Search results
Results From The WOW.Com Content Network
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.
For example, one may administer a test to a number of individuals. If it is assumed that each person's score (0 ≤ θ ≤ 1) is drawn from a population-level beta distribution, then an important statistic is the mean of this population-level distribution. The mean and sample size parameters are related to the shape parameters α and β via [3]
for i = 1, 2, ..., n, where a = 3/8 if n ≤ 10 and 0.5 for n > 10, 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.
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 ...
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.
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.
A simple example of this is for the center of nominal data: instead of using the mode (the only single-valued "center"), one often uses the empirical measure (the frequency distribution divided by the sample size) as a "center". For example, given binary data, say heads or tails, if a data set consists of 2 heads and 1 tails, then the mode is ...
An example of a geometric distribution arises from rolling a six-sided die until a "1" appears. Each roll is independent with a 1 / 6 {\displaystyle 1/6} chance of success. The number of rolls needed follows a geometric distribution with p = 1 / 6 {\displaystyle p=1/6} .