> 0 for all i. If τ(Y, λ, α) follows a truncated normal distribution, then Y is said to follow a Box–Cox distribution. Bickel and Doksum eliminated the need to use a truncated distribution by extending the range of the transformation to all y, as follows: where sgn(.) is the sign function. This change in definition has little practical import as long as is less than , which it usually is. Bickel and Doksum also proved that the parameter estimates are consistent and asymptotically normal under appropriate regularity conditions, though the standard Cramér–Rao lower bound can substantially underestimate the variance when parameter values are small relative to the noise variance. However, this problem of underestimating the variance may not be a substantive problem in many applications. Box–Cox transformation The one-parameter Box–Cox transformations are defined as and the two-parameter Box–Cox transformations as as described in the original article. Moreover, the first transformations hold for , and the second for . The parameter is estimated using the profile likelihood function and using goodness-of-fit tests. Confidence interval Confidence interval for the Box–Cox transformation can be asymptotically constructed using Wilks's theorem on the profile likelihood function to find all the possible values of that fulfill the following restriction: Example The BUPA liver data set contains data on liver enzymes ALT and γGT. Suppose we are interested in using log(γGT) to predict ALT. A plot of the data appears in panel (a) of the figure. There appears to be non-constant variance, and a Box–Cox transformation might help. The log-likelihood of the power parameter appears in panel (b). The horizontal reference line is at a distance of χ12/2 from the maximum and can be used to read off an approximate 95% confidence interval for λ. It appears as though a value close to zero would be good, so we take logs. Possibly, the transformation could be improved by adding a shift parameter to the log transformation. Panel (c) of the figure shows the log-likelihood. In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed. The final panel shows the transformed data with a superimposed regression line. Note that although Box–Cox transformations can make big improvements in model fit, there are some issues that the transformation cannot help with. In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic