geoR (version 1.9-4)

boxcox: The Box-Cox Transformed Normal Distribution

Description

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.

Usage

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

Value

The functions returns the following results:

rboxcox

a vector of random deviates.

dboxcox

a vector of densities.

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 of boxcoxfit (print, plot, lines).

mean

a vector of mean values at the normal scale.

sd

a vector of standard deviations at the normal scale.

Author

Paulo Justiniano Ribeiro Jr. ,
Peter J. Diggle .

Details

Denote $Y$ the variable at the original scale and $Y^{\prime }$ the transformed variable. The Box-Cox transformation is defined by:

$$Y^{\prime }=\{\begin{array}{l}log(Y)\text{ , if }\lambda =0\\ \frac{Y^{\lambda }-1}{\lambda }\text{ , otherwise}\end{array}$$.

An additional shifting parameter ${\lambda }_2$ can be included in which case the transformation is given by:

$$Y^{\prime }=\{\begin{array}{l}log(Y+{\lambda }_2)\text{ , }\lambda =0\\ \frac{(Y+{\lambda }_2{)}^{\lambda }-1}{\lambda }\text{ , otherwise}\end{array}$$.

The function rboxcox samples $Y^{\prime }$ 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 $Y^{\prime }\ge \frac{1}{\lambda }$ results in $Y=0$ in the original scale. Increasing the value of the mean and/or reducing the variance might help to avoid truncation.

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 is boxcoxfit. Other packages has BoxCox related functions such as boxcox in the package MASS and the function box.cox in the package car.

Examples

## 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)