Absolute Deviation Estimator Research Articles

Robust unit root tests with autoregressive errors

ABSTRACTThis article presents a new test for unit roots based on least absolute deviation estimation specially designed to work for time series with autoregressive errors. The methodology used is a bootstrap scheme based on estimating a model and then the innovations. The resampling part is performed under the null hypothesis and, as it is customary in bootstrap procedures, is automatic and does not rely on the calculation of any nuisance parameter. The validity of the procedure is established and the asymptotic distribution of the statistic proposed is proved to converge to the correct distribution. To analyze the performance of the test for finite samples, a Monte Carlo study is conducted showing a very good behavior in many different situations.

Robustness in sparse high-dimensional linear models: Relative efficiency and robust approximate message passing

Understanding efficiency in high dimensional linear models is a longstanding problem of interest. Classical work with smaller dimensional problems dating back to Huber and Bickel has illustrated the clear benefits of efficient loss functions. When the number of parameters $p$ is of the same order as the sample size $n$, $p\approx n$, an efficiency pattern different from the one of Huber was recently established. In this work, we study relative efficiency of sparsity linear models with $p\gg n$. In the interest of deriving the asymptotic mean squared error for $l_{1}$ regularized M-estimators, we propose a novel, robust and sparse approximate message passing algorithm (RAMP), that is adaptive to the error distribution. Our algorithm includes many non-quadratic and non-differentiable loss functions. We derive its asymptotic mean squared error and show its convergence, while allowing $p,n,s\to \infty$, with $n/p\in (0,1)$ and $n/s\in (1,\infty)$. We identify new patterns of relative efficiency regarding $l_{1}$ penalized $M$ estimators. We show that the classical information bound is no longer reachable, even for light–tailed error distributions. Moreover, we show new breakdown points regarding the asymptotic mean squared error. The asymptotic mean squared error of the $l_{1}$ penalized least absolute deviation estimator (P-LAD) breaks down at a critical ratio of the number of observations per number of sparse parameters in the case of light-tailed distributions; whereas, in the case of heavy-tailed distributions, the asymptotic mean squared error breaks down at a critical ratio of the optimal tuning parameter of P-LAD to the optimal tuning parameter of the $l_{1}$ penalized least square estimator.

Variable Selection for Semiparametric Varying Coefficient Partially Linear Errors-in-Variables (EV) Model with Missing Response

This paper focuses on the variable selection for semiparametric varying coefficient partially linear model when the covariates are measured with additive errors and the response is missing. An adaptive lasso estimator and the smoothly clipped absolute deviation estimator as a comparison for the parameters are proposed. With the proper selection of regularization parameter, the sampling properties including the consistency of the two procedures and the oracle properties are established. Furthermore, the algorithms and corresponding standard error formulas are discussed. A simulation study is carried out to assess the finite sample performance of the proposed methods.

Impact of Government Agricultural Policies on Major Staple Food Prices in Nigeria (1966 - 2011)

This study assessed the impact of government agricultural policies on the prices of major staple food in Nigeria. It described the trend of the major staple food prices and determined the effect of government agricultural policies on the major staple food prices in Nigeria. Secondary data were sourced from various organizations and used for this study. These data were transformed from their nominal value to real value and analyzed using descriptive statistics, unit root test, ordinary least square model, cochrane-orcutt, least absolute deviation estimators and simultaneous equation model. The study revealed that the prices of most of the major staple foods were at the peak value between 1991 and 1993 while their prices were at the lowest value between 1978 and 1983. The study observed that the price of cowpea is most volatile seconded by maize. The study further established that the prices of staple foods were inter-related. Also, it was showed that agricultural credit guarantee scheme funds (ACGSF), agricultural policies during structural adjustment period (SAP) and post-structural adjustment period (PSAP) were statistically significant while the agricultural input subsidy scheme was statistically insignificant. The study recommended painstaking formulation and implementation of agricultural policies and re-structuring of agricultural of input scheme.

