moment: Central Moments

In jeksterslabds/jeksterslabRdist: Jekster's Lab Distribution Utilities

Description

The sample central moment is defined by

m_j = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^j %(\#eq:dist-moments-sample-central)

where

• n is the sample size,

• x = \{x_1 … x_n\} is a set of values of a random variable X,

• \bar{x} is the sample mean of x, and

• j is the jth central moment.

Usage

moment(x, j)

Arguments

x	Numeric vector. Sample data.
j	Integer. jth moment. From 0 to 4.

Details

m_0 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^0 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( 1 \right) = \frac{1}{n} n = \frac{n}{n} = 1 %(\#eq:dist-moments-sample-central-zero)

m_1 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^1 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( 0 \right) = \frac{1}{n} 0 = \frac{0}{n} = 0 %(\#eq:dist-moments-sample-central-first)

m_2 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^2 %(\#eq:dist-moments-sample-central-second)

m_3 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^3 %(\#eq:dist-moments-sample-central-third)

m_4 = \frac{1}{n} ∑_{i = 1}^{n} ≤ft( x_i - \bar{x} \right)^4 %(\#eq:dist-moments-sample-central-fourth)

• The "zeroth" central moment is 1.

• The first central moment is 0.

• The second cental moment is the variance.

• The third central moment is used to define skewness.

• The fourth central moment is used to define kurtosis.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Wikipedia: Central Moment

Wikipedia: Standardized Moment

See Also

Other moments functions: cumulant(), kurt(), moments(), skew()

jeksterslabds/jeksterslabRdist documentation built on Aug. 9, 2020, 7:33 a.m.