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We consider determining change points in a time series of age-specific mortality and fertility curves observed over time. We propose two detection methods for identifying these change points. The first method uses a functional cumulative sum statistic to pinpoint the change point. The second method computes a univariate time series of integrated squared forecast errors after fitting a functional time-series model before applying a change-point detection method to the errors to determine the change point. Using Australian age-specific fertility and mortality data, we apply these methods to locate the change points and identify the optimal training period to achieve improved forecast accuracy., Comment: 23 pages, 3 figures, 4 tables

Fertility differentials by urban-rural residence and nativity of women in Australia significantly impact population composition at sub-national levels. We aim to provide consistent fertility forecasts for Australian women characterized by age, region, and birthplace. Age-specific fertility rates at the national and sub-national levels obtained from census data between 1981-2011 are jointly modeled and forecast by the grouped functional time series method. Forecasts for women of each region and birthplace are reconciled following the chosen hierarchies to ensure that results at various disaggregation levels consistently sum up to the respective national total. Coupling the region of residence disaggregation structure with the trace minimization reconciliation method produces the most accurate point and interval forecasts. In addition, age-specific fertility rates disaggregated by the birthplace of women show significant heterogeneity that supports the application of the grouped forecasting method., Comment: 34 pages, 6 figures, 3 tables

This paper introduces a robust estimation strategy for the spatial functional linear regression model using dimension reduction methods, specifically functional principal component analysis (FPCA) and functional partial least squares (FPLS). These techniques are designed to address challenges associated with spatially correlated functional data, particularly the impact of outliers on parameter estimation. By projecting the infinite-dimensional functional predictor onto a finite-dimensional space defined by orthonormal basis functions and employing M-estimation to mitigate outlier effects, our approach improves the accuracy and reliability of parameter estimates in the spatial functional linear regression context. Simulation studies and empirical data analysis substantiate the effectiveness of our methods, while an appendix explores the Fisher consistency and influence function of the FPCA-based approach. The rfsac package in R implements these robust estimation strategies, ensuring practical applicability for researchers and practitioners., Comment: 25 pages, 4 figures, 2 tables

A function-on-function regression model with quadratic and interaction effects of the covariates provides a more flexible model. Despite several attempts to estimate the model's parameters, almost all existing estimation strategies are non-robust against outliers. Outliers in the quadratic and interaction effects may deteriorate the model structure more severely than their effects in the main effect. We propose a robust estimation strategy based on the robust functional principal component decomposition of the function-valued variables and $\tau$-estimator. The performance of the proposed method relies on the truncation parameters in the robust functional principal component decomposition of the function-valued variables. A robust Bayesian information criterion is used to determine the optimum truncation constants. A forward stepwise variable selection procedure is employed to determine relevant main, quadratic, and interaction effects to address a possible model misspecification. The finite-sample performance of the proposed method is investigated via a series of Monte-Carlo experiments. The proposed method's asymptotic consistency and influence function are also studied in the supplement, and its empirical performance is further investigated using a U.S. COVID-19 dataset., Comment: 35 pages, 3 tables

We introduce a compositional power transformation, known as an {\alpha}-transformation, to model and forecast a time series of life-table death counts, possibly with zero counts observed at older ages. As a generalisation of the isometric log-ratio transformation (i.e., {\alpha} = 0), the {\alpha} transformation relies on the tuning parameter {\alpha}, which can be determined in a data-driven manner. Using the Australian age-specific period life-table death counts from 1921 to 2020, the {\alpha} transformation can produce more accurate short-term point and interval forecasts than the log-ratio transformation. The improved forecast accuracy of life-table death counts is of great importance to demographers and government planners for estimating survival probabilities and life expectancy and actuaries for determining annuity prices and reserves for various initial ages and maturity terms., Comment: 25 pages, 6 tables, 5 figures

