# Article

Full entry | PDF   (1.7 MB)
Keywords:
testing statistical hypothesis; locally most powerful tests
Summary:
The locally most powerful (LMP) tests of the hypothesis $H: \theta =\theta _0$ against one-sided as well as two-sided alternatives are compared with several competitive tests, as the likelihood ratio tests, the Wald-type tests and the Rao score tests, for several distribution shapes and for location, shape and vector parameters. A simulation study confirms the importance of the condition of local unbiasedness of the test, and shows that the LMP test can sometimes dominate the other tests only in a very restricted neighborhood of $H.$ Hence, we cannot recommend a universal application of the LMP tests in practice. The tests with a high Bahadur efficiency, though not exactly LMP, also seem to be good in the local sense.
References:
[1] Brown L. D., Marden J. M.: Local admissibility and local unbiasedness in hypothesis testing problems. Ann. Statist. 20 (1992), 832–852 DOI 10.1214/aos/1176348659 | MR 1165595 | Zbl 0767.62006
[2] Chibisov D. M.: Asymptotic expansions for some asymptotically pptimal tests. In: Proc. Prague Symp. on Asymptotic Statistics, Volume II (J. Hájek, ed.), Charles University, Prague 1973, pp. 37–68 MR 0400501
[3] Efron B.: Defining the curvature of a statistical problem (with application to second order efficiency). Ann. Statist. 3 (1975), 1189–1242 DOI 10.1214/aos/1176343282 | MR 0428531
[4] Gupta A. S., Vermeire L.: Locally optimal tests for multiparameter hypotheses. J. Amer. Statist. Assoc. 81 (1986), 819–825 DOI 10.1080/01621459.1986.10478340 | MR 0860517 | Zbl 0635.62020
[5] Isaacson S. L.: On the theory of unbiased tests of simple statistical hypothesis specifying the values of two or more parameters. Ann. Math. Statist. 22 (1951), 217–234 DOI 10.1214/aoms/1177729642 | MR 0041401
[6] Jurečková J.: $L_1$-derivatives, score function and tests. In: Statistical Data Analysis Based on the $L_1$-Norm and Related Methods (Y. Dodge, ed.), Birkhäuser, Basel 2002, pp. 183–189
[7] Kallenberg W. C. M.: The shortcomming of locally most powerful test in curved exponential families. Ann. Statist. 9 (1981), 673–677 DOI 10.1214/aos/1176345472 | MR 0615444
[8] Lehmann E. L.: Testing Statistical Hypothesis. Second edition. Chapman & Hall, New York 1994
[9] Littel R. C., Folks J. L.: A test of equality of two normal population means and variances. J. Amer. Statist. Assoc. 71 (1976), 968–971 DOI 10.1080/01621459.1976.10480978 | MR 0420945
[10] Peers H. W.: Likelihood ratio and associated test criteria. Biometrika 58 (1971), 577–587 DOI 10.1093/biomet/58.3.577 | Zbl 0245.62026
[11] Ramsey F. L.: Small sample power functions for nonparametric tests of location in the double exponential family. J. Amer. Statist. Assoc. 66 (1971), 149–151 DOI 10.1080/01621459.1971.10482236 | Zbl 0215.26402
[12] Witting H.: Mathematische Statistik I. Teubner–Verlag, Stuttgart 1985 MR 0943833 | Zbl 0581.62001
[13] Wong P. G., Wong S. P.: A curtailed test for the shape parameter of the Weibull distribution. Metrika 29 (1982), 203–209 DOI 10.1007/BF01893380 | MR 0685566 | Zbl 0492.62022

Partner of