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A Robust Rescaled Moment Test for Normality in Regression Journal of Mathematics and Statistics 5 (1): 54-62, 2009 ISSN 1549-3644 © 2009 Science Publications A Robust Rescaled Moment Test for Normality in Regression 1Md.Sohel Rana, 1Habshah Midi and 2A.H.M. Rahmatullah Imon 1Laboratory of Applied and Computational Statistics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia 2Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USA Abstract: Problem statement: Most of the statistical procedures heavily depend on normality assumption of observations. In regression, we assumed that the random disturbances were normally distributed. Since the disturbances were unobserved, normality tests were done on regression residuals. But it is now evident that normality tests on residuals suffer from superimposed normality and often possess very poor power. Approach: This study showed that normality tests suffer huge set back in the presence of outliers. We proposed a new robust omnibus test based on rescaled moments and coefficients of skewness and kurtosis of residuals that we call robust rescaled moment test. Results: Numerical examples and Monte Carlo simulations showed that this proposed test performs better than the existing tests for normality in the presence of outliers. Conclusion/Recommendation: We recommend using our proposed omnibus test instead of the existing tests for checking the normality of the regression residuals. Key words: Regression residuals, outlier, rescaled moments, skewness, kurtosis, jarque-bera test, robust rescaled moment test INTRODUCTION out that the powers of t and F tests are extremely sensitive to the hypothesized error distribution and may In regression analysis, it is a common practice over deteriorate very rapidly as the error distribution the years to use the Ordinary Least Squares (OLS) becomes long-tailed. Furthermore, Bera and Jarque [1] method mainly because of tradition and ease of have found that homoscedasticity and serial computation. The assumption of normality, that is, the independence tests suggested for normal errors may data are a random sample from a normal distribution is result in incorrect conclusions under non-normality. It the most important assumption for many statistical may be also essential to have proper knowledge of the procedures. Under a normal assumption of regression error distribution in prediction and in confidence limits errors, the OLS method has many desirable properties of predictions. Most of the standard results of this in both estimation of parameters and in testing of particular study are based on the normality assumption hypotheses. But in practice we often deal with data sets and the whole inferential procedure may be subjected to which are not normal in nature. Non-normality may error if there is a departure from this. In all, violation of occur because of their inherent random structure or the normality assumption may lead to the use of because of the presence of outliers. Nevertheless, suboptimal estimators, invalid inferential statements evidence is available that such departures can have and inaccurate predictions and for this reason test for unfortunate effects in a variety of situations. In normality has become an essential part of regression regression problems, the effect of departure from analysis. normality in estimation was studied by Huber [12] . In Several methods have been suggested in the testing hypotheses, the effect of departure from literature for assessing the assumption of normality. normality has been investigated by many There are a considerable amount of written papers statisticians [15] . When the errors are not normally relating to the performance of various tests for distributed, the estimated regression coefficients and normality in regression [5,8,9,13,18,20]. Among them the estimated error variances are no longer normal and chi- Jarque-Bera (JB) test [14] for normality (also known in square and consequently the t and F tests are not statistics the Bowman-Shenton test) has become very generally valid in finite samples. Koenker [16] pointed popular with the statisticians. The JB test statistic is a Corresponding Author: Sohel Rana, Laboratory of Applied and Computational Statistics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia 54 J. Math. & Stat., 5 (1): 54-62, 2009 sum of the sample coefficients of skewness and kurtosis where, E( ∈) = 0, V( ∈) = σ2I and I is an identity matrix and asymptotically follows a x2 distribution with two of order n. degrees of freedom. But the main shortcoming of the JB Now we establish general expressions for the test is that it possesses very poor power when the moments, coefficients of skewness and kurtosis of the