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The energy and the ECF tests are powerful tests that apply for testing univariate or multivariate normality and are statistically consistent against general alternatives. The normal distribution has the highest entropy of any distribution for a given standard deviation. There are a number of normality tests based on this property, the first ...
Multivariate normality tests check a given set of data for similarity to the multivariate normal distribution. The null hypothesis is that the data set is similar to the normal distribution, therefore a sufficiently small p -value indicates non-normal data.
The Shapiro–Wilk test tests the null hypothesis that a sample x 1, ..., x n came from a normally distributed population. The test statistic is = (= ()) = (¯), where with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample (not to be confused with ).
In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used: Bayesian information criterion; Kolmogorov–Smirnov test; Cramér–von Mises criterion; Anderson–Darling test; Berk-Jones tests [1] [2] Shapiro–Wilk test; Chi-squared test; Akaike information criterion ...
SPSS provides an F-ratio from four different methods: Pillai's trace, Wilks’ lambda, Hotelling's trace, and Roy's largest root. In general, Wilks’ lambda has been recommended as the most appropriate multivariate test statistic to use.
In statistics, the Jarque–Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The test is named after Carlos Jarque and Anil K. Bera. The test statistic is always nonnegative. If it is far from zero, it signals the data do not have a normal distribution.
This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (k − 1) degrees of freedom, where k is the number of random samples, which may vary in size and are each drawn from independent normal distributions. Bartlett's test is sensitive to departures from normality.
One generalisation of the problem involves multivariate normal distributions with unknown covariance matrices, and is known as the multivariate Behrens–Fisher problem. [16] The nonparametric Behrens–Fisher problem does not assume that the distributions are normal. [17] [18] Tests include the Cucconi test of 1968 and the Lepage test of 1971.
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