Heteroscedasticity diagnostics in varying-coefficient partially linear regression models and applications in analyzing Boston housing data

It is important to detect the variance heterogeneity in regression model because efficient inference requires that heteroscedasticity is taken into consideration if it really exists. For the varying-coefficient partially linear regression models, however, the problem of detecting heteroscedasticity has received very little attention. In this paper, we present two classes of tests of heteroscedasticity for varying-coefficient partially linear regression models. The first test statistic is constructed based on the residuals, in which the error term is from a normal distribution. The second one is motivated by the idea that testing heteroscedasticity is equivalent to testing pseudo-residuals for a constant mean. Asymptotic normality is established with different rates corresponding to the null hypothesis of homoscedasticity and the alternative. Some Monte Carlo simulations are conducted to investigate the finite sample performance of the proposed tests. The test methodologies are illustrated with a real data set example.     

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.     

固定效应模型基于个体LM的新可混合性检验

对固定效应模型,本文基于拉格朗日乘数（LM）原理提出了一种新的可混合性检验。不同于已有的LM型可混合性检验,这里使用每个截面个体的LM统计量构建可混合性检验统计量。数理分析表明,本文所提的方法有着渐进正态性,对于扰动项的异方差和非正态均稳健,且与PY检验（Pesaran&Yamagata,2008）渐近等价。Monte Carlo模拟实验表明,相对于PY检验及另外两种LM型的可混合性检验,对于不同大小的 ,本文提出的方法有着良好的水平表现和更优越的检验势。     

Two new estimators of entropy for testing normality

ABSTRACT

We present two new estimators for estimating the entropy of absolutely continuous random variables. Some properties of them are considered, specifically consistency of the first is proved. The introduced estimators are compared with the existing entropy estimators. Also, we propose two new tests for normality based on the introduced entropy estimators and compare their powers with the powers of other tests for normality. The results show that the proposed estimators and test statistics perform very well in estimating entropy and testing normality. A real example is presented and analyzed.     

A Modified Kolmogorov–Smirnov Test for Normality

In this article we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, given data are compared with a normal distribution that uses the sample mean and the sample variance. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data. This is like shifting and stretching the reference normal distribution so that it fits the data in the best possible way. A study of the power of the proposed test indicates that the test is able to discriminate between the normal distribution and distributions such as uniform, bimodal, beta, exponential, and log-normal that are different in shape but has a relatively lower power against the student's, t-distribution that is similar in shape to the normal distribution. We also compare the performance (both in power and sensitivity to outlying observations) of the proposed test with existing normality tests such as Anderson–Darling and Shapiro–Francia.     

On a new test for autocorrelation in regression models under nonnormality

A new test for autocorrelation in a general regression model under departures from the assumption of normality is derived by applying a beta distribution and bootstrap approximation, Critical values of the test, can be computed for each given design matrix, irrespective of the form of the underlying error distribution, Monte Carlo simulations are conducted in order to illustrate the performance of the test. Among others, it. is found that the suggested test is more robust and far more powerful than existing nonparametric tests.     

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.     

Testing normality in mixed models using a transformation method

Statistical inference often assumes the normality of the variables involved in the model under study. The existing tests are for independent observations and may not be readily extended to handle the case with correlated ones. In this paper, a transformation method is recommended for normality checking in the two-way analysis of variance model. Its sampling properties are investigated. Simulation studies are carried out to examine the performance of the proposed methodology.     

Non-parametric MANOVA approaches for non-normal multivariate outcomes with missing values

Between-group comparisons often entail many correlated response variables. The multivariate linear model, with its assumption of multivariate normality, is the accepted standard tool for these tests. When this assumption is violated, the non-parametric multivariate Kruskal–Wallis (MKW) test is frequently used. However, this test requires complete cases with no missing values in response variables. Deletion of cases with missing values likely leads to inefficient statistical inference. Here we extend the MKW test to retain information from partially observed cases. Results of simulated studies and analysis of real data show that the proposed method provides adequate coverage and superior power to complete case analyses.     

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.]     

Stringency-based ranking of normality tests

Many parametric statistical inferential procedures in finite samples depend crucially on the underlying normal distribution assumption. Dozens of normality tests are available in the literature to test the hypothesis of normality. Availability of such a large number of normality tests has generated a large number of simulation studies to find a best test but no one arrived at a definite answer as all depends critically on the alternative distributions which cannot be specified. A new framework, based on stringency concept, is devised to evaluate the performance of the existing normality tests. Mixture of t-distributions is used to generate the alternative space. The LR-tests, based on Neyman–Pearson Lemma, have been computed to construct a power envelope for calculating the stringencies of the selected normality tests. While evaluating the stringencies, Anderson–Darling (AD) statistic turns out to be the best normality test.     

