boxcox
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.
- Keywords
- distribution
Usage
rboxcox(n, lambda, lambda2 = NULL, mean = 0, sd = 1)dboxcox(x, lambda, lambda2 = NULL, mean = 0, sd = 1)
Arguments
- lambda
numerical value(s) for the transformation parameter \(\lambda\).
- lambda2
logical or numerical value(s) of the additional transformation (see DETAILS below). Defaults to
NULL.- n
number of observations to be generated.
- x
a vector of quantiles (
dboxcox) or an output ofboxcoxfit(print,plot,lines).- mean
a vector of mean values at the normal scale.
- sd
a vector of standard deviations at the normal scale.
Details
Denote \(Y\) the variable at the original scale and \(Y'\) the transformed variable. The Box-Cox transformation is defined by:
$$Y' = \left\{ \begin{array}{ll} log(Y) \mbox{ , if $\lambda = 0$} \cr \frac{Y^\lambda - 1}{\lambda} \mbox{ , otherwise} \end{array} \right.$$.
An additional shifting parameter \(\lambda_2\) can be included in which case the transformation is given by:
$$Y' = \left\{ \begin{array}{ll} log(Y + \lambda_2) \mbox{ , $\lambda = 0$ } \cr \frac{(Y + \lambda_2)^\lambda - 1}{\lambda} \mbox{ , otherwise} \end{array} \right.$$.
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
\(Y' \geq \frac{1}{\lambda}\) results in
\(Y = 0\) in the original scale.
Increasing the value of the mean and/or reducing the variance might help to avoid truncation.
Value
The functions returns the following results:
a vector of random deviates.
a vector of densities.
References
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.
Examples
# NOT RUN {
## 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)
# }