Local Walsh-average regression

Local polynomial regression is a useful nonparametric regression tool to explore fine data structures and has been widely used in practice. In this paper, we propose a new nonparametric regression technique called local composite-quantile-regression (CQR) smoothing in order to further improve local polynomial regression. Sampling properties of the proposed estimation procedure are studied. We derive the asymptotic bias, variance and normality of the proposed estimate. Asymptotic relative efficiency of the proposed estimate with respect to the local polynomial regression is investigated. It is shown that the proposed estimate can be much more efficient than the local polynomial regression estimate for various non-normal errors, while being almost as efficient as the local polynomial regression estimate for normal errors. Simulation is conducted to examine the performance of the proposed estimates. The simulation results are consistent with our theoretical findings. A real data example is used to illustrate the proposed method.

This chapter focuses on an alternative robust estimation/testing procedure for possibly infinite variance time series models. In the context of inference for heavy-tailed observation, least absolute deviations (LAD) estimators are known to be less sensitive to outliers than the classical least squares regression. This section generalizes the LAD regression-based inference procedure to the self-weighted version, which is a concept originally introduced by Ling (2005) for AR processes. Using the self-weighting method, we extend the generalized empirical likelihood (GEL) method to possibly infinite variance process, and construct feasible and robust estimation/testing procedures. The former half of this chapter provides a brief introduction to the LAD regression method for possibly infinite variance ARMA models, and construct the self-weighted GEL statistic following Akashi (2017). The desirable asymptotic properties of the proposed statistics will be elucidated. The latter half of this chapter illustrates an important application of the self-weighted GEL method to the change point problem of time series models.

In general linear modeling, an alternative to the method of least squares (LS) is the least absolute deviations (LAD) procedure. Although LS is more widely used, the LAD approach yields better estimates in the presence of outliers. In this paper, we examine the performance of LAD estimators for the parameters of the first‐order autoregressive model in the presence of outliers. A simulation study compared these estimates with those given by LS. The general conclusion is that LAD does not deal successfully with additive outliers. A simple procedure is proposed which allows exception reporting when outliers occur.

Observations at the first opposition are used to determine the orbits of 16 near-Earth asteroids with two or more observed oppositions. The orbits are improved by the differential method. This paper considers two modifications of the improvement technique, which are compared to the classical method based on the principle of the least square method (LSM). The first modification uses the principle of least absolute deviations (LAD). In the second modification, the differences O - C (between the observed and calculated positions) are transformed to fit into a new coordinate system whereby the axes go parallel and perpendicular to the asteroid's apparent path (the modified differential method (MDM)). The orbits determined from one opposition by the classical LSM, LAD, and MDM are compared to a more accurate orbit calculated by the LSM from all the available oppositions. The calculations show that in 13 cases out of 16, the asteroid orbits calculated by LAD are more accurate than those calculated by the classical LSM. The improvement by the modified differential method, which includes the O - C transformation, does not produce any perceptible increase in accuracy when compared to the orbits calculated by the classical method.

In order to identify the parameters of nonlinear Hammerstein model which are contaminated by colored noise and peak noise, the least absolute deviation (LAD) is selected as the objective function to solve the problem of large residual square when the identification data is disturbed by the impulse noise which obeys symmetrical alpha stable (SαS) distribution. However, LAD cannot meet the need of differentiability required by most algorithms. To improve robustness and to solve the nondifferentiable problem, an approximate least absolute deviation (ALAD) objective function is established by introducing a deterministic function to replace absolute value under certain situations. The proposed method is derived from ALAD criterion and extended stochastic gradient method. Due to the differentiability of the objective function, we can get a recursive identification algorithm which is simple and easy to calculate compared with LAD. The convergence of the proposed identification method is also proved by Lyapunov stability theory, and the simulation experiments show that the proposed method has higher accuracy and stronger robustness than the least square (LS) method in the identification of Hammerstein model with colored noise and impulse noise. The impact of impulse noise can be restrained effectively.

Bootstrap tests for unit roots based on LAD estimation

Least absolute deviations (LAD) estimation of linear time series models is considered under conditional heteroskedasticity and serial correlation. The limit theory of the LAD estimator is obtained without assuming the finite density condition for the errors that is required in standard LAD asymptotics. The results are particularly useful in application of LAD estimation to financial time series data.

Least absolute deviation (LAD) is a well-known criterion to fit statistical models, but little is known about LAD estimation in structural equation modeling (SEM). To address this gap, the authors use the LAD criterion in SEM by minimizing the sum of the absolute deviations between the observed and the model-implied covariance matrices. Using Monte Carlo simulations, the authors compare the performance of this LAD estimator along several dimensions (bias, efficiency, convergence, frequencies of improper solutions, and absolute percentage deviation) to the full information maximum likelihood (ML) and unweighted least squares (ULS) estimators in structural equation modeling. The results for LAD are mixed: There are special conditions under which the LAD estimator outperforms ML and ULS, but the simulation evidence does not support a general claim that LAD is superior to ML and ULS in small samples.

