recorded as deviations from a common median. It is observed that, some members of the proposed class of tests have smaller p-values than those of and -tests. As p-values are smaller for smaller subsample sizes, the choice of the test statistics with smaller subsample sizes is helpful in testing the difference in variability among two samples.
We conclude that, The proposed class of tests ( 1 , 2 ) based on U-Statistic as function of subsample extremes is distribution-free and its large values are significant for testing 0 against 1 .
The null distribution of the class of tests is symmetric and follows asymptotic normal distribution.
Some members of ( 1 , 2 ) yield smaller p-values than and tests.
The proposed class of tests have higher power for medium tailed distributions when ≤ 1.2 whereas, for larger values of 1 , 2 , 1 + , 2 + the power under light tailed distributions is higher for > 1.2.