**
Manish Goyal **
Department of Statistics, Panjab University, Chandigarh, India.

**
Narinder Kumar **
Department of Statistics, Panjab University, Chandigarh, India.

DOI https://dx.doi.org/10.33889/IJMEMS.2018.3.4-036

**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.

**Keywords-** Non-parametric tests, Umbrella alternatives, Null distribution, Pitman efficiency, Simulation study.

**Citation**

Goyal, M., & Kumar, N. (2018). A General Class of Tests for Testing Homogeneity of Location Parameters against Umbrella Alternatives. *International Journal of Mathematical, Engineering and Management Sciences*, *3*(4), 498-512. https://dx.doi.org/10.33889/IJMEMS.2018.3.4-036.

**Conflict of Interest**

**Acknowledgements**

The authors thank anonymous reviewers for their valuable suggestions, which led to an improvement of the paper.

**References**

Amita, & Kochar, S. C. (1989). Some distribution free tests for testing homogeneity of location parameters against ordered alternatives. Journal of Indian Statistical Association, 27, 1-8.

Archambault, W. A. T., Mack, G. A., & Wolfe, D. A. (1977). K‐sample rank tests using pair‐specific scoring functions. Canadian Journal of Statistics, 5(2), 195-207.

Bhat, S. V. (2009). Simple K-sample rank tests for umbrella alternatives. Research Journal of Mathematics and Statistics, 1(1), 27-29.

Büning, H., & Kössler, W. (1996). Robustness and efficiency of some tests for ordered alternatives in the c-sample location problem. Journal of Statistical Computation and Simulation, 55(4), 337-352.

Büning, H., & Kössler, W. (1997). Power of Some Tests for Umbrella Alternatives in the Multi‐Sample Location Problem. Biometrical Journal, 39(4), 481-494.

Deshpande, J. V., & Kochar, S. C. (1982). Some competitors of Wilcoxon-Mann-Whitney test for location alternatives. Journal of Indian Statistical Association, 19, 9-18.

Gökpinar, E., & Gökpinar, F. (2016). A modified Mack–Wolfe test for the umbrella alternative problem. Communications in Statistics-Theory and Methods, 45(24), 7226-7241.

Goyal, M., & Kumar, N. (2017). A general class of tests for testing homogeneity of location parameters against ordered alternatives. Proceedings in Mathematics and Statistics, under the title Logistics, Supply Chain and Financial Predictive Analytics: Theory and Practices, Springer (accepted).

Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods (Vol. 751). John Wiley & Sons.

Jonckheere, A. R. (1954). A distribution-free k-samples test against ordered alternatives. Biomerika, 41(1/2), 133-145.

Kössler, W. (2005). Some c-sample rank tests of homogeneity against ordered alternatives based on U-statistics. Journal of Nonparametric Statistics, 17(7), 777-795.

Kössler, W. (2006). Some c-sample rank tests of homogeneity against umbrella alternatives with unknown peak. Journal of Statistical Computation and Simulation, 76(1), 57-74.

Kumar, N. (1997). A class of two-sample tests for location based on sub-sample medians. Communication in Statistics - Theory and Methods, 26(4), 943-951.

Kumar, N. (2015). A class of nonparametric tests based on sample quantiles. Journal of Combinatorics, Information & System Sciences, 40(1-4), 211-223.

Kumar, N., & Goyal, M. (2018). Jonckheere type test procedure with optimal criterion under order restrictions. American Journal of Mathematical and Management Sciences. doi: https://doi.org/10.1080/01966324.2017.1404948.

Kumar, N., Gill, A. N., & Dhawan, A. K. (1994b). A class of distribution-free statistics for homogeneity against ordered alternatives. South African Statistical Journal, 28(1), 55-65.

Kumar, N., Gill, A. N., & Mehta, G. P. (1994a). Distribution-free test for homogeneity against ordered alternatives. Communication in Statistics – Theory and Methods, 23(4), 1247-1256.

Kumar, N., Singh, R. S., & Öztürk, Ö. (2003). A new class of distribution-free tests for location parameters. Sequential Analysis, 22(1-2), 107-128.

Lehmann, E. L. (1963). Robust estimation in analysis of variance. Annals of Mathematical Statistics, 34(3), 957-966.

Mack, G. A., & Wolfe, D. A. (1981). K-sample rank tests for umbrella alternatives. Journal of the American Statistical Association, 76(373), 175-181.

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of the random variables is stochastically greater than the other. Annals of Mathematical Statistics, 18(1), 50-60.

Patil, A. B. (2007). A Study on Distribution-Free Test Statistics. Thesis submitted to the Karnataka University, Dharwad.

Puri, M. L. (1965). Some distribution-free k-sample rank tests of homogeneity against ordered alternatives. Communications on Pure Applied Mathematics, 18(1-2), 51-63.

Rao, C. R. (1973). Linear Statistical Inference and its Applications. Wiley Eastern Ltd., New York.

Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. John Wiley and Sons, New York.

Simpson, D. G., & Margolin, B. H. (1986). Recursive nonparametric testing for dose-response relationships subject to downturns at high doses. Biometrika, 73(3), 589-596.

Terpstra, T. J. (1952). The asymptotic normality and consistency of Kendall’s test against trend, when ties are present in one ranking. Indigationes Mathematicae, 14(3), 327-333.

Wilcoxon, F. (1945). Individual comparisons by rank methods. Biometrics Bulletin, 1(6), 80-83.