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The Shapiro–Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. The test statistic is. where. with parentheses enclosing the subscript index i is the i th order statistic, i.e., the i th-smallest number in the sample (not to be confused with ). is the sample mean.
Illustration of the Kolmogorov–Smirnov statistic. The red line is a model CDF, the blue line is an empirical CDF, and the black arrow is the KS statistic.. Kolmogorov–Smirnov test (K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions that can be used to test whether a sample came from a ...
Kolmogorov–Smirnov test: this test only works if the mean and the variance of the normal distribution are assumed known under the null hypothesis, Lilliefors test: based on the Kolmogorov–Smirnov test, adjusted for when also estimating the mean and variance from the data, Shapiro–Wilk test, and; Pearson's chi-squared test.
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 ...
Lilliefors test. Lilliefors test is a normality test based on the Kolmogorov–Smirnov test. It is used to test the null hypothesis that data come from a normally distributed population, when the null hypothesis does not specify which normal distribution; i.e., it does not specify the expected value and variance of the distribution. [1]
Anderson–Darling test. The Anderson–Darling test is a statistical test of whether a given sample of data is drawn from a given probability distribution. In its basic form, the test assumes that there are no parameters to be estimated in the distribution being tested, in which case the test and its set of critical values is distribution-free.
Likelihood-ratio test. In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods.
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.