@article { ISI:000252359600010,
	title = {On tests for multivariate normality and associated simulation studies},
	journal = {JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION},
	volume = {77},
	number = {11-12},
	year = {2007},
	month = {NOV-DEC},
	pages = {1053-1068},
	publisher = {TAYLOR \& FRANCIS LTD},
	type = {Article},
	address = {4 PARK SQUARE, MILTON PARK, ABINGDON OX14 4RN, OXON, ENGLAND},
	abstract = {We study the empirical size and power of some recently proposed tests for multivariate normality (MVN) and compare them with the existing proposals that performed best in previously published studies. We show that the Royston{\textquoteright}s [Royston, J.P., 1983b, Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics, 32,121-133.] extension to the Shapiro and Wilk [Shapiro, S.S. and Wilk, M.B., 1965, An analysis of variance test for normality (complete samples). Biometrika, 52, 591-611.] test is unable to achieve the nominal significance level, and consider a subsequent extension proposed by Royston [Royston, J.P., 1992, Approximating the Shapiro-Wilk W-Test for non-normality. Statistics and Computing, 2, 117-119.] to correct this problem, which earlier studies appear to have ignored. A consistent and invariant test proposed by Henze and Zirkler [Henze, N. and Zirkler, B., 1990, A class of invariant consistent tests for multivariate normality. Communications in Statistics- Theory and Methods, 19, 3595-3617.] is found to have good power properties, particularly for sample sizes of 75 or more, while an approach suggested by Royston [Royston, J.P, 1992, Approximating the Shapiro-Wilk W-Test for non-normality. Statistics and Computing, 2, 117-119.] performs effectively at detecting departures from MVN for smaller sample sizes. We also compare our results to those of previous simulation studies, and discuss the challenges associated with generating multivariate data for such investigations.},
	keywords = {consistent tests, goodness-of-fit, invariant test, multivariate normality, power, size},
	issn = {0094-9655},
	doi = {10.1080/10629360600878449},
	author = {Farrell, Patrick J. and Salibian-Barrera, Mat{\'\i}as and Naczk, Katarzyna}
}
@article { ISI:000241551800014,
	title = {On the asymptotic distribution of Pearson{\textquoteright}s $X^2$ in cross-validation samples},
	journal = {Psychometrika},
	volume = {71},
	number = {3},
	year = {2006},
	month = {SEP},
	pages = {587-592},
	publisher = {Springer},
	type = {Article},
	abstract = {In categorical data analysis, two-sample cross-validation is used not only for model selection but also to obtain a realistic impression of the overall predictive effectiveness of the model. The latter is of particular importance in the case of highly parametrized models capable of capturing every idiosyncracy of the calibrating sample. We show that for maximum likelihood estimators or other asymptotically efficient estimators Pearson{\textquoteright}s X-2 is not asymptotically chi-square in the two-sample cross-validation framework due to extra variability induced by using different samples for estimation and goodness-off-it testing. We propose an alternative test statistic, X-xval(2) , obtained as a modification of X-2 which is asymptotically chi-square with C-1 degrees of freedom in cross-validation samples. Stochastically, X-xval(2) <= X-2. Furthermore, the use of X-2 instead of X-xval(2) with a chi(2)(C-1) reference distribution may provide an unduly poor impression of fit of the model in the cross-validation sample.},
	keywords = {contingency tables, goodness-of-fit, item response theory modeling, latent class analysis, quadratic form statistics},
	issn = {0033-3123},
	doi = {10.1007/s11336-005-1284-z},
	author = {Joe, Harry and Maydeu-Olivares, Albert}
}
