On tests for normality of experimental error in ridge regression

Considered are tests for normality of the errors in ridge regression. If an intercept is included in the model, it is shown that test statistics based on the empirical distribution function of the ridge residuals have the same limiting distribution as in the one-sample test for normality with estimated mean and variance. The result holds with weak assumptions on the behavior of the independent variables; asymptotic normality of the ridge estimator is not required.     

Testing residual normality in the ANOVA model

The use of single group skewness and kurtosis critical values for the assessment of residual normality in the ANOVA model is examined. Using single group critical values gives a conservative test of residual normality in multiple group designs. As the sample size per group increases, the empirical Type I error rates for the skewness and kurtosis tests of residual normality approach a. These results supplement previous work which has focused on testing residual normality in the linear regression model.     

Statistical Analysis of Parameter Estimation for 2-D Harmonics in Multiplicative and Additive Noise

This article considers the problem of parameter estimation for two dimensional (2-D) multi-component harmonics in non zero-mean multiplicative and additive noise. The least squares estimators (LSEs) are proposed to estimate the coherent model parameters, and some statistical results of the LSEs are obtained, including strong consistency, strong convergence rate, and asymptotic normality. Furthermore, the LSEs-based estimators are proposed to estimate the noncoherent model parameters, and the strong consistency and the asymptotic normality are also proved. Finally, some numerical experiments are performed to see how the asymptotic results work for finite sample sizes.     

A test of the normality assumption in ordered probit model

This paper presents a Lagrange multiplier test of the normality assumption underlying the ordered probit model. The test is presented both for the standard ordered probit model and a version in which censoring is present in the dependent variable. The test is then compared to normality tests proposed here compares favorably to tests based on artificial regression techinques.     

Checking Normality and Homoscedasticity in the General Linear Model Using Diagnostic Plots

Inference for the general linear model makes several assumptions, including independence of errors, normality, and homogeneity of variance. Departure from the latter two of these assumptions may indicate the need for data transformation or removal of outlying observations. Informal procedures such as diagnostic plots of residuals are frequently used to assess the validity of these assumptions or to identify possible outliers. A simulation-based approach is proposed, which facilitates the interpretation of various diagnostic plots by adding simultaneous tolerance bounds. Several tests exist for normality or homoscedasticity in simple random samples. These tests are often applied to residuals from a linear model fit. The resulting procedures are approximate in that correlation among residuals is ignored. The simulation-based approach accounts for the correlation structure of residuals in the linear model and allows simultaneously checking for possible outliers, non normality, and heteroscedasticity, and it does not rely on formal testing.

[Supplementary materials are available for this article. Go to the publisher's online edition of Communications in Statistics—Simulation and Computation® for the following three supplemental resource: a word file containing figures illustrating the mode of operation for the bisectional algorithm, QQ-plots, and a residual plot for the mussels data.]     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

Confidence intervals for impulse responses under departures from normality

Monte Carlo evidence shows that in structural VAR models with fat-tailed or skewed innovations the coverage accuracy of impulse response confidence intervals may deterorate substantially compared to the same model with Gaussian innovations. Empirical evidance suggests that such departures from normality are quite plausible for economic time series. The simulation results suggest that applied researchers are best off using nonparametric bootstrap intervals for impulse responses, regardless of whether or not there is evidence of fat tails or skewness in the error distribution. Allowing for departures from normality is shown to considerably weaken the evidence of the delayed overshooting puzzle in Eichenbaum and Evans (1995).     

A Geometric Examination of Linear Model Assumptions

From a geometric perspective, linear model theory relies on a single assumption, that (‘corrected’) data vector directions are uniformly distributed in Euclidean space. We use this perspective to explore pictorially the effects of violations of the traditional assumptions (normality, independence and homogeneity of variance) on the Type I error rate. First, for several non‐normal distributions we draw geometric pictures and carry out simulations to show how the effects of non‐normality diminish with increased parent distribution symmetry and continuity, and increased sample size. Second, we explore the effects of dependencies on Type I error rate. Third, we use simulation and geometry to investigate the effect of heterogeneity of variance on Type I error rate. We conclude, in a fresh way, that independence and homogeneity of variance are more important assumptions than normality. The practical implication is that statisticians and authors of statistical computing packages need to pay more attention to the correctness of these assumptions than to normality.     

基于贝叶斯推断的HIV非线性混合效应联合模型研究

 在纵向数据研究中,混合效应模型的随机误差通常采用正态性假设。而诸如病毒载量和CD4细胞数目等病毒性数据通常呈现偏斜性,因此正态性假设可能影响模型结果甚至导致错误的结论。在HIV动力学研究中,病毒响应值往往与协变量相关,且协变量的测量值通常存在误差,为此论文中联立协变量过程建立具有偏正态分布的非线性混合效应联合模型,并用贝叶斯推断方法估计模型的参数。由于协变量能够解释个体内的部分变化,因此协变量过程的模型选择对病毒载量的拟合效果有重要的影响。该文提出了一次移动平均模型作为协变量过程的改进模型,比较后发现当协变量采用移动平均模型时,病毒载量模型的拟合效果更好。该结果对协变量模型的研究具有重要的指导意义。     