Like density functions, period life-table death counts are nonnegative and have a constrained integral, and thus live in a constrained nonlinear space. Implementing established modelling and forecasting methods without obeying these constraints can be problematic for such nonlinear data. We introduce cumulative distribution function transformation to forecast the life-table death counts. Using the Japanese life-table death counts obtained from the Japanese Mortality Database (2024), we evaluate the point and interval forecast accuracies of the proposed approach, which compares favourably to an existing compositional data analytic approach. The improved forecast accuracy of life-table death counts is of great interest to demographers for estimating age-specific survival probabilities and life expectancy and actuaries for determining temporary annuity prices for different ages and maturities., Comment: 29 pages, 7 figures, 9 tables. arXiv admin note: text overlap with arXiv:1908.01446

Robust estimation for modern portfolio selection on a large set of assets becomes more important due to large deviation of empirical inference on big data. We propose a distributionally robust methodology for high-dimensional mean-variance portfolio problem, aiming to select an optimal conservative portfolio allocation by taking distribution uncertainty into account. With the help of factor structure, we extend the distributionally robust mean-variance problem investigated by Blanchet et al. (2022, Management Science) to the high-dimensional scenario and transform it to a new penalized risk minimization problem. Furthermore, we propose a data-adaptive method to estimate the quantified uncertainty size, which is the radius around the empirical probability measured by the Wasserstein distance. Asymptotic consistency is derived for the estimation of the population parameters involved in selecting the uncertainty size and the selected portfolio return. Our Monte-Carlo simulation results show that the chosen uncertainty size and target return from the proposed procedure are very close to the corresponding oracle version, and the new portfolio strategy is of low risk. Finally, we conduct empirical studies based on S&P index components to show the robust performance of our proposal in terms of risk controlling and return-risk balancing., Comment: 41 pages, 3 figures, 6 tables

Time series clustering is an important data mining task with a wide variety of applications. While most methods focus on time series taking values on the real line, very few works consider functional time series. However, functional objects frequently arise in many fields, such as actuarial science, demography or finance. Functional time series are indexed collections of infinite-dimensional curves viewed as random elements taking values in a Hilbert space. In this paper, the problem of clustering functional time series is addressed. To this aim, a distance between functional time series is introduced and used to construct a clustering procedure. The metric relies on a measure of serial dependence which can be seen as a natural extension of the classical quantile autocorrelation function to the functional setting. Since the dynamics of the series may vary over time, we adopt a fuzzy approach, which enables the procedure to locate each series into several clusters with different membership degrees. The resulting algorithm can group series generated from similar stochastic processes, reaching accurate results with series coming from a broad variety of functional models and requiring minimum hyperparameter tuning. Several simulation experiments show that the method exhibits a high clustering accuracy besides being computationally efficient. Two interesting applications involving high-frequency financial time series and age-specific mortality improvement rates illustrate the potential of the proposed approach., Comment: 43 pages, 5 figures, 10 tables. arXiv admin note: substantial text overlap with arXiv:2402.08687

We introduce a statistical method for modeling and forecasting functional panel data, where each element is a density. Density functions are nonnegative and have a constrained integral and thus do not constitute a linear vector space. We implement a center log-ratio transformation to transform densities into unconstrained functions. These functions exhibit cross-sectionally correlation and temporal dependence. Via a functional analysis of variance decomposition, we decompose the unconstrained functional panel data into a deterministic trend component and a time-varying residual component. To produce forecasts for the time-varying component, a functional time series forecasting method, based on the estimation of the long-range covariance, is implemented. By combining the forecasts of the time-varying residual component with the deterministic trend component, we obtain h-step-ahead forecast curves for multiple populations. Illustrated by age- and sex-specific life-table death counts in the United States, we apply our proposed method to generate forecasts of the life-table death counts for 51 states.

The age pension aims to assist eligible elderly Australians meet specific age and residency criteria in maintaining basic living standards. In designing efficient pension systems, government policymakers seek to satisfy the expectations of the overall aging population in Australia. However, the population's unique demographic characteristics at the state and territory level are often overlooked due to the lack of available data. We use the Hamilton-Perry model, which requires minimum input, to model and forecast the evolution of age-specific populations at the state level. We also integrate the obtained sub-national demographic information to determine sustainable pension ages up to 2051. We also investigate pension welfare distribution in all states and territories to identify disadvantaged residents under the current pension system. Using the sub-national mortality data for Australia from 1971 to 2021 obtained from AHMD (2023), we implement the Hamilton-Perry model with the help of functional time series forecasting techniques. With forecasts of age-specific population sizes for each state and territory, we compute the old age dependency ratio to determine the nationwide sustainable pension age., Comment: 32 pages, 15 figures, 1 table

