The function-on-function linear regression model in which the response and predictors consist of random curves has become a general framework to investigate the relationship between the functional response and functional predictors. Existing methods to estimate the model parameters may be sensitive to outlying observations, common in empirical applications. In addition, these methods may be severely affected by such observations, leading to undesirable estimation and prediction results. A robust estimation method, based on iteratively reweighted simple partial least squares, is introduced to improve the prediction accuracy of the function-on-function linear regression model in the presence of outliers. The performance of the proposed method is based on the number of partial least squares components used to estimate the function-on-function linear regression model. Thus, the optimum number of components is determined via a data-driven error criterion. The finite-sample performance of the proposed method is investigated via several Monte Carlo experiments and an empirical data analysis. In addition, a nonparametric bootstrap method is applied to construct pointwise prediction intervals for the response function. The results are compared with some of the existing methods to illustrate the improvement potentially gained by the proposed method.

Time series analysis is widely used in the fields of economics, ecology and medicine. Robust variable selection procedures through penalized regression have been gaining increased attention. In our work, a robust penalized regression estimator based on exponential squared loss for autoregressive (AR) models is proposed and discussed. The objective model with adaptive Lasso penalty realizes variable selection and parameter estimation simultaneously. Under some regular conditions, we establish the asymptotic and “Oracle” properties of the proposed estimator. In particular, the induced non-convex and non-differentiable mathematical programming problem offers challenges for solving algorithms. To solve this problem efficiently, we specially design a block coordinate descent (BCD) algorithm equipped with concave-convex process (CCCP) and provide a convergence guarantee. Numerical simulation studies are carried out to show that the proposed method is particularly robust and applicable compared with some recent methods when there are different types of noise or different intensity of noise. Furthermore, an application on a dataset of daily minimum temperature in Melbourne over 1981–1990 is performed.

Scalar-on-function regression allows for a scalar response to be dependent on functional predictors; however, not much work has been done when interaction effects between the functional predictors are included. In this paper, we introduce a multiple functional linear regression model with interaction terms. Meanwhile, we enforce the hierarchical structure constraint on the model, that is, interaction terms can be selected into the model only if the associated main effects are in the model. Based on the functional principal component analysis and group smoothly clipped absolute deviation (SCAD) penalty, we propose a new penalized estimation procedure to select the important functional predictors and interactions while automatically obeying the hierarchical structure. With appropriate selection of the tuning parameters, the rates of convergence of the proposed estimators and the consistency of the model selection procedure are established under some regularity conditions. At last, we illustrate the finite sample performance of our proposed methods with some simulation studies and a real data application.

Identifying key nodes, estimating the probability of connection between them, and distinguishing latent groups are some of the main objectives of social network analysis. In this paper, we propose a class of blockmodels to model stochastic equivalence and visualize groups in an unobservable space. In this setting, the proposed method is based on two approaches: latent distances and latent dissimilarities at the group level. The projection proposed in the paper is performed without needing to project individuals, unlike the main approaches in the literature. Our approach can be used in undirected or directed graphs and is flexible enough to cluster and quantify between and within-group tie probabilities in social networks. The effectiveness of the methodology in representing groups in latent spaces was analyzed under artificial datasets and in two case studies.

The presence of multicollinearity adversely affects the inferential properties of the maximum likelihood (ML) estimator in logistic regression model. It is a well-established fact that the use of restrictions lowers the effect of multicollinearity. In this article, an alternative to the ML estimator has been introduced by combining the exact prior information into the logistic $\mathit{r}-\mathit{k}$ class (Lrk) estimator. The estimator is named a logistic restricted $\mathit{r}-\mathit{k}$ class estimator. Another estimator, logistic restricted PCR estimator, is also developed as a special case of the LRrk estimator. The asymptotic mean squared error (MSE) matrix properties of the estimators are studied and necessary and sufficient conditions are derived. Further, a Monte Carlo simulation study is performed to compare the performance of the estimators in terms of the scalar MSE and the prediction MSE. It is found that the proposed estimators perform better than the existing estimators in most of the cases considered. Moreover, a numerical example has also been presented for comparing the performance of the estimators.

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in $(0,1)\cup (1,4/3)$ in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

In this paper, we mainly focus on the sums of non-negative integer-valued 1-dependent random variables and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as the Stein operator, uniform and non-uniform bounds on the solution of the Stein equation. Using Stein’s method, we obtain error bounds for the approximation problem considered. The obtained results can also be applied to the sums of m-dependent random variables via appropriate rearrangements of random variables. As special cases, we discuss two applications, namely, 2-runs and $({\mathit{k}}_1,{\mathit{k}}_2)$-runs, and compare our bounds with existing bounds.

In this paper, we mainly study the consistency of the nearest neighbor estimator of the density function based on m-extended negatively dependent samples. The weak consistency, strong consistency, uniformly strong consistency and the convergence rate are established under some mild conditions. The results obtained in this paper extend and improve some existing ones in the literature.

We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.

One of the most important assumptions in multiple regression analysis is the independence of the explanatory variables, however, this assumption is violated in several situations. In this work, we investigate regression equations when this independence does not hold and the explanatory variables are connected by many of elliptical copulas. We apply the proposed regression equation to study its heteroscedasticity diagnostic and using simulated data we also assess our regression model. A cross-validation procedure is carried out to ensure the unbiasedness of the results. Also, a real data analysis is presented as an application.