LAD Estimation with Applications in Time Series Analysis

The estimator of the vector parameter in a linear regression, known as the least absolute deviation (LAD) estimator, is defined by minimizing the sum of the absolute values of the residuals. However, the loss function lacks differentiability. In this study, we propose a convolution-type kernel smoothed least absolute deviation (SLAD) estimator based upon smoothing the objective function within the context of linear regression. Compared with the LAD estimator, the loss function of SLAD estimator is asymptotically differentiable, and the resulting SLAD estimator can yield a lower mean squared error. Furthermore, we demonstrate several interesting asymptotic properties of the SLAD method. Numerical studies and real data analysis confirm that the proposed SLAD method performs remarkably well under finite sample sizes.

It is well known that the least absolute deviation (LAD) estimators are more robust than the least squares estimators particularly in presence of heavy tail errors. We consider the LAD estimators of the unknown parameters of one-dimensional chirp signal model under independent and identically distributed error structure. The proposed estimators are strongly consistent and it is observed that the asymptotic distribution of the LAD estimators are normally distributed. We perform some simulation studies to verify the asymptotic theory for small sample sizes and the performance are quite satisfactory.

We consider estimates of the parameters of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models obtained using auxiliary information on latent variance which may be available from higher-frequency data, for example from an estimate of the daily quadratic variation such as the realized variance. We obtain consistent estimators of the parameters of the infinite Autoregressive Conditional Heteroskedasticity (ARCH) representation via a regression using the estimated quadratic variation, without requiring that it be a consistent estimate; that is, variance information containing measurement error can be used for consistent estimation. We obtain GARCH parameters using a minimum distance estimator based on the estimated ARCH parameters. With Least Absolute Deviations (LAD) estimation of the truncated ARCH approximation, we show that consistency and asymptotic normality can be obtained using a general result on LAD estimation in truncated models of infinite-order processes. We provide simulation evidence on small-sample performance for varying numbers of intra-day observations.

This study considers efficient mixture designs for the approximation of the response surface of a quantile regression model, which is a second degree polynomial, by a first degree polynomial in the proportions of q components. Instead of least squares estimation in the traditional regression analysis, the objective function in quantile regression models is a weighted sum of absolute deviations and the least absolute deviations (LAD) estimation technique should be used (Bassett and Koenker, 1982; Koenker and Bassett, 1978). Therefore, the standard optimal mixture designs like the D-optimal or A-optimal mixture designs for the least squared estimation are not appropriate. This study explores mixture designs that minimize the bias between the approximated 1st-degree polynomial and a 2nd-degree polynomial response surfaces by the LAD estimation. In contrast to the standard optimal mixture designs for the least squared estimation, the efficient designs might contain elementary centroid design points of degrees higher than two. An example of a portfolio with five assets is given to illustrate the proposed efficient mixture designs in determining the marginal contribution of risks by individual assets in the portfolio.

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.

LADE-based inferences for autoregressive models with heavy-tailed G-GARCH(1, 1) noise

To cope with the occlusion and intersection between targets and the environment, location prediction is employed in the visual tracking system. Target trace is fitted by sliding subsection polynomials based on least absolute deviation (LAD) estimation, and the future location of target is predicted with the fitted trace. Experiment results show that the proposed location prediction algorithm based on LAD estimation has significant robustness advantages over least square (LS) estimation, and it is more effective than LS-based methods in visual tracking.

The least absolute deviations (LAD) estimation for nonlinear regression models with randomly censored data is studied and the asymptotic properties of LAD estimators such as consistency, boundedness in probability and asymptotic normality are established. Simulation results show that for the problems with censored data, LAD estimation performs much more robustly than the least squares estimation.

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.

This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate$\root \of n $to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponentκ∈ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponentκ∈ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

LAD Asymptotics Under Conditional Heteroskedasticity with Possibly Infinite Error Densities

