We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form, that for linear covariance structures appears as the variance of a scaled projection of a random matrix that is of radial type, and a similar result is obtained for the corresponding sequence of estimators for the vector of variance components. These results are illustrated by the limiting behavior of estimators for a differentiable covariance structure in a variety of multivariate statistical models. We also derive a characterization for the influence function of corresponding functionals. Furthermore, we derive the limiting distribution and influence function of scale invariant mappings of such estimators and their corresponding functionals. As a consequence, the asymptotic relative efficiency of different estimators for the shape component of a structured covariance matrix can be compared by means of a single scalar and the gross error sensitivity of the corresponding influence functions can be compared by means of a single index. Similar results are obtained for estimators of the normalized vector of variance components. We apply our results to investigate how the efficiency, gross error sensitivity, and breakdown point of S-estimators for the normalized variance components are affected simultaneously by varying their cutoff value.

The analytical tools to quantify CO₂RR products are often slow and have high limits of detection. As a result, researchers are forced to extend the duration of their experiments to accumulate sufficient product and surpass these detection limits. This slows down research considerably, and the research scope often remains limited. To help speed up CO₂RR catalyst studies, we have developed a new differential electrochemical mass spectrometer (DEMS) setup and cell design that enables the quantification of major gaseous and liquid products significantly faster than conventional analytical techniques. Special attention was given to the hydrodynamics of the cell to avoid mass transfer limitations and the calibration of the setup to accurately quantify the major CO₂ reduction products. As proof of concept of the methodology, the products formed during CO₂RR on a polycrystalline Ag and Cu electrode in a 0.1-M KHCO₃ electrolyte at different potentials were measured and quantified.

A unified approach is provided for a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. Sufficient conditions are provided for the existence of the estimators and corresponding functionals, strong consistency and asymptotic normality is established, and robustness properties are derived in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some results are new and others are more general than existing ones in the literature. In this way, results on high breakdown estimation with high efficiency in a wide variety of multivariate models are completed and improved.

We provide a unified approach to S-estimation in balanced linear models with structured covariance matrices. Of main interest are S-estimators for linear mixed effects models, but our approach also includes S-estimators in several other standard multivariate models, such as multiple regression, multivariate regression and multivariate location and scatter. We provide sufficient conditions for the existence of S-functionals and S-estimators, establish asymptotic properties such as consistency and asymptotic normality, and derive their robustness properties in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some of our results are new and others are more general than existing ones in the literature. In this way, this manuscript completes and improves results on S-estimation in a wide variety of multivariate models.We illustrate our results by means of a simulation study and an application to data from a trial on the treatment of lead-exposed children.

We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate 0 in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate nm=.2mC1/, when 0 is m 2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods.

We consider Grenander-type estimators for a monotone function (Formula presented.), obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.

We consider the process Λ̂_n−Λ_n, where Λ_n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂_n is the least concave majorant of Λ_n. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of Λ̂_n−Λ_n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L_p-distance between Λ̂_n and Λ_n.

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate n^m∕(2m+1), where m≥2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.