Weighted least absolute deviations estimation for periodic ARMA models

This paper investigates the weighted least absolute deviations estimator (WLADE) for causal and invertible periodic autoregressive moving average (PARMA) models. Asymptotic normality of the estimator is derived under a fractional moment condition. A simulation study is given to assess the performance of the proposed WLADE.

Сравнение оценок максимального правдоподобия и наименьших модулей параметров процесса авторегрессии со случайными коэффициентами

The paper presents the calculation method of an asymptotic relative efficiency of the least absolute deviations estimate with respect to the maximum likelihood estimate for the parameter presented in the first order autoregressive model with a stochastic coefficient. The method is based on approximation of its asymptotic relative efficiency by Taylor series. The paper considers an example of the asymptotic relative efficiency calculation for the case where diffusion of the innovation process has an approximate Gaussian distribution (Tukey distribution). It is found out that if the assumptions of the Gaussian process are performed approximately, the maximum likelihood estimate is more efficient than the least absolute deviation estimate

LADE-Based Inference for ARMA Models With Unspecified and Heavy-Tailed Heteroscedastic Noises

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.

The LAD estimation of the change-point linear model with randomly censored data

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.

Bootstrap unit root test based on least absolute deviation estimation under dependence assumptions

In this paper, a bootstrap test based on the least absolute deviation (LAD) estimation for the unit root test in first-order autoregressive models with dependent residuals is considered. The convergence in probability of the bootstrap distribution function is established. Under the frame of dependence assumptions, the asymptotic behavior of the bootstrap LAD estimator is independent of the covariance matrix of the residuals, which automatically approximates the target distribution.

GARCH Model Estimation Using Estimated Quadratic Variation

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.

From KIDSCREEN-10 to CHU9D: creating a unique mapping algorithm for application in economic evaluation.

BackgroundThe KIDSCREEN-10 index and the Child Health Utility 9D (CHU9D) are two recently developed generic instruments for the measurement of health-related quality of life in children and adolescents. Whilst the CHU9D is a preference based instrument developed specifically for application in cost-utility analyses, the KIDSCREEN-10 is not currently suitable for application in this context. This paper provides an algorithm for mapping the KIDSCREEN-10 index onto the CHU9D utility scores.MethodsA sample of 590 Australian adolescents (aged 11–17) completed both the KIDSCREEN-10 and the CHU9D. Several econometric models were estimated, including ordinary least squares estimator, censored least absolute deviations estimator, robust MM-estimator and generalised linear model, using a range of explanatory variables with KIDSCREEN-10 items scores as key predictors. The predictive performance of each model was judged using mean absolute error (MAE) and root mean squared error (RMSE).ResultsThe MM-estimator with stepwise-selected KIDSCREEN-10 items scores as explanatory variables had the best predictive accuracy using MAE, whilst the equivalent ordinary least squares model had the best predictive accuracy using RMSE.ConclusionsThe preferred mapping algorithm (i.e. the MM-estimate with stepwise selected KIDSCREEN-10 item scores as the predictors) can be used to predict CHU9D utility from KIDSCREEN-10 index with a high degree of accuracy. The algorithm may be usefully applied within cost-utility analyses to generate cost per quality adjusted life year estimates where KIDSCREEN-10 data only are available.

Fuzzy logistic regression with least absolute deviations estimators

This study is an investigation of fuzzy logistic regression model for crisp input and fuzzy output data. The response variable is non-precise and is measured by linguistic terms. Especially this research develops least absolute deviations (LAD) method for modeling and compares the results with the least squares estimation (LSE) method. For these, two estimation methods, min---max method and fitting method, are provided in this research. This study presents new goodness-of-fit indices which are called measure of performance based on fuzzy distance $$(M_p)$$(Mp) and index of sensitivity $$(I_S)$$(IS). The study gives two numerical examples in real clinical studies about systematic lupus erythematosus and the other one in the field of nutrition to explain the proposed methods. In addition, we investigate the sensitivity of two estimation methods in the case of outliers by a numerical example.

