by Nikolai V. Shokhirev

Up: ABC Stat

- Definitions

- Distributions

- Multivariate correlations

- Principal Component Analysis (see the Linear algebra section for basic definitions)

- Panel Data Analysis

n-th moment | (1) | |

Norm | (2) | |

Mean | (3) | |

n-th central moment | (4) | |

Variance | (5) | |

Std deviation | (6) | |

Skewness | μ_{3 }/ σ^{3} |
(7) |

Kurtosis | μ_{4 }/ σ^{4} |
(8) |

Excess kurtosis | μ_{4 }/ σ^{4} - 3 |
(9) |

μ_{2 }= m_{2} - m_{1}^{2} |
(10) |

μ_{3 }= m_{3} - 3 m_{2} m_{1}+
2 m_{1}^{3} |
(11) |

μ_{4 }= m_{4} - 4 m_{3} m_{1}+
6 m_{2} m_{1}^{2} - 3 m_{1}^{4} |
(12) |

Power-exponential function:

Negative skew, left-skewed. |
Positive skew, right-skewed. |

See more examples in Distributions.

A high kurtosis distribution has a sharper "peak" and flatter "tails", while a low kurtosis distribution has a more rounded peak with wider "shoulders".

- Distributions with
**zero**excess kurtosis are called**mesokurtic**, or**mesokurtotic**. The most prominent example is the normal distribution. - A distribution with
**positive**excess kurtosis is called**leptokurtic**, or**leptokurtotic**. Laplace distribution (double exponential distribution), excess kurtosis = 3. - A distribution with
**negative**excess kurtosis is called**platykurtic**, or**platykurtotic**. Uniform distribution, excess kurtosis = -1.2.

Laplace distribution.

Uniform distribution.

See more examples in Distributions.

In practice, often the density functions *P*(*x*) are not
available and only a limited number of a random variable is
available: *x _{i}* ,

n-th moment | (13) |

Note, that *m _{n}* as the sum of random variable is a random
variable itself. It has its expectation (mean) value and standard deviation. An
estimator is called "unbiased" if its expectation value is equal to the exact
(population) value. For example,

Mean | m_{1} |
(14) |

Variance | (15) | |

Skewness | (16) | |

Excess kurtosis | (17) |

The above estimators are coded in my PasMatLib library and will be included soon in CppMatLib as well.

- http://en.wikipedia.org/wiki/Moment_%28mathematics%29
- http://en.wikipedia.org/wiki/Variance
- http://en.wikipedia.org/wiki/Bias_of_an_estimator
- http://en.wikipedia.org/wiki/Skewness
- http://en.wikipedia.org/wiki/Kurtosis

** ABC Tutorials**

Up: ABC Stat

- Definitions

- Distributions

- Multivariate correlations

- Principal Component Analysis

- Panel Data Analysis

©Nikolai V. Shokhirev, 2001-2008