Abstract The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities often relevant to applications. Notwithstanding a long history, a formulation providing high precision numerical evaluations for N large enough to probe asymptotic regimes, has not been provided. An exception is for the largest eigenvalue, where there is a formalism due to Chiani which uses a combination of a formulation as a positive real valued Pfaffian, and a recursive computation of the matrix elements. We augment this strategy by introducing a generating function for the conditioned gap probabilities. A finite Fourier series approach is then used to extract the sequence of marginal eigenvalue distributions as a linear combination of complex valued Pfaffians, with the latter then evaluated using an efficient numerical procedure available in the literature due to Wimmer. Applications are given to illustrating various asymptotic formulas, local central limit theorems, and central limit theorems, as well as to probing finite size corrections. Further, our data indicates that the mean values of the marginal distributions interlace with the zeros of the Hermite polynomial (Gaussian ensemble) and a particular Laguerre polynomial (Laguerre ensemble).

Abstract In statistical models for the analysis of time-to-event data, individual heterogeneity is usually accounted for by means of one or more random effects, also known as frailties. In the vast majority of the literature, the random effect is assumed to follow a continuous probability distribution. However, in some areas of application, a discrete frailty distribution may be more appropriate. We investigate and compare various existing families of discrete univariate and shared frailty models by taking as our focus the variance of the relative frailty distribution in survivors. The relative frailty variance ( $$\text {RFV}$$ RFV ) among survivors provides a readily interpretable measure of how the heterogeneity of a population, as represented by a frailty model, evolves over time. We explore the shape of the $$\text {RFV}$$ RFV for the purpose of model selection and review available discrete random effect distributions in this context. We find non-monotone trajectories of the $$\text {RFV}$$ RFV for discrete univariate and shared frailty models with multiple changes in slope over time, which is a property that seems to be absent for continuous frailty models discussed in the literature. We also show that, in contrast to continuous frailty models, the heterogeneity of data generated by a discrete time-invariant frailty distribution, such as the categorical k -point distribution, approaches either infinity or homogeneity in the long run. Through the one-to-one relationship of the $$\text {RFV}$$ RFV with the cross-ratio function in shared frailty models, our results also apply to patterns of association within a cluster. Extensions and contrasts to discrete time-varying frailty models are discussed.

We present sharp lower and upper bounds on the variances of generalized order statistics with arbitrarily fixed model parameters valid for arbitrary baseline distributions with finite second moments. The bounds are expresssed with use of the scale units being the variances of baseline distributions. Special cases of submodels are discussed. Moreover, explicit formulae for the density and distribution functions of generalized order statistics are presented.

Abstract Different types of omnibus goodness-of-fit tests for a binomial null hypothesis are developed, which are based on the counts’ probability generating function. For each statistic, closed-form asymptotics are derived and used to implement the tests in practice without the need for a bootstrap procedure. The finite-sample performance of the tests is analyzed by means of a comprehensive simulation study, and practical recommendations for the choice of the test statistic are given. The practical application of the tests is illustrated by a couple of real-world data examples.