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In this paper, we consider the problem of estimating the scale parameter of the inverse Rayleigh distribution based on general progressively Type-II censored samples and progressively Type-II censored samples. The pivotal quantity method is used to derive the estimator of the scale parameter. Besides, considering that the maximum likelihood estimator is tough to obtain for this distribution, we derive an explicit estimator of the scale parameter by approximating the likelihood equation with Taylor expansion. The interval estimation is also studied based on pivotal inference. Then we conduct Monte Carlo simulations and compare the performance of different estimators. We demonstrate that the pivotal inference is simpler and more effective. The further application of the pivotal quantity method is also discussed theoretically. Finally, two real data sets are analyzed using our methods. [ABSTRACT FROM AUTHOR]

There is no general exact analysis for the class of generalized mixed models, and asymptotic procedures are widely used. Importantly, under small sample conditions equivalent asymptotic procedures can yield conflicting inference when applied to the same data set [Aubin, E. C. Q., Cordeiro, G. M. (2000). Bartlett-corrected tests for normal linear models when the covariance matrix is nonscalar. Commun. Statist.—Theory Methods 29:2405–2426]. For the classical likelihood ratio test (LRT), Bartlett’s [Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proc. R. Soc. London. Ser. A, Math. Phys. Sci. 160(901):268–282] correction may be used to yield improved small sample performance. Zucker et al. [Zucker, D. M., Lieberman, O., Manor, O. (2000). Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. J. R. Statist. Soc., Ser. B 62:827–838] proposed and investigated methods for improved small sample inference in the mixed linear model using refined LRTs. The refinements included the use of a Bartlett correction and the Cox–Reid adjusted likelihood [Cox, D. R., Reid, N. (1987). Approximations to noncentral distributions. Can. J. Statist. 15(2):105–114], which using simulation studies (under a random-line model, and a two-period, four-treatment crossover design) were shown to yield Type I error rates very close to the nominal level. An alternative approach which has also been shown [Kowalchuk, R., Keselman, H. (2001). The analysis of repeated measurements with mixed-model Kenward Roger's adjusted F-tests. Paper presented at the Annual Meeting of the American Educational Research Association, Seattle, Washington] to have improved performance characteristics is a procedure involving t and F statistics for tests of fixed effects with modified degrees of freedom calculations detailed by Kenward and Roger [Kenward, M. G., Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53:983–997]. However, to date there has been no direct comparison of the Bartlett modified likelihood ratio and Kenward Roger procedures. This paper provides the results from a Monte Carlo simulation study examining the Type I error control and power profiles of modified likelihood ratio procedures including those described in Zucker et al. and proposed by Kenward and Roger for tests of fixed effect parameters in mixed linear models with data simulated from small sample unbalanced repeated measures designs, followed by results from a real data example. [ABSTRACT FROM AUTHOR]