The Power to See: A New Graphical Test of Normality

Abstract

Many statistical procedures assume the underlying data generating process involves Gaussian errors. Among the well-known procedures are ANOVA, multiple regression, linear discriminant analysis and many more. There are a few popular procedures that are commonly used to test for normality such as the KolmogorovSmirnov test and the ShapiroWilk test. Excluding the Kolmogorov-Smirnov testing procedure, these methods do not have a graphical representation. As such these testing methods offer very little insight as to how the observed process deviates from the normality assumption. In this paper we discuss a simple new graphical procedure which provides confidence bands for a normal quantile-quantile plot. These bands define a test of normality and are much narrower in the tails than those related to the Kolmogorov-Smirnov test. Correspondingly the new procedure has much greater power to detect deviations from normality in the tails. Department of StatisticsThe Wharton School University of Pennsylvania PA, USA AbstractMany statistical procedures assume the underlying data generating process involves Gaussian errors. Among the well-known procedures are ANOVA, multiple regression, linear discriminant analysis and many more. There are a few popular procedures that are commonly used to test for normality such as the Kolmogorov-Smirnov test and the ShapiroWilk test. Excluding the Kolmogorov-Smirnov testing procedure, these methods do not have a graphical representation. As such these testing methods offer very little insight as to how the observed process deviates from the normality assumption. In this paper we discuss a simple new graphical procedure which provides confidence bands for a normal quantile-quantile plot. These bands define a test of normality and are much narrower in the tails than those related to the KolmogorovSmirnov test. Correspondingly the new procedure has much greater power to detect deviations from normality in the tails.

A Study of Periodograms Standardized Using Training Datasets and Application to Exoplanet Detection

“…In the tails, however, this convergence is very slow (see [52], [67], [70] for illustrations). For this reason the test statistics (16), which compares favorably to HC and other goodness-of-fit (GOF) tests in some cases, was recently (and almost simultaneously) proposed by [52], [66], [68], [69]. As noted in [52], [53], this test was initially proposed by Berk and Jones (and called M + n ) in [71].…”

Comparing distributions by multiple testing across quantiles or CDF values

“…Fact 6 may also be used to quickly compute GOF p-values and uniform confidence bands; see Buja and Rolke (2006), Aldor-Noiman et al (2013), our code, and an earlier version of this paper for details.…”

“…The probability integral transform reduces the problem to order statistics from a standard uniform distribution, whose finite-sample distribution is known. Their one-sample uniform confidence band (from Section 5.1) was eventually detailed and published by Aldor-Noiman, Brown, Buja, Rolke, and Stine (2013). Buja and Rolke (2006) also construct a two-sample permutation test for equality.…”

“…The tradeoff is that iid sampling is required. However, the finite-sample sampling distribution (of the true CDF evaluated jointly at all order statistics) turns out to be equivalent to the finite-sample posterior distribution from the continuity-corrected Bayesian bootstrap in Banks (1988): in the iid case, the uniform confidence band of Aldor-Noiman et al (2013) is also a Bayesian uniform credible band. 5 We are hopeful that further work will show that our iid assumption may be "relaxed" by using the Bayesian bootstrap to allow sampling weights (as in Lo, 1993) and clustering (as in Cameron, Gelbach, and Miller, 2008) while maintaining exact finite-sample properties in the iid case.…”

Visualizing Type II Error in Normality Tests

“…In this regard, Rosenkrantz (2000) derives bounds for the theoretical quantile function suggesting a test that has the same visual representation advantages of the KS test. In a similar manner, Aldor-Noiman et al (2013) propose a new method which provides simultaneous confidence bands for a normal quantile-quantile plot that are narrower in the extremes than those associated with the KS test, so higher powers are obtained, keeping the visualization benefit. This paper presents a visual and easy to understand way to show Type II error in normality tests.…”