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The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
| λ = −1 | approximately Cauchy |
| λ = 0 | exactly logistic |
| λ = 0.14 | approximately normal N(0, 2.142) |
| λ = 0.5 | strictly concave (-shaped) |
| λ = 1 | exactly uniform U(−1, 1) |
| λ = 2 | exactly uniform U(−½, ½) |
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 λ at or near 0.14, then the data can be modeled with a normal distribution. Values of λ less than this imply a heavy-tailed distribution (with −1 approximating a Cauchy). That is, as the optimal value of lambda goes from 0.14 to −1, increasingly heavy tails are implied. Similarly, as the optimal value of λ becomes greater than 0.14, shorter tails are implied.
Since 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 distributions. A histogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.
Read more about this topic: Tukey Lambda Distribution
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