We study the modeling and forecasting of high-dimensional functional time series (HDFTS), which can be cross-sectionally correlated and temporally dependent. We introduce a decomposition of the HDFTS into two distinct components: a deterministic component and a residual component that varies over time. The decomposition is derived through the estimation of two-way functional analysis of variance. A functional time series forecasting method, based on functional principal component analysis, is implemented to produce forecasts for the residual component. By combining the forecasts of the residual component with the deterministic component, we obtain forecast curves for multiple populations. We apply the model to age- and sex-specific mortality rates in the United States, France, and Japan, in which there are 51 states, 95 departments, and 47 prefectures, respectively. The proposed method is capable of delivering more accurate point and interval forecasts in forecasting multi-population mortality than several benchmark methods considered., Comment: 31 pages, 6 figures

Intraday financial data often take the form of a collection of curves that can be observed sequentially over time, such as intraday stock price curves. These curves can be viewed as a time series of functions observed on equally spaced and dense grids. Due to the curse of dimensionality, high-dimensional data poses challenges from a statistical aspect; however, it also provides opportunities to analyze a rich source of information so that the dynamic changes within short-time intervals can be better understood. We consider a sieve bootstrap method of Paparoditis and Shang (2022) to construct one-day-ahead point and interval forecasts in a model-free way. As we sequentially observe new data, we also implement two dynamic updating methods to update point and interval forecasts for achieving improved accuracy. The forecasting methods are validated through an empirical study of 5-minute cumulative intraday returns of the S&P/ASX All Ordinaries Index., Comment: 25 pages, 10 figures, 2 tables

In this paper, we consider detecting and estimating breaks in heterogeneous mean functions of high-dimensional functional time series which are allowed to be cross-sectionally correlated and temporally dependent. A new test statistic combining the functional CUSUM statistic and power enhancement component is proposed with asymptotic null distribution theory comparable to the conventional CUSUM theory derived for a single functional time series. In particular, the extra power enhancement component enlarges the region where the proposed test has power, and results in stable power performance when breaks are sparse in the alternative hypothesis. Furthermore, we impose a latent group structure on the subjects with heterogeneous break points and introduce an easy-to-implement clustering algorithm with an information criterion to consistently estimate the unknown group number and membership. The estimated group structure can subsequently improve the convergence property of the post-clustering break point estimate. Monte-Carlo simulation studies and empirical applications show that the proposed estimation and testing techniques have satisfactory performance in finite samples.

An important issue in functional time series analysis is whether an observed series comes from a purely random process. We extend the BDS test, a widely-used nonlinear independence test, to the functional time series. Like the BDS test in the univariate case, the functional BDS test can act as the model specification test to evaluate the adequacy of various prediction models and as a nonlinearity test to detect the existence of nonlinear structures in a functional time series after removing the linear structure exhibited. We show that the test statistic from the functional BDS test has the same asymptotic properties as those in the univariate case and provides the recommended range of its hyperparameters. Additionally, empirical data analysis features its applications in evaluating the adequacy of the fAR(1) and fGARCH(1,1) models in fitting the daily curves of cumulative intraday returns (CIDR) of the VIX index. We showed that the functional BDS test remedies the weakness of the existing independence test in the literature, as the latter is restricted in detecting linear structures, thus, can neglect nonlinear temporal structures., Comment: 35 pages, 3 figures

We introduce methods and theory for fractionally cointegrated curve time series. We develop a variance-ratio test to determine the dimensions associated with the nonstationary and stationary subspaces. For each subspace, we apply a local Whittle estimator to estimate the long-memory parameter and establish its consistency. A Monte Carlo study of finite-sample performance is included, along with two empirical applications.

