A great literature has been established for both feature screening under various situations and the analysis of case-cohort studies. Also several methods have been proposed for feature screening when one faces case-cohort studies but all of the existing methods apply only to right-censored failure time data. In this paper, we discuss feature screening for case-cohort studies in the presence of interval censoring, a type of censoring that occurs more generally and includes right censoring as a special case. For the problem, two methods, a model-based procedure and a model-free procedure, are proposed. The former, an inverse probability weighted joint likelihood screening approach, is developed under the proportional hazards model, the most commonly used model for failure time data analysis. The latter is a weighted smooth partial rank screening method that allows for both ultrahigh-dimensional covariates and covariate-dependent censoring. For the proposed methods, the sure screening properties are established under mild regularity conditions. An extensive simulation study is conducted to assess the performance of the proposed methods and indicates that they work well in practical situations.

We consider nonparametric estimation in Wicksell's problem, which has applications in astronomy for estimating the distribution of star positions in a galaxy and in material sciences for determining a material's 3D microstructure from 2D cross sections. We focus on the isotonised version of the plug-in estimator (IIE) for the cdf F of the spheres' squared radii. This estimator is fully automatic, requiring no tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function F to be estimated. We also prove a local asymptotic minimax lower bound in this non-standard setting, with log ⁡ n / n -asymptotics and where the functional F to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator.

ABSTRACT This research proposes a novel semi-parametric elliptical distribution model for application in semi-supervised learning tasks. We use labelled and unlabelled data to develop a pseudo maximum likelihood method for estimation and classification. The proposed estimator outperforms the estimator based solely on labelled data and achieves the semi-parametric efficiency bound with a suitable size of unlabelled data. We efficiently maximise the objective function by utilising low-dimensional groupwise pseudo-likelihood functions in a block coordinate descent manner while ensuring numerical stability and convergence through appropriate bandwidth selectors and initial parameter estimates. Additionally, the study comprehensively investigates the impact of labelled and unlabelled data on the pseudo maximum likelihood estimator and classifier. Simulation studies and empirical data applications illustrate the superiority of our methodology.

The kernel distribution estimator (KDE) is proposed based on residuals of the innovation distribution in the autoregressive moving-average (ARMA) time series. The deviation between KDE and the innovation distribution function is shown to converge to the Brownian bridge, leading to the construction of a Kolmogorov–Smirnov smooth simultaneous confidence band for the innovation distribution function. Additionally, an empirical cumulative distribution function (CDF) based on prediction residuals is introduced for the multi-step-ahead prediction error distribution function. This empirical process weakly converges to a Gaussian process with a specific covariance function. Furthermore, a quantile estimator is derived from the empirical CDF of prediction residuals, and multi-step-ahead prediction intervals (PIs) for future observations are established using these estimated quantiles. The PIs achieve the nominal prediction level asymptotically for the finite variance ARMA model. Simulation studies and analysis of crude oil price data confirm the validity of the asymptotic theory.

We investigate data-driven bandwidth selection within the confines of robust (resistant) kernel smoothing. While several approaches presently exist, they require user defined robustness parameters. We discuss identification issues within this setting and propose several tractable avenues to fully operationalise this approach. Simulations reveal that the proposed selection methods perform well relative to competing approaches and a small empirical example illustrates its usefulness.

This paper introduces generalised smooth asymmetric kernel estimators for smooth functionals with non-negative support. More precisely, for x ∈ [ 0 , ∞ ) , and for a functional, Φ ( x , F ) of the distribution function F, we develop estimators of the functional Φ and its derivatives. The proposed estimator can be seen as the solution to a minimisation problem in the polynomial space L 2 ( q ) , where q is an asymmetric density function. The framework presented here covers several classical nonparametric functional estimators and is linked with estimation using hierarchical kernels. We establish the asymptotic properties of the proposed estimators in the general framework. Furthermore, special attention is paid to comparing the asymptotic mean integrated square error (AMISE) of the proposed estimator with that of other classical symmetric/asymmetric density estimators. Additionally, a comparison of finite sample behaviour is conducted for both density estimation and hazard rate estimation via simulation and real data application.

In many modern applications, a dependent functional response is observed for each subject over repeated time, leading to longitudinal functional data. In this paper, we propose a novel statistical procedure to test whether the mean function varies over time. Our approach relies on reducing the dimension of the response using data-driven orthogonal projections, and employs likelihood-based hypothesis testing. We investigate the methodology theoretically and discuss a computationally efficient implementation. The proposed test maintains the Type-1 error rate, and shows excellent power to detect departures from the null hypothesis in finite sample simulation studies. We apply our method to the longitudinal diffusion tensor imaging study of multiple sclerosis (MS) patients to formally assess whether the brain's healthy tissue, as summarised by the fractional anisotropy (FA) profile, degrades over time during the study period.

We propose a novel approach for model-free time series forecasting. Unlike most existing methods, the proposed method does not rely on parametric error distributions nor assume parametric forms of the mean function, leading to broad applicability. We achieve such generality by establishing a simple but powerful representation of a time series { X t ; t ∈ Z } with sup t E | X t | < ∞ , that is, X t has a solution which is a linear combination of infinite past values. Then using the obtained solution a prediction algorithm is presented, with large sample theoretical guarantees. Simulation studies show favourable performance of the proposed method compared with popular parametric and neural networks methods, and suggest its superiority when the sample size is small. An application to practical time series is discussed.

The halfspace depth is a well-studied tool of nonparametric statistics in multivariate spaces. We introduce a flag halfspace – an intermediary between a closed halfspace and its interior – and demonstrate that the halfspace depth can be equivalently formulated also in terms of flag halfspaces. Flag halfspaces allow us to derive theoretical results regarding the halfspace depth without the need to differentiate absolutely continuous measures from measures containing atoms, as was frequently done previously. Flag halfspaces are used to state results on the dimensionality of the halfspace median set for random samples. We prove that under mild conditions, the dimension of the sample halfspace median set of d-variate data cannot be d−1 and that for d = 2, the sample halfspace median set must be either a two-dimensional convex polygon or a data point.

ABSTRACT Statistical depth, a commonly used analytic tool in nonparametric statistics, has been extensively studied for multivariate and functional observations over the past few decades. Although various forms of depth were introduced, they are mainly procedure based whose definitions are independent of the generative model for observations. To address this problem, we introduce a generative model-based approach to define statistical depth for both multivariate and functional data. The proposed model-based depth framework permits simple computation via a bootstrap sampling and improves the depth estimation accuracy. When applied to functional data, the proposed depth can capture important features such as continuity, smoothness or phase variability, depending on the defining criteria. We propose efficient algorithms to compute the proposed depths and establish estimation consistency. Through simulations and real data, we demonstrate that the proposed functional depths reveal important statistical information such as those captured by the median and quantiles, and detect outliers.