An empirical study of the type I error rate and power for some selected normal-theory and nonparametric tests of the independence of two sets of variables

Normal-theory tests of the hypothesis of no relationship among two sets of variables require assumptions of independence, hamoscedasticity, and normality. If, however, the assumption of normality is not tenable, there are few guidelines for properly using these tests. Historically, the lack of a comprehensive hypothesis-testing framework in the nonparametric case has provided few alternatives to normal-theory procedures. Fortunately, this situation has changed with the introduction of nonparametric, general linear model-based tests that can be used with existing computing packages. Multivariate-nonparametric tests due to Puri and Sen (1969, 1971, 1985) and Conover and Iman (1981) are outlined, and the results of a simulation study of the performance of three nonparametric and one normal-theory test of the hypothesis of no relationship among two sets of variables are presented. These results suggest that multivariate-nonparametric tests should be considered for a variety of data conditions. especially heavy-tailed and badly skewed data for small samples and a large number of variates.     

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.     

Empirical smoothing lack-of-fit tests for variance function

This paper discusses a nonparametric empirical smoothing lack-of-fit test for the functional form of the variance in regression models. The proposed test can be treated as a nontrivial modification of Zheng's nonparametric smoothing test, Koul and Ni's minimum distance test for the mean function in the classic regression models. The paper establishes the asymptotic normality of the proposed test under the null hypothesis. Consistency at some fixed alternatives and asymptotic power under some local alternatives are also discussed. A simulation study is conducted to assess the finite sample performance of the proposed test. Simulation study also shows that the proposed test is more powerful and computationally more efficient than some existing tests.     

A class of test statistics for testing whether new is better than used

A class of test statistics Is proposed for testing that a life distribution Is exponential against that it is new better than used and not exponential, Consistency, unbiasedness and the asymptotic normality for the one class of class of test are proved. The efflication are compeled and compared with some otner statinties. The proposed test is shown to perform well against other tests.     

Normality testing: two new tests using L-moments

Establishing that there is no compelling evidence that some population is not normally distributed is fundamental to many statistical inferences, and numerous approaches to testing the null hypothesis of normality have been proposed. Fundamentally, the power of a test depends on which specific deviation from normality may be presented in a distribution. Knowledge of the potential nature of deviation from normality should reasonably guide the researcher's selection of testing for non-normality. In most settings, little is known aside from the data available for analysis, so that selection of a test based on general applicability is typically necessary. This research proposes and reports the power of two new tests of normality. One of the new tests is a version of the R-test that uses the L-moments, respectively, L-skewness and L-kurtosis and the other test is based on normalizing transformations of L-skewness and L-kurtosis. Both tests have high power relative to alternatives. The test based on normalized transformations, in particular, shows consistently high power and outperforms other normality tests against a variety of distributions.     

Shapiro–Wilk test for skew normal distributions based on data transformations

A probability property that connects the skew normal (SN) distribution with the normal distribution is used for proposing a goodness-of-fit test for the composite null hypothesis that a random sample follows an SN distribution with unknown parameters. The random sample is transformed to approximately normal random variables, and then the Shapiro–Wilk test is used for testing normality. The implementation of this test does not require neither parametric bootstrap nor the use of tables for different values of the slant parameter. An additional test for the same problem, based on a property that relates the gamma and SN distributions, is also introduced. The results of a power study conducted by the Monte Carlo simulation show some good properties of the proposed tests in comparison to existing tests for the same problem.     

Jarque–Bera Test and its Competitors for Testing Normality – A Power Comparison

For testing normality we investigate the power of several tests, first of all, the well-known test of Jarque & Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro & Wilk (1965) as well as tests of Kolmogorov–Smirnov and Cramér-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque–Bera test and the Kolmogorov–Smirnov and Cramér-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters μ and σ have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters μ and σ and different proportions of contamination. It turns out that for the Jarque–Bera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque–Bera test is poor for distributions with short tails, especially if the shape is bimodal – sometimes the test is even biased. In this case a modification of the Cramér-von Mises test or the Shapiro–Wilk test may be recommended.     

Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing

This paper uses Monte Carlo simulation analysis to study the finite-sample behavior of bootstrap estimators and tests in the linear heteroskedastic model. We consider four different bootstrapping schemes, three of them specifically tailored to handle heteroskedasticity. Our results show that weighted bootstrap methods can be successfully used to estimate the variances of the least squares estimators of the linear parameters both under normality and under nonnormality. Simulation results are also given comparing the size and power of the bootstrapped Breusch-Pagan test with that of the original test and of Bartlett and Edgeworth-corrected tests. The bootstrap test was found to be robust against unfavorable regression designs.