Definitions, Formulae

by Nikolai V. Shokhirev

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- Panel Data Analysis

Definitions 

 n-th moment    (1)
 Norm   (2)
 Mean   (3)
 n-th central moment   (4)
 Variance   (5)
 Std deviation   (6)
 Skewness  μ3 / σ3 (7)
Kurtosis  μ4 / σ4 (8)
Excess kurtosis  μ4 / σ4 - 3 (9)

Useful relations

 μ2 m2 - m12 (10)
 μ3 m3 - 3 m2 m1+ 2 m13  (11)
 μ4 m4 - 4 m3 m1+ 6 m2 m12 - 3 m14  (12)

 

Skewness example

Power-exponential function:

 

 Negative skew, left-skewed.

  Positive skew, right-skewed.

See more examples in Distributions.

 

Kurtosis terminology

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

Laplace distribution.

Uniform distribution.

See more examples in Distributions.

 

Sample estimators

In practice, often the density functions P(x) are not available and only a limited number of a random variable is available: xi , i = 1, . . . , N . So-called sample estimators can be used instead of (1):

 n-th moment      (13) 

Note, that mn 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, E(m1) = μ, is an unbiased estimator; E(m2 - m12) = σ2 N/(N - 1), is biased (although it tends to σ2 as N → ∞).

 

Unbiased estimators

 Mean  m1 (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.

References

  1. http://en.wikipedia.org/wiki/Moment_%28mathematics%29
  2. http://en.wikipedia.org/wiki/Variance
  3. http://en.wikipedia.org/wiki/Bias_of_an_estimator
  4. http://en.wikipedia.org/wiki/Skewness
  5. http://en.wikipedia.org/wiki/Kurtosis

ABC Tutorials
Up: ABC Stat
- Definitions
- Distributions
- Multivariate correlations
- Principal Component Analysis
- Panel Data Analysis

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