Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.     

Maximum Likelihood Estimation of a Poisson Parameter in a Model of Equioverlapping Samples:A Simulation Study

In a model of equioverlapping samples maximum likelihood estimation of a Poisson parameter is examined and compared with two linear unbiased estimations by mean squared error. Since a likelihood estimator is not explicitly available in general, a simulation study has been performed and the results are illustrated     

Relations for m-generalized order statistics via an Opial-type inequality

An Opial-type inequality is applied to obtain relations for expectations of functions of m-generalized order statistics (m-gOSs), their distribution functions, as well as moment-generating functions. Respective inequalities for common order statistics and record values are contained as particular cases.     

Asymptotic Equivalence of Seat Bias Models

Apportionment methods round vote proportions to integer numbers of seats in a parliament. An important issue is whether a given apportionment method treats larger and smaller parties equally or gives rise to seat biases. In this paper two models of quantifying seat biases of popular apportionment methods are compared to each other. The models are found to result in the same asymptotic behaviour when the number of seats available for apportionment grows large.