Scalar-on-function logistic regression, where the response is a binary outcome and the predictor consists of random curves, has become a general framework to explore a linear relationship between the binary outcome and functional predictor. Most of the methods used to estimate this model are based on the least-squares type estimators. However, the least-squares estimator is seriously hindered by outliers, leading to biased parameter estimates and an increased probability of misclassification. This paper proposes a robust partial least squares method to estimate the regression coefficient function in the scalar-on-function logistic regression. The regression coefficient function represented by functional partial least squares decomposition is estimated by a weighted likelihood method, which downweighs the effect of outliers in the response and predictor. The estimation and classification performance of the proposed method is evaluated via a series of Monte Carlo experiments and a strawberry puree data set. The results obtained from the proposed method are compared favorably with existing methods., Comment: 23 pages, 8 figures, to appear at the Communications in Statistics - Theory and Methods

The scalar-on-function regression model has become a popular analysis tool to explore the relationship between a scalar response and multiple functional predictors. Most of the existing approaches to estimate this model are based on the least-squares estimator, which can be seriously affected by outliers in empirical datasets. When outliers are present in the data, it is known that the least-squares-based estimates may not be reliable. This paper proposes a robust functional partial least squares method, allowing a robust estimate of the regression coefficients in a scalar-on-multiple-function regression model. In our method, the functional partial least squares components are computed via the partial robust M-regression. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments and two chemometric datasets: glucose concentration spectrometric data and sugar process data. The results produced by the proposed method are compared favorably with some of the classical functional or multivariate partial least squares and functional principal component analysis methods., Comment: 31 pages, 6 figures, to appear at the Journal of Chemometrics

We propose modeling raw functional data as a mixture of a smooth function and a high-dimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is inadequate to recover the smooth curve or capture the data variation in some situations. These include cases where there is a large amount of measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. A factor-augmented smoothing model is proposed to address these challenges, and an iterative numerical estimation approach is implemented in practice. Including the factor model component in the proposed method solves the aforementioned problems since a few common factors often drive the variation that cannot be captured by the smoothing model. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. Specifically, we show that the smoothing coefficients projected on the complement space of the factor loading matrix are asymptotically normal. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies illustrate that these factor adjustments are essential in improving estimation accuracy and avoiding the curse of dimensionality. The superiority of our model is also shown in modeling Australian temperature data., Comment: 88 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:2102.02580

Modelling and forecasting homogeneous age-specific mortality rates of multiple countries could lead to improvements in long-term forecasting. Data fed into joint models are often grouped according to nominal attributes, such as geographic regions, ethnic groups, and socioeconomic status, which may still contain heterogeneity and deteriorate the forecast results. Our paper proposes a novel clustering technique to pursue homogeneity among multiple functional time series based on functional panel data modelling to address this issue. Using a functional panel data model with fixed effects, we can extract common functional time series features. These common features could be decomposed into two components: the functional time trend and the mode of variations of functions (functional pattern). The functional time trend reflects the dynamics across time, while the functional pattern captures the fluctuations within curves. The proposed clustering method searches for homogeneous age-specific mortality rates of multiple countries by accounting for both the modes of variations and the temporal dynamics among curves. We demonstrate that the proposed clustering technique outperforms other existing methods through a Monte Carlo simulation and could handle complicated cases with slow decaying eigenvalues. In empirical data analysis, we find that the clustering results of age-specific mortality rates can be explained by the combination of geographic region, ethnic groups, and socioeconomic status. We further show that our model produces more accurate forecasts than several benchmark methods in forecasting age-specific mortality rates.

Using the ordinal pattern concept in permutation entropy, we propose a model sufficiency test to study a given model's point prediction accuracy. Compared to some classical model sufficiency tests, such as the Broock et al.'s (1996) test, our proposal does not require a sufficient model to eliminate all structures exhibited in the estimated residuals. When the innovations in the investigated data's underlying dynamics show a certain structure, such as higher-moment serial dependence, the Broock et al.'s (1996) test can lead to erroneous conclusions about the sufficiency of point predictors. Due to the structured innovations, inconsistency between the model sufficiency tests and prediction accuracy criteria can occur. Our proposal fills in this incoherence between model and prediction evaluation approaches and remains valid when the underlying process has non-white additive innovation., Comment: 32 pages, 5 figures, to appear at the Journal of Forecasting

