BCNormal function

Box-Cox Normal Distribution

Density, cumulative distribution, quantile functions and random number generation for the distribution that becomes normal after the Box-Cox transformation. Note that this is based on the original Box-Cox paper.

dbcnorm(q, mu = 0, sigma = 1, lambda = 0, log = FALSE)

pbcnorm(q, mu = 0, sigma = 1, lambda = 0)

qbcnorm(p, mu = 0, sigma = 1, lambda = 0)

rbcnorm(n = 1, mu = 0, sigma = 1, lambda = 0)

Arguments

• q: vector of quantiles.
• mu: vector of location parameters (means).
• sigma: vector of scale parameters.
• lambda: the value of the Box-Cox transform parameter.
• log: if TRUE, then probabilities are returned in logarithms.
• p: vector of probabilities.
• n: number of observations. Should be a single number.

Returns

Depending on the function, various things are returned (usually either vector or scalar):

• dbcnorm returns the density function value for the provided parameters.
• pbcnorm returns the value of the cumulative function for the provided parameters.
• qbcnorm returns quantiles of the distribution. Depending on what was provided in p, mu and sigma, this can be either a vector or a matrix, or an array.
• rbcnorm returns a vector of random variables generated from the bcnorm distribution. Depending on what was provided in mu and sigma, this can be either a vector or a matrix or an array.

Details

The distribution has the following density function:

f(y) = y^(lambda-1) 1/sqrt(2 pi) exp(-((y^lambda-1)/lambda -mu)^2 / (2 sigma^2))

Both pbcnorm and qbcnorm are returned for the lower tail of the distribution.

In case of lambda=0, the values of the log normal distribution are returned. In case of lambda=1, the values of the normal distribution are returned with mu=mu+1.

All the functions are defined for non-negative values only.

Examples

x <- dbcnorm(c(-1000:1000)/200, 0, 1, 1)
plot(c(-1000:1000)/200, x, type="l")

x <- pbcnorm(c(-1000:1000)/200, 0, 1, 1)
plot(c(-1000:1000)/200, x, type="l")

qbcnorm(c(0.025,0.975), 0, c(1,2), 1)

x <- rbcnorm(1000, 0, 1, 1)
hist(x)

References

• Box, G. E., & Cox, D. R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2), 211–252. Retrieved from https://www.jstor.org/stable/2984418

See Also

Distributions

Author(s)

Ivan Svetunkov, ivan@svetunkov.com