1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions
1.3.6.6.15. |
Tukey-Lambda Distribution |
The Tukey-Lambda distribution has the
shape parameter
. As with other probability
distributions, the Tukey-Lambda distribution can be transformed with a
location parameter,
, and a
scale parameter,
. Since the general form of
probability functions can be expressed in
terms of the standard distribution, all subsequent formulas in this
section are given for the standard form of the function.
The following is the plot of the Tukey-Lambda probability density
function for four values of .
The following is the plot of the Tukey-Lambda cumulative distribution
function with the same values of
as the pdf plots above.
The following is the plot of the Tukey-Lambda percent point function
with the same values of as the
pdf plots above.
|
= -1
|
approximately Cauchy |
|
= 0
|
exactly logistic |
|
= 0.14
|
approximately normal |
|
= 0.5
|
U-shaped |
|
= 1
|
exactly uniform (from -1 to +1) |
The most common use of this distribution is to generate a Tukey-Lambda
PPCC plot of a data set. Based on the ppcc
plot, an appropriate model for the data is suggested. For example, if
the maximum correlation occurs for a value of
at or near 0.14, then the data
can be modeled with a normal distribution. Values of
less than this imply a
heavy-tailed distribution (with -1 approximating a Cauchy). That is, as
the optimal value of goes from
0.14 to -1, increasingly heavy tails are implied. Similarly, as
the optimal value of becomes
greater than 0.14, shorter tails are implied.
As the Tukey-Lambda distribution is a symmetric distribution, the use of the Tukey-Lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributuins. A histogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.