This paper proposes a new AR-sieve bootstrap approach on high-dimensional time series. The major challenge of classical bootstrap methods on high-dimensional time series is two-fold: the curse dimensionality and temporal dependence. To tackle such difficulty, we utilise factor modelling to reduce dimension and capture temporal dependence simultaneously. A factor-based bootstrap procedure is constructed, which conducts AR-sieve bootstrap on the extracted low-dimensional common factor time series and then recovers the bootstrap samples for original data from the factor model. Asymptotic properties for bootstrap mean statistics and extreme eigenvalues are established. Various simulations further demonstrate the advantages of the new AR-sieve bootstrap under high-dimensional scenarios. Finally, an empirical application on particulate matter (PM) concentration data is studied, where bootstrap confidence intervals for mean vectors and autocovariance matrices are provided.

In this study, we propose a function-on-function linear quantile regression model that allows for more than one functional predictor to establish a more flexible and robust approach. The proposed model is first transformed into a finite-dimensional space via the functional principal component analysis paradigm in the estimation phase. It is then approximated using the estimated functional principal component functions, and the estimated parameter of the quantile regression model is constructed based on the principal component scores. In addition, we propose a Bayesian information criterion to determine the optimum number of truncation constants used in the functional principal component decomposition. Moreover, a stepwise forward procedure and the Bayesian information criterion are used to determine the significant predictors for including in the model. We employ a nonparametric bootstrap procedure to construct prediction intervals for the response functions. The finite sample performance of the proposed method is evaluated via several Monte Carlo experiments and an empirical data example, and the results produced by the proposed method are compared with the ones from existing models., Comment: 24 pages, 8 figures, to appear at the Mathematical Modelling and Analysis

We study the importance of group structure in grouped functional time series. Due to the non-uniqueness of group structure, we investigate different disaggregation structures in grouped functional time series. We address a practical question on whether or not the group structure can affect forecast accuracy. Using a dynamic multivariate functional time series method, we consider joint modeling and forecasting multiple series. Illustrated by Japanese sub-national age-specific mortality rates from 1975 to 2016, we investigate one- to 15-step-ahead point and interval forecast accuracies for the two group structures., Comment: 29 pages, 8 figures, to appear at the Journal of Data Science

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., Comment: 27 pages, 8 figures, to appear in the Brazilian Journal of Probability and Statistics

In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method., Comment: 30 pages, 5 figures, to appear at the Journal of Agricultural, Biological and Environmental Statistics

We propose a dual-factor model for high-dimensional functional time series (HDFTS) that considers multiple populations. The HDFTS is first decomposed into a collection of functional time series (FTS) in a lower dimension and a group of population-specific basis functions. The system of basis functions describes cross-sectional heterogeneity, while the reduced-dimension FTS retains most of the information common to multiple populations. The low-dimensional FTS is further decomposed into a product of common functional loadings and a matrix-valued time series that contains the most temporal dynamics embedded in the original HDFTS. The proposed general-form dual-factor structure is connected to several commonly used functional factor models. We demonstrate the finite-sample performances of the proposed method in recovering cross-sectional basis functions and extracting common features using simulated HDFTS. An empirical study shows that the proposed model produces more accurate point and interval forecasts for subnational age-specific mortality rates in Japan. The financial benefits associated with the improved mortality forecasts are translated into a life annuity pricing scheme.

This paper presents static and dynamic versions of univariate, multivariate, and multilevel functional time-series methods to forecast implied volatility surfaces in foreign exchange markets. We find that dynamic functional principal component analysis generally improves out-of-sample forecast accuracy. More specifically, the dynamic univariate functional time-series method shows the greatest improvement. Our models lead to multiple instances of statistically significant improvements in forecast accuracy for daily EUR-USD, EUR-GBP, and EUR-JPY implied volatility surfaces across various maturities, when benchmarked against established methods. A stylised trading strategy is also employed to demonstrate the potential economic benefits of our proposed approach., Comment: 52 pages, 5 figures, to appear at the International Journal of Forecasting

