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Propagations of nonlinear quantum dust acoustic waves (QDAWs) in an inhomogeneous magnetized dusty plasma model which is three-dimensional collisionless, and having number density gradients of its ...

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H0 is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ2 of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing Ho under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results

Four general classes of partially balanced designs for 2n factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ2 of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H0 is derived under defined local alternatives. Optimal designs d1 maximizing λ2 in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.

In am earlier paper, the authors have presented the methodology for consideration of specified treatment contrasts in paired comparisons. An iterative method of solution of derived likelihood equations was given. This paper considers the iterative method. It is proved that, subject to a mild condition, the iterative scheme converges to a solution of the likelihood equations that is unique and yields the maximum of the likelihood function on a parameter space constrained by the condition that certain treatment contrasts are null.

Treatment contrasts in paired comparison experiments are considered. Given that specified orthonormal treatment contrasts are null, remaining treatment parameters, orthonormal contrasts among them, and their logarithms are estimated. The estimators are shown to be consistent and, when suitably scaled and centered, to have distribution functions that are normal in the limit for large sample sizes. Hypothesis tests on treatment contrasts are obtained. The limiting distribution functions of related likelihood ratio tests are derived for defined local alternatives and their noncentrality parameters are shown. The theoretical results are applied to experiments with factorial treatment combinations and some optimal design results are obtained.

SYNOPTIC ABSTRACTTests of hypotheses stated in terms of linear constraints on the parameters involved in the distribution functions of random variables are considered. A sequence of local alternatives and a set of assumptions incorporating the possibility that the information matrix is singular are presented. Restricted likelihood equations are derived and existence of likelihood estimators emerging from solving these equations is demonstrated. Asymptotic properties of the obtained estimators, together with the asymptotic noncentral chi-square distribution of the likelihood ratio statistic −2 log λ, are developed. Numerical examples illustrate the applications of the results.