Box-Cox distribution with and without offset

The BIS.Net Team BIS.Net Team

The Box–Cox distribution, aka a power distribution, is the resultant distribution after applying a Box-Cox transformation to normalize non-normal data. The transformed data is a truncated Normal distribution.

Once transformed it is possible to fit curves, compute capability indexes and other statistics

The probability density function for the one parameter function is:

Box Cox formula 1

Where y > 0, m is the location parameter, s the dispersion, ƒ the family parameter, I is the indicator function, phi the cumulative distribution function of the standard normal distribution, and sgn the sign function.

The one-parameter Box–Cox transformation is defined as

Box Cox formula 2

and the two-parameter Box-Cox transformations is defined as

Box Cox formula 3

Lambda and Lambda1 are the transformation parameters and lambda2 an offset.

Lambda is normally calculated using a profile likelihood function. BIS.Net Process Performance uses a highly efficient machine powered algorithm to find lambda and the offset subject to minimizing the Anderson Darling statistic.

The Box-Cox distribution is applied when it is not possible to fit one of the currently known distributions. There is no guarantee that the normalization will work.

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