We propose a novel method to extract global and local features of functional time series. The global features concerning the dominant modes of variation over the entire function domain, and local features of function variations over particular short intervals within function domain, are both important in functional data analysis. Functional principal component analysis (FPCA), though a key feature extraction tool, only focus on capturing the dominant global features, neglecting highly localized features. We introduce a FPCA-BTW method that initially extracts global features of functional data via FPCA, and then extracts local features by block thresholding of wavelet (BTW) coefficients. Using Monte Carlo simulations, along with an empirical application on near-infrared spectroscopy data of wood panels, we illustrate that the proposed method outperforms competing methods including FPCA and sparse FPCA in the estimation functional processes. Moreover, extracted local features inheriting serial dependence of the original functional time series contribute to more accurate forecasts. Finally, we develop asymptotic properties of FPCA-BTW estimators, discovering the interaction between convergence rates of global and local features.

In this research, a functional time series model was introduced to predict future realizations of river flow time series. The proposed model was constructed based on a functional time series's correlated lags and the essential exogenous climate variables. Rainfall, temperature, and evaporation variables were hypothesized to have substantial functionality in river flow simulation. Because an actual time series model is unspecified and the input variables' significance for the learning process is unknown in practice, it was employed a variable selection procedure to determine only the significant variables for the model. A nonparametric bootstrap model was also proposed to investigate predictions' uncertainty and construct pointwise prediction intervals for the river flow curve time series. Historical datasets at three meteorological stations (Mosul, Baghdad, and Kut) located in the semi-arid region, Iraq, were used for model development. The prediction performance of the proposed model was validated against existing functional and traditional time series models. The numerical analyses revealed that the proposed model provides competitive or even better performance than the benchmark models. Also, the incorporated exogenous climate variables have substantially improved the modeling predictability performance. Overall, the proposed model indicated a reliable methodology for modeling river flow within the semi-arid region., Comment: 42 pages, 13 figures, to appear at the Journal of Hydrology

We propose modeling raw functional data as a mixture of a smooth function and a highdimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is not adequate to recover the smooth curve or capture the data variation in some situations. These include cases where there is a large amount of measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. To address these challenges, a factor-augmented smoothing model is proposed, and an iterative numerical estimation approach is implemented in practice. Including the factor model component in the proposed method solves the aforementioned problems since a few common factors often drive the variation that cannot be captured by the smoothing model. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. Specifically, we show that the smoothing coefficients projected on the complement space of the factor loading matrix is asymptotically normal. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies illustrate that these factor adjustments are essential in improving estimation accuracy and avoiding the curse of dimensionality. The superiority of our model is also shown in modeling Canadian weather data and Australian temperature data.

We propose a double bootstrap procedure for reducing coverage error in the confidence intervals of descriptive statistics for independent and identically distributed functional data. Through a series of Monte Carlo simulations, we compare the finite sample performance of single and double bootstrap procedures for estimating the distribution of descriptive statistics for independent and identically distributed functional data. At the cost of longer computational time, the double bootstrap with the same bootstrap method reduces confidence level error and provides improved coverage accuracy than the single bootstrap. Illustrated by a Canadian weather station data set, the double bootstrap procedure presents a tool for visualising the distribution of the descriptive statistics for the functional data., Comment: 22 pages, 9 figures, 1 table, to appear at the Journal of Statistical Computation and Simulation

We consider forecasting functional time series of extreme values within a generalised extreme value distribution (GEV). The GEV distribution can be characterised using the three parameters (location, scale and shape). As a result, the forecasts of the GEV density can be accomplished by forecasting these three latent parameters. Depending on the underlying data structure, some of the three parameters can either be modelled as scalars or functions. We provide two forecasting algorithms to model and forecast these parameters. To assess the forecast uncertainty, we apply a sieve bootstrap method to construct pointwise and simultaneous prediction intervals of the forecasted extreme values. Illustrated by a daily maximum temperature dataset, we demonstrate the advantages of modelling these parameters as functions. Further, the finite-sample performance of our methods is quantified using several Monte-Carlo simulated data under a range of scenarios., Comment: 21 pages, 4 figures, 1 table, to appear at Communication in Statistics: Case Studies and Data Analysis

A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method., Comment: 34 pages, 6 figures, 2 tables, to appear at Computational Statistics