A Numeric Comparison
Recall that the CDF measures the proportion of probability to the left of any particular point. Because the difference is so pronounced in the center of the distribution we have chosen, we will check CDF values for the Stable distribution and compare those values to ones obtained under the normal assumption. The table below indicates that the difference can be considerable.
Normal Probability |
Stable Probability |
Difference |
|
-2.5 |
0.187008 |
0.057323 |
0.129636 |
-1.5 |
0.301357 |
0.192809 |
0.108548 |
-0.5 |
0.439581 |
0.464561 |
-0.024980 |
0.5 |
0.585678 |
0.719111 |
-0.133432 |
1.5 |
0.720697 |
0.859537 |
-0.138840 |
2.5 |
0.829802 |
0.922593 |
-0.092790 |
Returning to our original metaphor of a game with known probabilities, suppose it became known that the benevolent and omniscient dictator (your financial advisor), provided you with probabilities calculated based on the assumption of normality. Suppose that you accepted those numbers and on that basis decided the amount you were willing to pay to enter the game (invest). Then suppose that a different financial analyst came along with the ability to estimate parameters under stable assumption? If the price you are willing to pay - the amount you are willing to invest - is dependent on the risk, as it always is, the table above indicates that you would be willing to invest very different amounts under the different assumptions.
Taking our lottery metaphor to the next step, we might conclude that the assumption of normality is a tax on people who can't estimate stable parameters!!!
For a short tour of both ancient roots and recent developments in the market for heavy tail thinking, see the Why Bother? page.
The next step begins a technical discussion of the mathematics of Stable Distributions and their implementation with Mathematica.
