statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

, the Lilliefors test is a normality test based on the

Kolmogorov–Smirnov test In statistics, the 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 compare a sample with ...

. It is used to test the

null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is ...

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 In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

and

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...

of the distribution. It is named after Hubert Lilliefors, professor of statistics at

George Washington University , mottoeng = "God is Our Trust" , established = , type = Private federally chartered research university , academic_affiliations = , endowment = $2.8 billion (2022) , presi ...

. A variant of the test can be used to test the null hypothesis that data come from an exponentially distributed population, when the null hypothesis does not specify which

exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

The test

The test proceeds as follows: # First estimate the population mean and population variance based on the data. # Then find the maximum discrepancy between the

empirical distribution function In statistics, an empirical distribution function (commonly also called an empirical Cumulative Distribution Function, eCDF) is the distribution function associated with the empirical measure of a sample. This cumulative distribution function ...

and the

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Eve ...

(CDF) of the normal distribution with the estimated mean and estimated variance. Just as in the Kolmogorov–Smirnov test, this will be the test statistic. # Finally, assess whether the maximum discrepancy is large enough to be

statistically significant In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...

, thus requiring rejection of the null hypothesis. This is where this test becomes more complicated than the Kolmogorov–Smirnov test. Since the hypothesised CDF has been moved closer to the data by estimation based on those data, the maximum discrepancy has been made smaller than it would have been if the null hypothesis had singled out just one normal distribution. Thus the "null distribution" of the test statistic, i.e. its

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

assuming the null hypothesis is true, is stochastically smaller than the Kolmogorov–Smirnov distribution. This is the Lilliefors distribution. To date, tables for this distribution have been computed only by

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deter ...

s. In 1986 a corrected table of critical values for the test was published.

References

Sources

* Conover, W.J. (1999), "Practical nonparametric statistics", 3rd ed. Wiley : New York.

External links

Lilliefors test
in R
Lilliefors test
in Python
Lilliefors test
on

Mathworks MathWorks is an American privately held corporation that specializes in mathematical computing software. Its major products include MATLAB and Simulink, which support data analysis and simulation. History The company's key product, MATLAB, wa ...

{{DEFAULTSORT:Lilliefors Test Normality tests

The test

See also

References

Sources

External links