can someone explain graphically what a fatter tail is? let’s say we have x and y axis. the # of observations are on the y scale and the distribution percentages are on the x scale from 0 to 100. when we have leptokurtosis and we have fatter tails. what do the tails look like that make them fatter?
fat tails = more risk
It’s better to think of the standard normal curve with mean = 0 and st dev = 1. With this distribution, we know P(|x|>3) = approx 1%. With fat tails that number would be higher. In a graphical sense there would more area under the ends of both the positive and negative tails.
You could also think of the curve as being “elevated” over the x-axis. If the curve is less “elevated” over the x-axis, thinner tails. If the curve is more "elevated over the x-axis, fatter tails. Just how I think about it…
fatter tales = flattened bell curve. pretend someone stepped on it. thinner tales = a more “peaked” bell curve.
interpretation-wise, chances that very good (very high x) or very bad (very low x) events will happen is higher. image-wise, fatter tail doesn’t mean flattened bell curve, in fact just the opposite.
soxboys21 Wrote: ------------------------------------------------------- > You could also think of the curve as being > “elevated” over the x-axis. > > If the curve is less “elevated” over the x-axis, > thinner tails. > > If the curve is more "elevated over the x-axis, > fatter tails. > > Just how I think about it… this is what i was thinking too…so if we have leptokurtosis, the tails are more elevated and of course the peak is more elevated than normal…
exotic Wrote: ------------------------------------------------------- > interpretation-wise, chances that very good (very > high x) or very bad (very low x) events will > happen is higher. > > image-wise, fatter tail doesn’t mean flattened > bell curve, in fact just the opposite. err ya, late night brain-cramp on my part. what he said.
Yes, a counter-intuitive aspect of fat tails is that the fatness of the extreme tails tend to come at the expense of the moderate tail, not the center. So you have a higher peak in the middle, thicker tails at the ends, ad a thinner part (compared to the standard normal curve) in the moderate region. It can throw you sometimes. The fact that the tails are thick does *not* mean that the very middle is shorter, in fact, the very middle tends to be taller.
when you hear fat tails, it has more to do with behavior in the tails than in the middle. Some distributions have a higher peak, some don’t. What’s really important is the decay of density function for large deviations from the mean. For example, density of standard normal distribution is (1/sqrt(2*pi))*exp(-x^2/2). It’s easy to see that decay is even faster than exponential because of exp(-x^2). If decay is exponential, distribution has thin tails. If decay is polynomial (as x^(-n) for some n), then the distribution has fat tails. For example, Cauchy distribution is a good example of very fat tails with density function of c/(x^2+1) //decay of c*x^(-2).