Abstract
In this paper, a general class of non-parametric tests for testing homogeneity of location parameter against umbrella alternatives is proposed. Testing for umbrella alternatives has many applications in the field of biology, medicine, botany, dose level testing, engineering, economics, psychology, zoology. As an example, the effectiveness of a drug is likely to increase with increase of dose up to a certain level and then its effect begins to decrease. The proposed test is based on linear combination of two-sample U-statistics. The null distribution of the test statistics is developed. We compare the test with some other competing tests in terms of Pitman asymptotic relative efficiency. To see execution of the test, a numerical example is provided. Simulation study is carried out to assess the power of proposed class of tests.
Highlights
Suppose there are k (k ≥ 2) independent random samples Xl1, ... , Xlnl; l = 1, ... , k, of size nl with absolute continuous cumulative distribution function of lth sample as Fl(x) = F(x − θl), l = 1, ... , k
It is of interest to check whether all the samples are from a common distribution or there is an umbrella pattern in their location parameter θl′s
For testing of umbrella alternative, Mack and Wolfe (1981) firstly proposed a test which is further modified by a number
Summary
We proposed a general class of distribution-free tests for k-sample location problem with umbrella alternative based on linear combination of two-sample test statistics proposed by Kumar (2015). 3. Distribution of the Proposed Test Statistics The expectation of Uc(,ld,l;+i,j1) is: E [U(l,l+1)] =
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