In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of discrete, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of ${\mathbb{R}}^d$, which provides a quantitative measure of estimation accuracy. In addition, the uniform convergence with rate and the normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions on the data. By demonstrating that the wavelet methodology remains robust under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations.

In this research, we formulated an asymptotic theory for single index regression applied to locally stationary functional time series. Our approach involved introducing estimators featuring a regression function that exhibited smooth temporal changes. We rigorously established the uniform convergence rates for kernel estimators, specifically the Nadaraya-Watson (NW) estimator for the regression function. Additionally, we provided a central limit theorem for the NW estimator. Finally, the theory was supported by a comprehensive simulation study to investigate the finite-sample performance of our proposed method.

$U$-statistics represent a fundamental class of statistics used to model quantities derived from responses of multiple subjects. These statistics extend the concept of the empirical mean of a $d$-variate random variable $X$ by considering sums over all distinct $m$-tuples of observations of $X$. Within this realm, W. Stute ^[134] introduced conditional $U$-statistics, a generalization of the Nadaraya-Watson estimators for regression functions, and demonstrated their strong point-wise consistency. This paper presented a first theoretical examination of the Dirichlet kernel estimator for conditional $U$-statistics on the $dm$-dimensional simplex. This estimator, being an extension of the univariate beta kernel estimator, effectively addressed boundary biases. Our analysis established its asymptotic normality and uniform strong consistency. Additionally, we introduced a beta kernel estimator specifically tailored for conditional $U$-statistics, demonstrating both weak and strong uniform convergence. Our investigation considered the expansion of compact sets and various sequences of smoothing parameters. For the first time, we examined conditional $U$-statistics based on mixed categorical and continuous regressors. We presented new findings on conditional $U$-statistics smoothed by multivariate Bernstein kernels, previously unexplored in the literature. These results are derived under sufficiently broad conditions on the underlying distributions. The main ingredients used in our proof were truncation methods and sharp exponential inequalities tailored to the $U$-statistics in connection with the empirical processes theory. Our theoretical advancements significantly contributed to the field of asymmetric kernel estimation, with potential applications in areas such as discrimination problems, $\ell$-sample conditional $U$-statistics, and the Kendall rank correlation coefficient. Finally, we conducted some simulations to demonstrate the small sample performances of the estimators.

Local polynomial fitting exhibits numerous compelling statistical properties, particularly within the intricate realm of multivariate analysis. However, as functional data analysis gains prominence as a dynamic and pertinent field in data science, the exigency arises for the formulation of a specialized theory tailored to local polynomial fitting. We explored the intricate task of estimating the regression function operator and its partial derivatives for stationary mixing random processes, denoted as $(Y_i,X_i)$, using local higher-order polynomial fitting. Our key contributions include establishing the joint asymptotic normality of the estimates for both the regression function and its partial derivatives, specifically in the context of strongly mixing processes. Additionally, we provide explicit expressions for the bias and the variance-covariance matrix of the asymptotic distribution. Demonstrating uniform strong consistency over compact subsets, along with delineating the rates of convergence, we substantiated these results for both the regression function and its partial derivatives. Importantly, these findings rooted in reasonably broad conditions that underpinned the underlying models. To demonstrate practical applicability, we leveraged our results to compute pointwise confidence regions. Finally, we extended our ideas to the nonparametric conditional distribution, and obtained its limiting distribution.

Analyzing the real impact of spatial dependency in financial time series data is crucial to financial risk management. It has been a challenging issue in the last decade. This is because most financial transactions are performed via the internet and the spatial dependency between different international stock markets is not standard. The present paper investigates functional expectile regression as a spatial financial risk model. Specifically, we construct a nonparametric estimator of this functional model for the functional single index regression (FSIR) structure. The asymptotic properties of this estimator are elaborated over general spatial settings. More precisely, we establish Borel-Cantelli consistency (BCC) of the constructed estimator. The latter is obtained with the precision of the convergence rate. A simulation investigation is performed to show the easy applicability of the constructed estimator in practice. Finally, real data analysis about the financial data (Euro Stoxx-50 index data) is used to illustrate the effectiveness of our methodology.

In the present study, we address the nonparametric estimation challenge related to the regression function within the Single Functional Index Model in the random censoring framework. The principal achievement of this investigation lies in the establishment of the asymptotic characteristics of the estimator, including rates of almost complete convergence. Moreover, we establish the asymptotic normality of the constructed estimator under mild conditions. Subsequently, we provide the application of our findings towards the construction of confidence intervals. Lastly, we illuminate the finite-sample performance of both the model and the estimation methodology through the analysis of simulated data and a real-world data example.

In recent years, there has been a notable shift in focus towards the analysis of non-stationary time series, driven largely by the complexities associated with delineating significant asymptotic behaviors inherent to such processes. The genesis of the theory of locally stationary processes arises from the quest for asymptotic inference grounded in nonparametric statistics. This paper endeavors to formulate a comprehensive framework for conducting inference within the realm of locally stationary functional time series by harnessing the conditional $U$-statistics methodology as propounded by W. Stute in 1991. The proposed methodology extends the Nadaraya-Watson regression function estimations. Within this context, a novel estimator was introduced for the single index conditional $U$-statistics operator, adept at accommodating the non-stationary attributes inherent to the data-generating process. The primary objective of this paper was to establish the weak convergence of conditional $U$-processes within the domain of locally stationary functional mixing data. Specifically, the investigation delved into scenarios of weak convergence involving functional explanatory variables, considering both bounded and unbounded sets of functions while adhering to specific moment requirements. The derived findings emanate from broad structural specifications applicable to the class of functions and models under scrutiny. The theoretical insights expounded in this study constitute pivotal tools for advancing the domain of functional data analysis.