On the GLR and UMP tests in the family with support dependent on the parameter
Abstract
Some general results about the GLR tests, for testing simple hypothesis versus two-sided hypothesis, in the family with support dependent on the parameter, are obtained. In addition, we show that such GLR tests are equivalent to the UMP tests in the same problems. Moreover, we derive the general form of the UMP tests for testing an interval hypothesis versus two-sided alternative.References
R. Aaberge, and L.C. Zhang, A class of exact UMP unbiased tests for conditional symmetry in small-sample square contingency tables, Journal of Applied Statistics, vol. 32, pp. 333–339, 2005.
D. Birkes, Generalized likelihood ratio tests and uniformly most powerful tests, The American Statistician, vol. 44, pp. 163–166, 1990.
A. Birnbaum, Admissible test for the mean of a rectangular distribution, The Annals of Mathematical Statistics, vol. 25, pp. 157–161, 1954.
D. Edelman, A note on uniformly most powerful two-sided tests, The American Statistician, vol. 44, pp. 219–220, 1990.
D.G. Kabe, and A.G. Laurent, On some nuisance parameter free uniformly most powerful tests, Biometrical Journal, vol. 23, pp. 245–250, 1981.
E.L. Lehmann, and C. Stein, Most powerful tests of composite hypotheses I. normal distribution, The Annals of Mathematical Statistics, vol. 19, pp. 495–516, 1948.
E.L. Lehmann, and J.P. Romano, Testing Statistical Hypotheses, Third Ed., Springer, New York, 2005.
P. Majerski, and Z. Szkutnik, Approximations to most powerful invariant tests for multi normality against some irregular alternatives, TEST, vol. 19, pp. 113–130, 2010.
M.P. McDermott, and Y. Wang, Construction of uniformly more powerful tests for hypotheses about linear inequalities, Journal of Statistical Planning and Inference, vol. 107, pp. 207–217, 2002.
S. Migliorati, Uniformly most powerful tests for two-sided hypotheses, Metron, vol. LX-N, pp. 107–114, 2002.
K. Nomakuchi, A note on the uniformly most powerful tests in the presence of nuisance parameters, Annals of the Institute of Statistical Mathematics, vol. 44, pp. 141–145, 1992.
J.W. Pratt, Admissible one-sided tests for the mean of a rectangular distribution, The Annals of Mathematical Statistics, vol. 29, pp. 1268–1271, 1958.
A. Sayyareh, G. Barmalzan, and A. Haidari, Two sided uniformly most powerful test for Pitman family, Applied Mathematical Sciences, vol. 74, pp. 3649–3660, 2011.
H. Scherb, Determination of uniformly most powerful tests in discrete sample spaces, Metrika, vol. 53, pp. 71–84, 2001.
K. Takeuchi, A note on the test for the location parameter of an exponential distribution, The Annals of Mathematical Statistics, vol. 40, pp. 1838–1839, 1969.
Y. Wang, L.J. Young, and D.E. Johnson, A UMPU test for comparing means of two negative binomial distributions, Communications in Statistics - Simulation and Computation, vol. 30, pp. 1053–1075, 2001.
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