Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect based on finite samples and using TDA methods that ignore the probabilistic mechanism that generates the data. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behaviour of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results.

In this paper, we primarily focus on the edge frequency polygon estimator of , which represents the probability density function of a sequence of φ-mixing random variables . We establish the uniformly strong consistency and the convergence rate of asymptotic normality for the edge frequency polygon estimator under suitable conditions. Notably, the convergence rate achieves , which is more precise compared to the corresponding rate mentioned in the existing literature. Additionally, we present simulation studies to validate the theoretical results.

The paper focuses on the automatic selection of the grouped explanatory variables in a high-dimensional model, when the model blue error is asymmetric. After introducing the model and notations, we define the adaptive group LASSO expectile estimator for which we prove the oracle properties: the sparsity and the asymptotic normality. Afterwards, the results are generalized by considering the asymmetric -norm loss function. The theoretical results are obtained in several cases with respect to the number of variable groups. This number can be fixed or dependent on the sample size n, with the possibility that it is of the same order as n. Note that these new estimators allow us to consider weaker assumptions on the data and on the model errors than the usual ones. Simulation study demonstrates the competitive performance of the proposed penalized expectile regression, especially when the samples size is close to the number of explanatory variables and model errors are asymmetrical. An application on air pollution data is considered.

This paper concerns the study of a non-smooth logistic regression function. The focus is on a high-dimensional binary response case by penalizing the decomposition of the unknown logit regression function on a wavelet basis of functions evaluated on the sampling design. Sample sizes are arbitrary (not necessarily dyadic) and we consider general designs. We study separable wavelet estimators, exploiting sparsity of wavelet decompositions for signals belonging to homogeneous Besov spaces, and using efficient iterative proximal gradient descent algorithms. We also discuss a level by level block wavelet penalization technique, leading to a type of regularization in multiple logistic regression with grouped predictors. Theoretical and numerical properties of the proposed estimators are investigated. A simulation study examines the empirical performance of the proposed procedures, and real data applications demonstrate their effectiveness.

Multi-centre study is increasingly used for borrowing strength from multiple research groups to obtain reproducible study findings. Regression analysis is widely used for analysing multi-group studies, however, some of the regression predictors are nonlinear and/or often measured with batch effects. Also, the group compositions are potentially heterogeneous across different centres. The conventional pooled data analysis can cause biased regression estimates. This paper proposes an integrated partially linear regression model (IPLM) to account for predictor's nonlinearity, general batch effect, group composition heterogeneity, and potential measurement-error in covariates simultaneously. A local linear regression-based approach is employed to estimate the nonlinear component and a regularization procedure is introduced to identify the predictors' effects. The IPLM-based method has estimation consistency and variable-selection consistency. Moreover, it has a fast computing algorithm and its effectiveness is supported by simulation studies. A multi-centre Alzheimer's disease research project is provided to illustrate the proposed IPLM-based analysis.

In this paper, we discuss the joint estimation and prediction of unobserved order statistics based on a Type-II censored sample from a location-scale family. Using the concept of Loewner order, we simplify the derivations made earlier, and also strengthen in the process some of the existing results. We then study the efficiency of the methods and finally examine the determination of optimal number of order statistics to be observed as well as the performance of non-linear predictors.

This article proposes a method of optimizing control chart (sequential test) to detect an abnormal change in a sequence of finite or even small samples with the unknown change-point and the unknown post-change probability distribution. We not only introduced a performance measure for a given charting statistic to evaluate the detection effect of a control chart, but also constructed an optimal control chart under the measure. The effect of optimization method was illustrated by numerical simulations of three optimized Shewhart, CUSUM and EWMA control charts, and a real data example.

In this paper, we introduce a special multichannel model in the class of multichannel sinusoidal model. In multichannel sinusoidal model, the inherent frequencies from distinct channels are the same with different amplitudes. The underlying assumption here is that there is a fundamental frequency that is the same in each channel and the effective frequencies are harmonics of this fundamental frequency. We name this model as multichannel fundamental frequency with harmonics model. It is assumed that the errors in individual channel are independently and identically distributed whereas the signal from different channels are correlated. We propose generalized least squares estimators which become the maximum likelihood estimators also, when the error distribution of the different channels follows a multivariate Gaussian distribution. The proposed estimators are strongly consistent and asymptotically normally distributed. We have provided the implementation of the generalized least squares estimators in practice. Special attention has been taken when the number of channels is two and both have equal number of components. Simulation experiments have been carried out to observe the performances of the proposed estimators. Real data sets have been analysed using a two-channel fundamental frequency model.

The paper is devoted to the problem of parameter estimation in a multivariate optional semimartingale regression model. The family of optional semimartingales is a rich class of stochastic processes that contains càdlàg semimartingales. In general, such processes do not admit càdlàg modifications, i.e. right-continuous with finite left-limits. The weighted least squares estimator is derived, and its strong consistency is proved under general conditions on regressors. Furthermore, sequential least squares estimates are systematically studied. It is shown that such estimates have a nice statistical property called fixed accuracy. Sequential estimation procedure developed in the paper works without restrictions on dimensions of unknown parameter and of observation process. The paper contains several examples of multivariate regressions to demonstrate our results and proposed techniques.

In this work, we investigate spacings based on order statistics obtained from continuous distribution functions. At the beginning of the paper, we present distributional results for spacings and a method of classification of distributions according to their tails. Then we use this method to derive asymptotic results for spacings. By applying some special versions of Borel–Cantelli lemma, we obtain their strong limit results. At the end of the paper, we present some illustrative examples.