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Strong consistency of the local linear estimator for a generalized regression function with dependent functional data

Danilo Hiroshi Matsuoka, Hudson da Silva Torrent

Abstract

In this study, we focus on a generalized nonparametric scalar-on-function regression model for heterogeneously distributed and strongly mixing data. We provide almost complete convergence rates for the local linear estimator of the regression function. We show that, under our conditions, the pointwise and uniform convergence rates are the same on a compact set. On the other hand, when the data is dependent, it is proved that the convergence rate can be slower than those obtained for independent data. A simulation study shows the good performance and finite sample properties of the functional local linear estimator (FLL) in comparison to the local constant estimator (FLC). In addition, a one step ahead energy consumption forecasting exercise illustrates that the forecasts of the FLL estimator are significantly more accurate than those of the FLC.

Strong consistency of the local linear estimator for a generalized regression function with dependent functional data

Abstract

In this study, we focus on a generalized nonparametric scalar-on-function regression model for heterogeneously distributed and strongly mixing data. We provide almost complete convergence rates for the local linear estimator of the regression function. We show that, under our conditions, the pointwise and uniform convergence rates are the same on a compact set. On the other hand, when the data is dependent, it is proved that the convergence rate can be slower than those obtained for independent data. A simulation study shows the good performance and finite sample properties of the functional local linear estimator (FLL) in comparison to the local constant estimator (FLC). In addition, a one step ahead energy consumption forecasting exercise illustrates that the forecasts of the FLL estimator are significantly more accurate than those of the FLC.
Paper Structure (13 sections, 14 theorems, 94 equations, 3 figures)

This paper contains 13 sections, 14 theorems, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and estimation
  3. Asymptotics
  4. Preliminaries
  5. Pointwise consistency
  6. Assumptions
  7. Uniform consistency
  8. Assumptions
  9. Application to Wiener processes and a simulation study
  10. Real Data Application
  11. Conclusion
  12. Lemmas
  13. Propositions

Key Result

Theorem 1

Suppose that assumptions A1-A10 are fullfiled. Then

Figures (3)

  • Figure 1: MSPE - Simulation Study The figure displays the boxplot of the mean squared predictive error (MSPE), as described in equation \ref{['eq8_']}, for both FLC and FLL estimators. Three values for the coefficient of the AR(1) process that characterizes the error sequence are presented: $0, 1 / 3$ and $2 / 3$.
  • Figure 2: Time Series of Energy Consumption The figure displays the sample of hourly energy consumption data from America Electric Power (AEP). The data ranges from 2004-10-01 to 2018-08-02, giving $T = 5054$ days of observations. A rolling window scheme is considered with window length equal to $W = 1081$. Thus, we generate $T_{out} = 3973$ one step ahead forecasts of the daily energy consumption.
  • Figure 3: CSFE - Energy Forecast Application The figure displays the cumulative squared forecast error (CSFE) as described in equation \ref{['eq16_']}. Increasing CSFE implies better predictive performance of the FLL estimator compared to the FLC estimator, while decreasing CSFE implies otherwise.

Theorems & Definitions (19)

  • Definition 1: Strong mixing
  • Definition 2: Asymptotic orders
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Definition 3: Kolmogorov's entropy
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • ...and 9 more