Introduction: Time series models are generally categorized as a data-driven method or mathematically-based method. These models are known as one of the most important tools in modeling and forecasting of hydrological processes, which are used to design and scientific management of water resources projects. On the other hand, a better understanding of the river flow process is vital for appropriate streamflow modeling and forecasting. One of the main concerns of hydrological time series modeling is whether the hydrologic variable is governed by the linear or nonlinear models through time. Although the linear time series models have been widely applied in hydrology research, there has been some recent increasing interest in the application of nonlinear time series approaches. The threshold autoregressive (TAR) method is frequently applied in modeling the mean (first order moment) of financial and economic time series. Thise type of the model has not received considerable attention yet from the hydrological community. The main purposes of this paper are to analyze and to discuss stochastic modeling of daily river flow time series of the study area using linear (such as ARMA: autoregressive integrated moving average) and non-linear (such as two- and three- regime TAR) models. Material and Methods: The study area has constituted itself of four sub-basins namely, Saghez Chai, Jighato Chai, Khorkhoreh Chai and Sarogh Chai from west to east, respectively, which discharge water into the Zarrineh Roud dam reservoir. River flow time series of 6 hydro-gauge stations located on upstream basin rivers of Zarrineh Roud dam (located in the southern part of Urmia Lake basin) were considered to model purposes. All the data series used here to start from January 1, 1997, and ends until December 31, 2011. In this study, the daily river flow data from January 01 1997 to December 31 2009 (13 years) were chosen for calibration and data for January 01 2010 to December 31 2011 (2 years) were chosen for validation, subjectively. As data have seasonal cycles, statistical indices (such as mean and standard deviation) of daily discharge were estimated using Fourier series. Then ARMA and two- and three-regime SETAR models applied to the standardized daily river flow time series. Some performance criteria were used to evaluate the models accuracy. In other words, in this paper, linear and non-linear models such as ARMA and two- and three-regime SETAR models were fitted to observed river flows. The parameters associated to the models, e.g. the threshold value for the SETAR model was estimated. Finally, the fitted linear and non-linear models were selected using the Akaike Information Criterion (AIC), Root Mean Square (RMSE) and Sum of Squared Residuals (SSR) criteria. In order to check the adequacy of the fitted models the Ljung-Box test was used. Results and Discussion: To a certain degree the result of the river flow data of study area indicates that the threshold models may be appropriate for modeling and forecasting the streamflows of rivers located in the upstream part of Zarrineh Roud dam. According to the obtained evaluation criteria of fitted models, it can be concluded the performance of two- and three- regime SETAR models are slightly better than the ARMA model in all selected stations. As well as, modeling and comparison of SETAR models showed that the three-regime SETAR model have evaluation criteria better than two-regime SETAR model in all stations except Ghabghablou station. Conclusion: In the present study, we attempted to model daily streamflows of Zarrineh Rood Basin Rivers located in the south of Urmia Lake by applying ARMA and two- and three-regime SETAR models. This is mainly because very few efforts and rather less attention have been paid to this non-linear approach in hydrology and water resources engineering generally. Therefore, two types of data-driven models were used for modeling and forecasting daily streamflow: (i) deseasonalized ARMA-type model, and (ii) Threshold Autoregressive model, including Self-Existing TAR (SETAR) model. Each ARMA and SETAR models were fitted to daily streamflow time series of the rivers located in the study area. In general, it can be concluded that the overall performance of SETAR model is slightly better than ARMA model. Furthermore, SETAR model is very similar AR model, therefor, it can be easily used in water resources engineering field. On the other hand, due to apply these non-linear models, the number of estimated parameters in comparison with linear models has decreased.

Pakistan is considered among the top five countries with the highest CO2 emissions globally. This calls for pragmatic policy implementation by all stakeholders to bring finality to this alarming situation since it contributes greatly to global warming, thereby leading to climate change. This study is an attempt to make a comparative analysis of the linear time series models with nonlinear time series models to study CO2 emission data in Pakistan. These linear and nonlinear time series models were used to model and forecast future values of CO2 emissions for a short period. To assess and select the best model among these linear and nonlinear time series models, we used the root mean square error (RMSE) and the mean absolute error (MAE) as performance indicators. The outputs showed that the nonlinear machine learning models are the best among all other models, having the lowest RMSE and MAE values. Based on the forecasted value of the nonlinear machine learning neural network autoregressive model, Pakistan's CO2 emissions will be 1.048 metric tons per capita by 2028. The increasing trend in emissions is a frightening and clear warning, suggesting that innovative policies must be initiated to reduce the trend. We encourage the Pakistan government to price CO2 emissions by companies and entities per ton, adapt electricity production from hydro, wind, and different sources with no emissions of CO2, initiate rigorous planting of more trees in the populated areas of Pakistan as forest covers, provide incentives to companies, organisations, institutions, and households to come out with clean technologies or use technologies with no CO2 emissions or those with lower ones, and fund more studies to develop clean and innovative technologies with less or no CO2 emissions.

This article develops a systematic procedure of statistical inference for the auto-regressive moving average (ARMA) model with unspecified and heavy-tailed heteroscedastic noises. We first investigate the least absolute deviation estimator (LADE) and the self-weighted LADE for the model. Both estimators are shown to be strongly consistent and asymptotically normal when the noise has a finite variance and infinite variance, respectively. The rates of convergence of the LADE and the self-weighted LADE are n− 1/2, which is faster than those of least-square estimator (LSE) for the ARMA model when the tail index of generalized auto-regressive conditional heteroskedasticity (GARCH) noises is in (0, 4], and thus they are more efficient in this case. Since their asymptotic covariance matrices cannot be estimated directly from the sample, we develop the random weighting approach for statistical inference under this nonstandard case. We further propose a novel sign-based portmanteau test for model adequacy. Simulation study is carried out to assess the performance of our procedure and one real illustrating example is given. Supplementary materials for this article are available online.

Uncertain time series analysis is a method of predicting future values by analyzing imprecise observations. In this paper, the least absolute deviation (LAD) method is applied to solve for the unknown parameters of the uncertain max-autoregressive (UMAR) model. The predicted value and confidence interval of the future data are calculated using the fitted UMAR model. Moreover, the relative change rate of parameter is proposed to test the robustness of different estimation methods. Then, two comparative analyses demonstrate the LAD estimation can handle outliers better than the least squares (LS) estimation and the necessity of introducing the UMAR model. Finally, a numerical example displays the LAD estimation in detail to verify the effectiveness of the method. The LAD estimation is also applied to a collection of actual data with cereal yield.