Lack of fit tests based on sums of ordered residuals for linear models

Christensen & Lin ( 2015 ) suggested two lack of fit tests to assess the adequacy of a linear model based on partial sums of residuals. In particular, their tests evaluated the adequacy of the mean function. Their tests relied on asymptotic results without requiring small sample normality. We propose four new tests, find their asymptotic distributions, and propose an alternative simulation method for defining tests that is remarkably robust to the distribution of the errors. To assess their strengths and weaknesses, the Christensen & Lin ( 2015 ) tests and the new tests were compared in different scenarios by simulation. In particular, the new tests include two based on partial sums of absolute residuals. Previous partial sums of residuals tests have used signed residuals whose values when summed can cancel each other out. The use of absolute residuals requires small sample normality, but allows detection of lack of fit that was previously not possible with partial sums of residuals.     

Adjusted Pearson residuals in beta regression models

In this paper, matrix formulae of order n^?1, where n is the sample size, for the first two moments of Pearson residuals are obtained in beta regression models. Adjusted Pearson residuals are also obtained, having, to this order, expected value zero and variance one. Monte Carlo simulation results are presented illustrating the behaviour of both adjusted and unadjusted residuals.     

Goodness-of-fit tests for parametric models in censored regression

The authors propose a goodness-of-fit test for parametric regression models when the response variable is right-censored. Their test compares an estimation of the error distribution based on parametric residuals to another estimation relying on nonparametric residuals. They call on a bootstrap mechanism in order to approximate the critical values of tests based on Kolmogorov-Smirnov and Cramér-von Mises type statistics. They also present the results of Monte Carlo simulations and use data from a study about quasars to illustrate their work.     

Testing Monotonicity of Regression Functions – An Empirical Process Approach

We propose several new tests for monotonicity of regression functions based on different empirical processes of residuals and pseudo‐residuals. The residuals are obtained from an unconstrained kernel regression estimator whereas the pseudo‐residuals are obtained from an increasing regression estimator. Here, in particular, we consider a recently developed simple kernel‐based estimator for increasing regression functions based on increasing rearrangements of unconstrained non‐parametric estimators. The test statistics are estimated distance measures between the regression function and its increasing rearrangement. We discuss the asymptotic distributions, consistency and small sample performances of the tests.     

Comments on “bootstrapping time series models”

We take issue with the main suggestion in Li and Maddala (LiMa) that bootstrapping residuals is always the preferred approach and question some of their guidelines. We show that it can be potentially misleading to mimic the autocovariance structure of residuals, since it can be very different from that of true errors. We emphasize that the residuals are sensitive to model misspecification and generally not a part of the information set. We make constructive suggestions and propose a semiparametric method.     

Variance Estimation in Spatial Regression Using a Non-parametric Semivariogram Based on Residuals

Abstract.  The empirical semivariogram of residuals from a regression model with stationary errors may be used to estimate the covariance structure of the underlying process. For prediction (kriging) the bias of the semivariogram estimate induced by using residuals instead of errors has only a minor effect because the bias is small for small lags. However, for estimating the variance of estimated regression coefficients and of predictions, the bias due to using residuals can be quite substantial. Thus we propose a method for reducing this bias. The adjusted empirical semivariogram is then isotonized and made conditionally negative-definite and used to estimate the variance of estimated regression coefficients in a general estimating equations setup. Simulation results for least squares and robust regression show that the proposed method works well in linear models with stationary correlated errors.     

Oracle model selection for correlated data via residuals

This paper concerns model selection for autoregressive time series when the observations are contaminated with trend. We propose an adaptive least absolute shrinkage and selection operator (LASSO) type model selection method, in which the trend is estimated by B-splines, the detrended residuals are calculated, and then the residuals are used as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an adaptive LASSO model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The intensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements.     

The Sensitivity of Detrended Long-Memory Processes

Long-memory tests are often complicated by the presence of deterministic trends. Hence, an additional step of detrending the data is necessary. The typical way to detrend a suspected long-memory series is to use OLS or BSP residuals. Applying the method of sensitivity analysis we address the of question of how robust these residuals are in presence of potential long memory components. Unlike short-memory ARMA process long-memory I(d) processes causes sensitivity to OLS/BSP residuals. Therefore, we develop a finite sample measure of the sensitivity of a detrended series based on the residuals. Based on our sensitivity measure we propose a “rule of thumb” for practitioners to choose between the two methods of detrending, has been provided in this article.     

Residual analysis for spatial point processes (with discussion)

Summary.  We define residuals for point process models fitted to spatial point pattern data, and we propose diagnostic plots based on them. The residuals apply to any point process model that has a conditional intensity; the model may exhibit spatial heterogeneity, interpoint interaction and dependence on spatial covariates. Some existing ad hoc methods for model checking (quadrat counts, scan statistic, kernel smoothed intensity and Berman's diagnostic) are recovered as special cases. Diagnostic tools are developed systematically, by using an analogy between our spatial residuals and the usual residuals for (non-spatial) generalized linear models. The conditional intensity λ plays the role of the mean response. This makes it possible to adapt existing knowledge about model validation for generalized linear models to the spatial point process context, giving recommendations for diagnostic plots. A plot of smoothed residuals against spatial location, or against a spatial covariate, is effective in diagnosing spatial trend or co-variate effects. Q – Q -plots of the residuals are effective in diagnosing interpoint interaction.     

Local Fourier tests for structural change based on residuals

We propose a structural change test based on the recursive residuals with the local Fourier series estimators. The statistical properties of the proposed test are derived and the empirical properties are shown via simulation. We also consider other structural change tests based on CUSUM, MOSUM, moving estimates (ME), and empirical distribution functions with the recursive residuals and the ordinary residuals. Empirical powers are calculated in various structural change models for the comparison of those tests. These structural change tests are applied to South Korea's gross domestic product (GDP), South Korean Won to US Dollar currency exchange rates, and South Korea's Okun's law.     