Introduction: Macroeconomic problems such as inflation and exchange rates are often highlighted as benchmarks for achieving economic progress. The stability of both must be monitored by the government in order to control the inflation rate and exchange rate. This instability is a phenomenon of fluctuation, namely the phenomenon of the rise and fall of the exchange rate of a currency based on demand and supply. Given the large impact of exchange rate fluctuations on the economy, the prediction of the wage exchange rate against the US dollar is considered necessary because it is useful to anticipate and minimize bad possibilities that arise. Method: Methods that can be used to analyze fluctuating currency exchange rate data are nonparametric regression, Nadaraya-Watson estimator, Gaussian kernel function, and Local Polynomial Regression. Results and Discussion: The results of a nonparametric regression comparison between the Nadaraya-Watson estimator, Gaussian kernel function, and local polynomial regression were obtained by MAPE of 2.508% and 0.179%, respectively. This shows that the best model uses the local polynomial regression method and predicted USD exchange rate data against IDR using the best model, namely Local polynomial Regression where the MAPE value is less than 10%, which means the prediction rate is very good. Conclusion: The nonparametric regression method of the Nadaraya-Watrson estimator, Gaussian kernel function, and local polynomial regression shows that the best model uses the local polynomial regression method.

Geographically Weighted Regression (GWR) is an analytical method for data that contains spatial heterogeneity effects. However, parameter estimation in the GWR model has a weakness, namely it is prone to outliers and can cause the parameter estimation to be biased. This can be overcome by the Robust Geographically Weighted Regression (RGWR) method which is more robust against the presence of outliers. This method is suitable for Gross Regional Domestic Product (GRDP) data in West Java Province, which contains outliers and also has spatial effects. The data used in this study are secondary data obtained from the Central Statistics Agency (BPS) of West Java Province. The purpose of this study is to compare the Robust Geographically Weighted Regression (RGWR) method with the Least Absolute Deviation (LAD) Estimation and M-estimation and also to find out the factors that affect the Gross Regional Domestic Product (GRDP) in West Java Province in 2021 based on the model resulting from. Selection of the best model is seen based on the value of the coefficient of determination (R2) and Mean Squared of Error (MSE). The research results show that the Robust Geographically Weighted Regression (RGWR) method with M-estimation is much more effective in estimating the distribution of GRDP in West Java Province in 2021, seen from the larger coefficient of determination and the smaller Mean Square Error (MSE). The variables that have a significant influence on GRDP in West Java Province in 2021 are the variables of foreign investment and local income.

We analyzed readily available Federal Aviation Administration (FAA) data on daily average flight times between major US airports. By subtracting out expected flight times and organizing the resulting deviations by origin and destination, we were able to identify unusual conditions and trace their sources. This approach facilitates detection of trends and anomalies in the National Airspace System (NAS) and opens up new possibilities for automated next-day analysis of operational problems. We arranged the deviations data as a two-way table and decomposed the average deviation in each cell as Deviation = System Effect + Origin Effect + Destination Effect + En route Effect. Least absolute deviations (LAD) estimation was chosen as the best method to analyze the FAA data. We studied the distributions of the four components of deviations as well as various relationships between them. This approach revealed some interesting phenomena, including the fact that certain airports work in a regime of being either a good origin or a good destination on any given day and the fact that geographically close airports track each other’s problems in terms of their terminal effects.

Approximation to the distribution of LAD estimators for censored regression by random weighting method

The widely used least absolute deviation (LAD) estimator with the smoothly clipped absolute deviation (SCAD) penalty function (abbreviated as LAD-SCAD) is known to produce corrupt estimates in the presence of outlying observations. The problem becomes more complicated when the number of predictors diverges. To overcome these problems, the LAD-SCAD based on sure independence screening (SIS) technique is put forward. The SIS method uses the rank correlation screening (RCS) algorithm in the pre-screening step and the traditional Pathwise coordinate descent algorithm for computing the sequence of the regularization parameters in the post screening step for onward model selection. It is now evident that the rank correlation is less robust against outliers. Motivated by these inadequacies, we propose to improvise the LAD-SCAD estimator using robust wrapped correlation screening (WCS) method by replacing the rank correlation in the SIS method with robust wrapped correlation. The proposed estimator is denoted as WCS+LAD-SCAD and will be employed for variable selection. The simulation study and real-life data examples show that the proposed procedure produces more efficient results compared to the existing methods.

Nonparametric methods of regression seek to describe the relationship between an explanatory and response variable while making only the most general type of assumptions about the functional form of this relationship. Techniques for computing nonparametric regression functions include kernel estimation, local polynomial regression, and spline smoothing. In each case, there is a balance to be reached between the smoothness of the regression function and its goodness of fit to the data. Simple nonparametric regression models (with one explanatory variable) can be extended to deal with multiple covariates by employing so‐called generalized additive models.

In this paper we examine the application of the Least Absolute Deviations (LAD) method for ridge-type parameter estimation of Seemingly Unrelated Regression Equations (SURE) models. The methodology is aimed to deal with the SURE models with non-Gaussian error terms and highly collinear predictors in each equation. Some biasing parameters used in the literature are taken and the efficiency of both Least Squares (LS) ridge estimation and the LAD ridge estimation of the SURE models, through the Mean Squared Error (MSE) of parameter estimators, is evaluated.