Abstract
Several graphical methods for testing univariate composite normality from an i.i.d. sample are presented. They are endowed with correct simultaneous error bounds and yield size-correct tests. As all are based on the empirical CDF, they are also consistent for all alternatives. For one test, called the modified stabilized probability test, or MSP, a highly simplified computational method is derived, which delivers the test statistic and also a highly accurate p-value approximation, essentially instantaneously. The MSP test is demonstrated to have higher power against asymmetric alternatives than the well-known and powerful Jarque-Bera test. A further size-correct test, based on combining two test statistics, is shown to have yet higher power. The methodology employed is fully general and can be applied to any i.i.d. univariate continuous distribution setting.
Highlights
We consider testing the composite null hypothesis that an independent, identically distributed (i.i.d.)set of data comes from some normal distribution; the actual values of the location and scale terms, μ and σ, are not part of the null hypothesis
In light of the growing recognition of the ubiquity of non-Gaussian processes, but, in financial econometrics and quantitative risk management, it might seem that normality testing is becoming less relevant. This is, not the case: for any proposed model that results in a set of i.i.d. data from a particular distribution, the fitted cumulative distribution function (CDF) can be applied to the data points, and the inverse CDF of the normal distribution can be applied, so that the tests can be used
In the case of multivariate models with hundreds or thousands of financial assets, in which non-Gaussian generalized autoregressive conditional heteroskedasticity (GARCH) types of filters are applied for the location and time-varying scale of each return series, yielding approximate i.i.d
Summary
We consider testing the composite null hypothesis that an independent, identically distributed (i.i.d.). Our method is clearly related to that of [8,9], though their computational method is different Those authors: (i) concentrate mostly on the case in which the full distribution is specified as opposed to the composite case; (ii) in the composite case, they compare different estimates for the two parameters of the normal distribution and use them as part of the fully-specified distribution, as opposed to use of the simulation method considered to adjust for parameter estimation; (iii) require simulation (as does our method), but do not simplify the calculations (as we do), so that the graphic and test statistics (and, in our case, a p-value) are delivered instantaneously, i.e., in our case, simulation is no longer required; and (iv) neither claim nor demonstrate that their resulting graphical test is size-correct in the composite null case.
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