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For a sample of n values, a method of moments estimator of the population excess kurtosis can be defined as = = = (¯) [= (¯)] where m 4 is the fourth sample moment about the mean, m 2 is the second sample moment about the mean (that is, the sample variance), x i is the i th value, and ¯ is the sample mean.
Let X and Y each be normally distributed with correlation coefficient ρ. The cokurtosis terms are (,,,) = +(,,,) = (,,,) =Since the cokurtosis depends only on ρ, which is already completely determined by the lower-degree covariance matrix, the cokurtosis of the bivariate normal distribution contains no new information about the distribution.
In statistics, the method of moments is a method of estimation of population parameters.The same principle is used to derive higher moments like skewness and kurtosis. It starts by expressing the population moments (i.e., the expected values of powers of the random variable under consideration) as functions of the parameters of interest.
HOS are particularly used in the estimation of shape parameters, such as skewness and kurtosis, as when measuring the deviation of a distribution from the normal distribution. In statistical theory , one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulants and joint ...
For instance, the Laplace distribution has a kurtosis of 6 and weak exponential tails, but a larger 4th L-moment ratio than e.g. the student-t distribution with d.f.=3, which has an infinite kurtosis and much heavier tails. As an example consider a dataset with a few data points and one outlying data value.
and Φ −1 is the standard normal quantile function. If the data are consistent with a sample from a normal distribution, the points should lie close to a straight line. As a reference, a straight line can be fit to the points. The further the points vary from this line, the greater the indication of departure from normality.
For medium size samples (<), the parameters of the asymptotic distribution of the kurtosis statistic are modified [37] For small sample tests (<) empirical critical values are used. Tables of critical values for both statistics are given by Rencher [ 38 ] for k = 2, 3, 4.
In the following, { x i } denotes a sample of n observations, g 1 and g 2 are the sample skewness and kurtosis, m j ’s are the j-th sample central moments, and ¯ is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as √ β 1 and β 2 respectively.