Box–Cox symmetric distributions and applications to nutritional data

Abstract

We introduce and study the Box–Cox symmetric class of distributions, which is useful for modeling positively skewed, possibly heavy-tailed, data. The new class of distributions includes the Box–Cox t, Box–Cox Cole-Green (or Box–Cox normal), Box–Cox power exponential distributions, and the class of the log-symmetric distributions as special cases. It provides easy parameter interpretation, which makes it convenient for regression modeling purposes. Additionally, it provides enough flexibility to handle outliers. The usefulness of the Box–Cox symmetric models is illustrated in a series of applications to nutritional data.

Appendix

In this “Appendix,” we give the first and second derivatives of the log-likelihood function with respect to the parameters. Let $z=h(y;\mu ,\sigma ,\lambda )$, where $h(y;\mu ,\sigma ,\lambda )$ is given in (1), $\varpi =-2r'(z^2)/r(z^2)$, and $\xi =r((\sigma \lambda )^{-2}) / R((\sigma |\lambda |)^{-1}).$ We have

$$\begin{aligned} \displaystyle \frac{\partial z}{\partial \mu }= & {} -\frac{1}{\mu \sigma } \left( \frac{y}{\mu }\right) ^\lambda \xrightarrow [\lambda \rightarrow 0]{} -\frac{1}{\mu \sigma }, \\ \displaystyle \frac{\partial z}{\partial \lambda }= & {} \frac{1}{\sigma \lambda ^2} \left\{ 1+ \left( \frac{y}{\mu }\right) ^{\lambda } \left[ -1 + \lambda \log \left( \frac{y}{\mu }\right) \right] \right\} \xrightarrow [\lambda \rightarrow 0]{} \frac{1}{2 \sigma }\left[ \log \left( \frac{y}{\mu }\right) \right] ^2,\\ \displaystyle \frac{\partial ^2 z}{\partial \mu ^2}= & {} \frac{(\lambda +1)}{\mu ^2 \sigma } \left( \frac{y}{\mu }\right) ^\lambda \xrightarrow [\lambda \rightarrow 0]{} \frac{1}{\mu ^2 \sigma },\\ \displaystyle \frac{\partial ^2 z}{\partial \lambda ^2}= & {} \frac{1}{\sigma \lambda ^3}\left\{ -2+ \left( \frac{y}{\mu }\right) ^{\lambda } \left[ 2- 2 \lambda \log \left( \frac{y}{\mu }\right) +\lambda ^2 \left( \log \left( \frac{y}{\mu }\right) \right) ^2 \right] \right\} \\&\times \xrightarrow [\lambda \rightarrow 0]{} \frac{1}{3 \sigma } \left[ \log \left( \frac{y}{\mu } \right) \right] ^3,\\ \displaystyle \frac{\partial ^2 z}{\partial \mu \partial \lambda }= & {} -\frac{1}{\mu \sigma } \left( \frac{y}{\mu }\right) ^\lambda \log \left( \frac{y}{\mu }\right) \xrightarrow [\lambda \rightarrow 0]{} -\frac{1}{\mu \sigma }\log \left( \frac{y}{\mu }\right) . \end{aligned}$$

Let $\ell$ denote the log-likelihood for a single observation y. We have

$$\begin{aligned} \ell = (\lambda -1) \log y - \lambda \log \mu - \log \sigma +\log r(z^2)- \log R\left( \frac{1}{\sigma |\lambda |}\right) , \end{aligned}$$

if $\lambda \ne 0$; the last term in $\ell$ is zero if $\lambda =0$. The first derivatives of $\ell$ are given by