Improved autoregressive forecasts in the presence of non-normal errors

This paper is concerned with obtaining more accurate point forecasts in the presence of non-normal errors. Specifically, we apply the residual augmented least-squares (RALS) estimator to autoregressive models to utilize the additional moment restrictions embodied in non-normal errors. Monte Carlo experiments are performed to compare our RALS forecasts to forecasts based on the ordinary least-squares estimator and the least absolute deviations (LAD) estimator. We find that the RALS approach provides superior forecasts when the data are skewed. Compared to the LAD forecast, the RALS forecast has smaller mean squared prediction errors in the baseline case with normal errors.

Least absolute deviation estimation for general fractionally integrated autoregressive moving average time series models

We introduce a new class of ARFIMA models, which removes the restrictions that the roots of AR and MA polynomials are outside the unit circle. We establish consistency and asymptotic normality of the least absolute deviation estimator under non-Gaussian setting.

Multiscale reconstruction algorithm for compressed sensing

Compressed sensing (CS) method has attracted increasing attention owing to providing a novel insight for signal and image processing technology. Acquiring high-quality reconstruction results plays a crucial role in successful applications of CS method. This paper presents a multiscale reconstruction model that simultaneously considers the inaccuracy properties on the measurement data and the measurement matrix. Based on the wavelet analysis method, the original inverse problem is decomposed into a sequence of inverse problems, which are solved successively from the largest scale to the original scale. An objective functional, that integrate the beneficial advantages of the least trimmed sum of absolute deviations (LTA) estimation and the combinational M-estimation, is proposed. An iteration scheme that incorporates the advantages of the homotopy method and the evolutionary programming (EP) algorithm is designed for solving the proposed objective functional. Numerical simulations are implemented to validate the feasibility of the proposed reconstruction method.

Efficient Mixture Design Fitting Quadratic Surface with Quantile Responses Using First-degree Polynomial

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.

Weighted least absolute deviations estimation for ARFIMA time series with finite or infinite variance

In this paper a weighted least absolute deviations (WLAD) estimator for fractionally autoregressive integrated moving average (ARFIMA) models is proposed, in which stationary and non-stationary cases are discussed. The asymptotic normality of their local estimators is derived under a fractional moment condition. A Wald test for a linear hypothesis has been constructed, its limiting distribution is presented, and a simulation study is given to evaluate the performance of the proposed WLAD estimator under the stationary case.

Asymptotic theory for LAD estimation of moderate deviations from a unit root

An asymptotic result is given for the least absolute deviations (LAD) estimation of autoregressive time series with a root of the form ρn=1+c/kn, where kn increases to infinity at a rate slower than n. For c<0, a nkn rate of convergence and asymptotic normality for the serial correlation coefficient are provided. While in the case of c>0, the serial correlation coefficient is shown to have a Cauchy limit distribution with a knρnn convergence rate. The results are complementary to the limit theory of least squares (LS) estimator which has been established in Phillips and Magdalinos (2007a).

Exact Computation of Censored Least Absolute Deviations Estimators

We show that exact computation of Powell's (1984) censored least absolute deviations (CLAD) estimator may be achieved by formulating the estimator as a solvable linear Mixed Integer Programming (MIP) problem with disjunctive constraints. We apply our approach to two previously used censored datasets and find that standard heuristic/approximate approaches to computation of the estimator can lead to substantial errors and misleading economic conclusions. Additionally, we present a small Monte Carlo study which illustrates that MIP computation using widely available solvers is practical and efficient for data sets of sizes typically encountered in econometric applications.

Weighted Wilcoxon Estimators in Nonlinear Regression

SummaryIn this paper, we consider the estimation of parameters of a general near regression model. An estimator that minimises the weighted Wilcoxon dispersion function is considered and its asymptotic properties established under mild regularity conditions similar to those used in least squares and least absolute deviations estimation. As in linear models, the procedure provides estimators that are robust and highly efficient. The estimates depend on the choice of a weight function and diagnostics which differentiate between nonlinear fits are provided along with appropriate benchmarks. The behavior of these estimates is discussed on a real data set. A simulation study verifies the robustness, efficiency and validity of these estimates over several error distributions including the normal and a family of contaminated normal distributions.