Search results
Results from the WOW.Com Content Network
Larger kurtosis indicates a more serious outlier problem, and may lead the researcher to choose alternative statistical methods. D'Agostino's K-squared test is a goodness-of-fit normality test based on a combination of the sample skewness and sample kurtosis, as is the Jarque–Bera test for normality.
This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denoted Skew[X]. If σ is finite and μ is finite too, then skewness can be expressed in terms of the non-central moment E[X 3] by expanding the previous formula:
The first is the square of the skewness: β 1 = γ 1 where γ 1 is the skewness, or third standardized moment. The second is the traditional kurtosis, or fourth standardized moment: β 2 = γ 2 + 3. (Modern treatments define kurtosis γ 2 in terms of cumulants instead of moments, so that for a normal distribution we have γ 2 = 0 and β 2 = 3.
The shape of a distribution may be considered either descriptively, using terms such as "J-shaped", or numerically, using quantitative measures such as skewness and kurtosis.
Explicit expressions for the skewness and kurtosis are lengthy. [8] As β {\displaystyle \beta } tends to infinity the mean tends to α {\displaystyle \alpha } , the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below).
One disadvantage of L-moment ratios for estimation is their typically smaller sensitivity. 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.
The probability density function is the partial derivative of the cumulative distribution function: (;,) = (;,) = / (+ /) = (() / + / ()) = ().When the location parameter μ is 0 and the scale parameter s is 1, then the probability density function of the logistic distribution is given by
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.