Pareto Distribution

Where do you meet this distribution?

• Meteorology, Seismology : Model of extreme event (Extreme value theory)
• Risk management -- Operational risk

Shape of Distribution

Basic Properties

• Two parameters $a, b$ are required.

  $$a>0, b>0$$

• Continuous distribution defined on semi-infinite range $x>b$

• This distribution is always asymmetric.

Probability

• Cumulative distribution function

  $$F(x)=1-\left(\frac{b}{x}\right)^a$$

• Probability density function

  $$f(x)=\frac{ab^a}{x^{a+1}}$$

• How to compute these on Excel.

	A	B
1	Data	Description
2	5	Value for which you want the distribution
3	8	Value of parameter A
4	2	Value of parameter B
5	Formula	Description (Result)
6	=1-POWER(A4/A2,A3)	Cumulative distribution function for the terms above
7	=A3*A4^A3/POWER(A2,A3+1)	Probability density function for the terms above

Quantile

• Inverse function of cumulative distribution function

  $$F^{-1}(P)=\frac{b}{(1-P)^{1/a}}$$

• How to compute this on Excel.

	A	B
1	Data	Description
2	0.7	Probability associated with the distribution
3	1.7	Value of parameter A
4	0.9	Value of parameter B
5	Formula	Description (Result)
6	=A4/POWER(1-A2,1/A3)	Inverse of the cumulative distribution function for the terms above

Characteristics

Mean -- Where is the "center" of the distribution? (Definition)

• Mean of the distribution is given as

  $$\frac{ab}{a-1}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter Alpha
3	2	Value of parameter Beta
4	Formula	Description (Result)
5	=A2*A3/(A2-1)	Mean of the distribution for the terms above

Standard Deviation -- How wide does the distribution spread? (Definition)

• Variance of the distribution is given as

  $$\frac{ab^2}{(a-1)^2(a-2)}$$

  Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter A
3	2	Value of parameter B
4	Formula	Description (Result)
5	=A3/(A2-1)*SQRT(A2/(A2-2))	Standard deviation of the distribution for the terms above

Skewness -- Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as

  $$\sqrt{\frac{a-2}{a}}\frac{2(a+1)}{a-3}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter A
3	Formula	Description (Result)
4	=SQRT((A2-2)/A2)*2*(A2+1)/(A2-3)	Skewness of the distribution for the terms above

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as

  $$\frac{6(a^3+a^2-6a-2)}{a(a-3)(a-4)}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	8	Value of parameter A
3	Formula	Description (Result)
4	=6*(A2^3+A2^2-6*A2-2)/(A2*(A2-3)*(A2-4))	Kurtosis of the distribution for the terms above

Random Numbers

• Random number x is generated by inverse function method, which is for uniform random U,

  $$x=\frac{b}{(1-U)^{1/a}}$$

• How to generate random numbers on Excel.

	A	B
1	Data	Description
2	0.5	Value of parameter A
3	2	Value of parameter B
4	Formula	Description (Result)
5	=A3/POWER(1-NTRAND(100),1/A2)	100 Pareto deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

NtRand Functions

Not supported yet

Reference

• Wolfram Mathworld -- Pareto distribution
• Wikipedia -- Pareto distribution
• Statistics Online Computational Resource