BoxCox: The Box-Cox Transformed Normal Distribution

boxcoxR Documentation

The Box-Cox Transformed Normal Distribution


Functions related with the Box-Cox family of transformations. Density and random generation for the Box-Cox transformed normal distribution with mean equal to mean and standard deviation equal to sd, in the normal scale.


rboxcox(n, lambda, lambda2 = NULL, mean = 0, sd = 1)

dboxcox(x, lambda, lambda2 = NULL, mean = 0, sd = 1)



numerical value(s) for the transformation parameter lambda.


logical or numerical value(s) of the additional transformation (see DETAILS below). Defaults to NULL.


number of observations to be generated.


a vector of quantiles (dboxcox) or an output of boxcoxfit (print, plot, lines).


a vector of mean values at the normal scale.


a vector of standard deviations at the normal scale.


Denote Y the variable at the original scale and Y' the transformed variable. The Box-Cox transformation is defined by:

Y' = log(Y) if lambda = 0 , Y' = ((Y ^ lambda) - 1)/lambda otherwise


An additional shifting parameter lambda2 can be included in which case the transformation is given by:

Y' = log(Y + lambda2) if lambda = 0 , Y' = (((Y + lambda2) ^ lambda) - 1)/lambda otherwise


The function rboxcox samples Y' from the normal distribution using the function rnorm and backtransform the values according to the equations above to obtain values of Y. If necessary the back-transformation truncates the values such that Y' >= -1/lambda results in Y = 0 in the original scale. Increasing the value of the mean and/or reducing the variance might help to avoid truncation.


The functions returns the following results:


a vector of random deviates.


a vector of densities.


Paulo Justiniano Ribeiro Jr.,
Peter J. Diggle


Box, G.E.P. and Cox, D.R.(1964) An analysis of transformations. JRSS B 26:211–246.

See Also

The parameter estimation function boxcoxfit, the function boxcox in the package MASS and the function box.cox in the package car.


## Simulating data
simul <- rboxcox(100, lambda=0.5, mean=10, sd=2)
## Comparing models with different lambdas,
## zero  means and unit variances
curve(dboxcox(x, lambda=-1), 0, 8)
for(lambda in seq(-.5, 1.5, by=0.5))
  curve(dboxcox(x, lambda), 0, 8, add = TRUE)

geoR documentation built on Aug. 9, 2022, 5:11 p.m.

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