Kernel density estimators with circular data have been studied extensively for decades, as they allow flexible estimation even when the shape of the underlying density is complex. Many recent studies have examined bias correction methods; however, these methods are limited by the order when trying to improve the convergence rate of the bias, even if the true density is sufficiently smooth. To overcome this limitation, the present study considers a new bias correction approach based on the characteristic functions of the underlying circular density. We introduce wrapped flat-top kernels, which are generated by wrapping the standard flat-top kernels defined on the real line onto the circumference of a unit circle. The asymptotic mean squared errors of the wrapped flat-top kernel density estimators are then derived. The results show that the convergence rate of these estimators is faster than that of previously introduced estimators. Furthermore, wrapped flat-top kernel density estimators achieve $$\sqrt{n}$$ -consistency under the characteristic function of finite support, such as the circular uniform and cardioid distributions. We confirm these theoretical results in numerical experiments. In empirical analyses, we also show that wrapped flat-top kernel density estimators effectively capture the shape of data. Therefore, such estimators are expected to allow flexible and accurate estimation in circular data analysis.

Abstract The process of measuring a continuous variable is typically subject to imprecision due to causes as varied as measurement errors or rounding, and it is the duty of practitioners to take account of this imprecision when performing different statistical inference tasks. In this paper, we address the problem of accounting for this imprecision in the context of confidence intervals for parameters of probability distributions. For such purpose, we formalize the notions of inner and outer confidence interval, both of which generalize the classical notion of confidence interval in the presence of imprecision. Different properties of both mathematical constructs are here studied and their explicit expressions are provided in five prominent cases in the field of Statistics.

Abstract Finding the eigenvalues connected to the covariance operator of a centered Hilbert-space valued Gaussian process is genuinely considered a hard problem in several mathematical disciplines. In Statistics this problem arises for instance in the asymptotic null distribution of goodness-of-fit test statistics of weighted $$L^2$$ -type as well as in the limit distribution of degenerate U -statistics. For this problem we present the Rayleigh–Ritz method to approximate the eigenvalues. The usefulness of these approximations is shown by high lightening implications such as critical value approximation and theoretical comparison of test statistics by means of Bahadur efficiencies.