Degree-based goodness-of-fit tests for heterogeneous random graph models: Independent and exchangeable cases

The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness of fit for a given random graph model. In this paper, we introduce goodness-of-fit tests for two classes of models. First, we consider the case of independent graph models such as the heterogeneous Erdös-Rényi model in which the edges have different connection probabilities. Second, we consider a generic model for exchangeable random graphs called the W-graph. The stochastic block model and the expected degree distribution model fall within this framework. We prove the asymptotic normality of the degree mean square under these independent and exchangeable models and derive formal tests. We study the power of the proposed tests and we prove the asymptotic normality under specific sparsity regimes. The tests are illustrated on real networks from social sciences and ecology, and their performances are assessed via a simulation study.     

ON THE HILDRETH-HOUCK ESTIMATOR FOR RANDOM COEFFICIENT REGRESSION MODELS

This paper is concerned with the linear regression model in which the coefficients are random variables. The Hildreth-Houck method is considered for estimating the model. Sufficient conditions for the consistency of the Hildreth-Houck estimator are discussed and its asymptotic normality is established.     

Asymptotics and bootstrap for random-effects panel data transformation models

This article investigates the asymptotic properties of quasi-maximum likelihood (QML) estimators for random-effects panel data transformation models where both the response and (some of) the covariates are subject to transformations for inducing normality, flexible functional form, homoskedasticity, and simple model structure. We develop a QML-type procedure for model estimation and inference. We prove the consistency and asymptotic normality of the QML estimators, and propose a simple bootstrap procedure that leads to a robust estimate of the variance-covariance (VC) matrix. Monte Carlo results reveal that the QML estimators perform well in finite samples, and that the gains by using the robust VC matrix estimate for inference can be enormous.     

On a new family of shifted logarithmic transformations

Most parametric statistical methods are based on a set of assumptions: normality, linearity and homoscedasticity. Transformation of a metric response is a popular method to meet these assumptions. In particular, transformation of the response of a linear model is a popular method when attempting to satisfy the Gaussian assumptions on the error components in the model. A particular problem with common transformations such as the logarithm or the Box–Cox family is that negative and zero data values cannot be transformed. This paper proposes a new transformation which allows negative and zero data values. The method for estimating the transformation parameter consider an objective criteria based on kurtosis and skewness for achieving normality. Use of the new transformation and the method for estimating the transformation parameter are illustrated with three data sets.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Estimating Variance Components and Random Effects Using the Box-Cox Transformation in the Linear Mixed Model

The linear mixed model assumes normality of its two sources of randomness: the random effects and the residual error. Recent research demonstrated that a simple transformation of the response targets normality of both sources simultaneously. However, estimating the transformation can lead to biased estimates of the variance components. Here, we provide guidance regarding this potential bias and propose a correction for it when such bias is substantial. This correction allows for accurate estimation of the random effects when using a transformation to achieve normality. The utility of this approach is demonstrated in a study of sleep-wake behavior in preterm infants.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Dimension test approach of heteroscedasticity in the linear model

Heteroscedasticity testing has a long history and is still an important matter in the linear model. There exist many types of tests, but they are limited in use to their own specific cases and sensitive to normality. Here, we propose a dimension test approach to heteroscedasticity. The proposed test overcomes the shortcomings of the existing methods, so that it is robust to normality and is unified in sense that it is applicable in the linear model with multi-dimensional response. Numerical studies confirm that the proposed test is favorable over the existing tests with moderate sample sizes, and real data analysis is presented.     

Estimation of harmonic component in regression with cyclically dependent errors

This paper deals with the estimation of hidden periodicities in a non-linear regression model with stationary noise displaying cyclical dependence. Consistency and asymptotic normality are established for the least-squares estimates.     

An investigation of Bayesian inference approach to model validation with non-normal data

Quantitative model validation is playing an increasingly important role in performance and reliability assessment of a complex system whenever computer modelling and simulation are involved. The foci of this paper are to pursue a Bayesian probabilistic approach to quantitative model validation with non-normality data, considering data uncertainty and to investigate the impact of normality assumption on validation accuracy. The Box–Cox transformation method is employed to convert the non-normality data, with the purpose of facilitating the overall validation assessment of computational models with higher accuracy. Explicit expressions for the interval hypothesis testing-based Bayes factor are derived for the transformed data in the context of univariate and multivariate cases. Bayesian confidence measure is presented based on the Bayes factor metric. A generalized procedure is proposed to implement the proposed probabilistic methodology for model validation of complicated systems. Classic hypothesis testing method is employed to conduct a comparison study. The impact of data normality assumption and decision threshold variation on model assessment accuracy is investigated by using both classical and Bayesian approaches. The proposed methodology and procedure are demonstrated with a univariate stochastic damage accumulation model, a multivariate heat conduction problem and a multivariate dynamic system.     

Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors

In this article, the asymptotic normality and strong consistency of the least square estimators for the unknown parameters in the simple linear errors in variables model are established under the assumptions that the errors are stationary negatively associated sequences.