sample size is small or moderate [21]. To overcome this true errors assuming only that the errors are problem Imon [13] modified this test by suitably rescaling uncorrelated, zero mean and identically distributed and the moment estimators obtained from the least squares the existence of their first four moments. Let us define residuals. His proposed Rescaled Moment (RM) test the k-th moment about the origin of the i-th error by: improves the estimation of moments, coefficients of µ= ∈k = skewness and kurtosis and gives better power than the kE( i ), k 1,2,... (3) JB test for normality. However, both the JB and the RM test statistics rely on coefficients of sample skewness As E( ∈) = 0, on the null model, we need not and kurtosis which are very sensitive to outliers. For distinguish here between moments about the origin and this reason some robust tests of normality are being those about the mean. Simple forms of the coefficients proposed by some authors [9,8]. Gel and Gastwirth [8] of skewness and kurtosis are given by: propose a modification of the JB test utilizing a robust µ µ estimate of spread, namely the average absolute β=3 β= 4 13 , 2 2 (4) deviation from the median (MAAD), in the µ 2 µ 2 2 denominators of skewness and kurtosis. This Robust Jarque-Bera (RJB) test statistic asymptotically follows a In practice population moments are often estimated 2 [8] x distribution with two degrees of freedom and by sample moments. We generally define the k-th provides higher or similar power in detecting heavy- sample moment by: tailed alternatives compared to the JB. Since the RJB is designed as a general statistical test for normality, we k ()∈=1 () ∈−∈ = mk∑ i ,k1,2,... (5) suspect that it may not perform well in regression n i analysis. In this study we propose a robust rescaled moment test (RRM) for normality designed for and hence the raw coefficients of skewness and kurtosis regression models extending the idea of Imon [13] and [8] can be defined as: Gel and Gastwirth . The main objective of the proposed method is to suggest a normality test that is m()∈ m() ∈ S()∈=3 ,K() ∈= 4 (6) fairly robust in the presence of outliers and also 3 ∈ 2 [m (∈ )] 2 [m2 ( )] performs best in general. The empirical evidence shows 2 that the proposed RRM test offers substantial improvements over the existing tests of normality. Much work has been done in producing omnibus tests for normality, combining S and K in one test MATERIALS AND METHODS statistic. So a large deviation either of S from 0 or K from 3 should give a significant result, regardless of The robust rescaled moments test: The simplest form which one deviates from normal values. D'Agostino [6] of the general linear model is given by: and Pearson first suggested this kind of test. However, it requires an assumption of independence y= xT β+∈ , i = 1,2,...,n (1) between S and K which is only asymptotically correct. i i i [2] Bowman and Shenton suggested a normality test, Where: popularly known as Jarque-Bera test, with the test yi = The i-th observed response statistic: xi = A p ×1 vector of predictors 2 ()K− 3 2 β = A p ×1 vector of unknown finite parameters =S + JB 2 2 (7) ∈ = Uncorrelated random errors with mean 0 and σ()S σ () K variance σ2 Where: T T 2 2 Writing Y = (y 1,y 2,….y n) , X = (x 1,x 2,…x n) and σ (S) = 6/n and σ (K) = 24/n are the asymptotic T ∈ = ( ∈1, ∈2,… ∈n) the model is: variance of S and K respectively S = Under normality asymptotically, is distributed Y = X β+∈ (2) as x2 distribution with 2 degrees of freedom 55 J. Math. & Stat., 5 (1): 54-62, 2009 Gel and Gastwirth [8] give a robust version of the The residual vector can also be expressed in terms Jarque-Bera test using a robust estimate of spread of unobservable errors as: which is less influenced by outliers in the denominators of the sample estimates of skewness and kurtosis. They ∈=ˆ [I − X(XX)T− 1 X] T ∈= H ∈ (12) consider the average absolute deviation from the [7] sample median (MAAD) proposed by Gastwirth and We shall follow Chatterjee and Hadi [3] by referring is defined by: to H= [I − X(XX)T− 1 X T ] as the residual hat matrix. The n elements hij of this matrix will be termed as residual hat =A − Jn∑ |XM| i (8) elements, which play a very important role in linear n = i 1 regression. The quantities (1− h ) are often referred to ii as leverage values which measure how far the input where, A = π / 2 . The robust sample estimates of vector x are from the rest of the data. Let the k-th skewness and kurtosis are mˆ / J 3 and mˆ / J 4 , where i 3 n 4 n moment of the i-th OLS residual be defined as: ˆ ˆ m3 and m4 are the 3rd and 4th order of the estimated sample moments respectively, which lead the µ∈=∈( ) k = kˆ iE( ˆ i ) , k 1,2,..
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