Deletion residuals in the detection of heterogeneity of variances in linear regression

The heterogeneity of error variance often causes a huge interpretive problem in linear regression analysis. Before taking any remedial measures we first need to detect this problem. A large number of diagnostic plots are now available in the literature for detecting heteroscedasticity of error variances. Among them the ‘residuals’ and ‘fits’ (R–F) plot is very popular and commonly used. In the R–F plot residuals are plotted against the fitted responses, where both these components are obtained using the ordinary least squares (OLS) method. It is now evident that the OLS fits and residuals suffer a huge setback in the presence of unusual observations and hence the R–F plot may not exhibit the real scenario. The deletion residuals based on a data set free from all unusual cases should estimate the true errors in a better way than the OLS residuals. In this paper we propose ‘deletion residuals’ and the ‘deletion fits’ (DR–DF) plot for the detection of the heterogeneity of error variances in a linear regression model to get a more convincing and reliable graphical display. Examples show that this plot locates unusual observations more clearly than the R–F plot. The advantage of using deletion residuals in the detection of heteroscedasticity of error variance is investigated through Monte Carlo simulations under a variety of situations.     

High-correlated residuals improved estimation in the high-dimensional SUR model

This article focuses on estimating regression coefficients in the high-dimensional seemingly unrelated regression model. When the number of equations exceeds that of the observations, both the maximum likelihood estimator and Zellner’s two-stage estimator do not exist. As an alternative, we propose a two-stage conditional expectation improved estimator. The new estimator is further improved by the high-correlated residuals, and the high correlation is determined by hypothesis testings. Simulations show that the new estimator outperforms the ordinary least-squares estimator in terms of mean square errors, especially when high-correlated residuals exist between the equations.     

An evaluation of bootstrap methods for outlier detection in least squares regression

Outlier detection is a critical part of data analysis, and the use of Studentized residuals from regression models fit using least squares is a very common approach to identifying discordant observations in linear regression problems. In this paper we propose a bootstrap approach to constructing critical points for use in outlier detection in the context of least-squares Studentized residuals, and find that this approach allows naturally for mild departures in model assumptions such as non-Normal error distributions. We illustrate our methodology through both a real data example and simulated data.     

An autocorrelation criterion for bandwidth selection in nonparametric regression ∗

In this note, we propose a new method for selecting the bandwidth parameter in non-parametric regression. While standard criteria, such as cross-validation, are based on the true regression curve about which we know little, we propose a criterion which focuses on the true errors about which assumptions may be made. Our proposal is to choose the bandwidth for which the residuals are as uncorrelated as possible. We use the Box-Pierce statistic as the objective to be minimized. In doing so, the behaviour of our residuals will be close to that of the true errors under the hypothesis of independent errors. A simulation study shows that our method succeeds in capturing the main features of the regression curve, in the sense that the number of turning-points of the curve is correctly estimated most of the time.     

ANOVA-like analysis of matched series of studies with a common structure

It is shown how to condense the information contained in a series of studies, each constituted by an objects by variables matrix and a pair of weight matrices, into a structure vector and a sum of sums of squares of residuals. Based on this condensation we propose to carry out ANOVA-like inference for matched series of studies associated with the level combinations of some factors. It is shown how to validate the assumptions underlying the inference. An application to the results of local elections in Portugal is given.     

A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

Strong convergence rate of the least median absolute estimator in linear regression models

We propose a least median of absolute (LMA) estimator for a linear regression model, based on minimizing the median absolute deviation of the residuals. Under some regularity conditions on the design points and disturbances, the strong convergence rate of the LMA estimator is established.     

Monitoring Variability and Analyzing Multivariate Autocorrelated Processes

Traditional multivariate quality control charts are based on independent observations. In this paper, we explain how to extend univariate residual charts to multivariate cases and how to combine the traditional statistical process control (SPC) approaches to monitor changes in process variability in a dynamic environment. We propose using Alt's (1984) W chart on vector autoregressive (VAR) residuals to monitor the variability for multivariate processes in the presence of autocorrelation. We study examples jointly using the Hotelling T² chart on VAR residuals, the W chart, and the Portmanteau test to diagnose the types of shift in process parameters.     

Identification of Multiple Outliers in Logistic Regression

The use of logistic regression modeling has seen a great deal of attention in the literature in recent years. This includes all aspects of the logistic regression model including the identification of outliers. A variety of methods for the identification of outliers, such as the standardized Pearson residuals, are now available in the literature. These methods, however, are successful only if the data contain a single outlier. In the presence of multiple outliers in the data, which is often the case in practice, these methods fail to detect the outliers. This is due to the well-known problems of masking (false negative) and swamping (false positive) effects. In this article, we propose a new method for the identification of multiple outliers in logistic regression. We develop a generalized version of standardized Pearson residuals based on group deletion and then propose a technique for identifying multiple outliers. The performance of the proposed method is then investigated through several examples.     

Residual-based specification of the random-effects distribution for cluster data

We propose a method for specifying the distribution of random effects included in a model for cluster data. The class of models we consider includes mixed models and frailty models whose random effects and explanatory variables are constant within clusters. The method is based on cluster residuals obtained by assuming that the random effects are equal between clusters. We exhibit an asymptotic relationship between the cluster residuals and variations of the random effects as the number of observations increases and the variance of the random effects decreases. The asymptotic relationship is used to specify the random-effects distribution. The method is applied to a frailty model and a model used to describe the spread of plant diseases.