$$\begin{aligned}&\frac{\partial \ell }{\partial \mu }= -\frac{\lambda }{\mu }-\varpi z\frac{\partial z}{\partial \mu },\\&\displaystyle \frac{\partial \ell }{\partial \sigma }=\left\{ \begin{array}{l@{\quad }l} \displaystyle {-\frac{1}{\sigma }+ \frac{\varpi z^2}{\sigma }+ \frac{\xi }{\sigma ^2 |\lambda |}}, &{} \quad {\hbox {if} \quad \lambda \ne 0}, \\ \displaystyle {-\frac{1}{\sigma }+ \frac{\varpi z^2}{\sigma }}, &{} \quad { \hbox {if} \quad \lambda = 0,} \end{array} \right. \\&\frac{\partial \ell }{\partial \lambda }= \log \left( \frac{y}{\mu }\right) -\varpi z\frac{\partial z}{\partial \lambda }+ \hbox {sign}(\lambda )\frac{ \xi }{\sigma \lambda ^2}. \end{aligned}$$

The second derivatives of $\ell$ are given by

$$\begin{aligned}&\frac{\partial ^2 \ell }{\partial \mu ^2}= \frac{\lambda }{\mu ^2}-\left( z \frac{\hbox {d}\varpi }{\hbox {d}z} +\varpi \right) \left( \frac{\partial z}{\partial \mu }\right) ^2-\varpi z \frac{\partial ^2 z}{\partial \mu ^2},\\&\displaystyle \frac{\partial ^2 \ell }{\partial \sigma ^2}=\left\{ \begin{array}{l@{\quad }l} \displaystyle {\frac{1}{\sigma ^2}- \frac{z^3}{\sigma ^2} \frac{\hbox {d}\varpi }{\hbox {d}z}- \frac{3 \varpi z^2}{\sigma ^2}+\frac{1}{\sigma ^2 |\lambda |} \frac{\partial \xi }{\partial \sigma }-\frac{2 \xi }{\sigma ^3 |\lambda |}}, &{} {\hbox {if} \quad \lambda \ne 0}, \\ \displaystyle {\frac{1}{\sigma ^2}- \frac{z^3}{\sigma ^2} \frac{\hbox {d}\varpi }{\hbox {d}z}- \frac{3 \varpi z^2}{\sigma ^2}}, &{} { \hbox {if} \quad \lambda = 0,} \end{array} \right. \\&\frac{\partial ^2 \ell }{\partial \lambda ^2}= -\left( z \frac{\hbox {d}\varpi }{\hbox {d}z} +\varpi \right) \left( \frac{\partial z}{\partial \lambda }\right) ^2-\varpi z \frac{\partial ^2 z}{\partial \lambda ^2}+\hbox {sign}(\lambda )\left( \frac{1}{\sigma \lambda ^2} \frac{\partial \xi }{\partial \lambda }-\frac{2 \xi }{\sigma \lambda ^3}\right) ,\\&\frac{\partial ^2 \ell }{\partial \mu \partial \sigma }=\frac{z}{\sigma } \frac{\partial z}{\partial \mu } \left( z \frac{\hbox {d}\varpi }{\hbox {d}z} + 2 \varpi \right) ,\\&\frac{\partial ^2 \ell }{\partial \mu \partial \lambda }= -\frac{1}{\mu }-\left( z \frac{\hbox {d}\varpi }{\hbox {d}z} +\varpi \right) \frac{\partial z}{\partial \mu }\frac{\partial z}{\partial \lambda }-\varpi z \frac{\partial ^2 z}{\partial \mu \partial \lambda },\\&\frac{\partial ^2 \ell }{\partial \sigma \partial \lambda }= \frac{z}{\sigma } \frac{\partial z}{\partial \lambda } \left( z \frac{\hbox {d}\varpi }{\hbox {d}z}+ 2 \varpi \right) + \frac{1}{ \sigma ^2 |\lambda |} \frac{\partial \xi }{\partial \lambda } - \hbox {sign}(\lambda ) \frac{\xi }{\sigma ^2 \lambda ^2}. \end{aligned}$$

The first and second derivatives of $\ell$ are obtained after plugging the derivatives of z given above.

Note that the first derivatives of $\ell$ depend on the weighting function $\varpi$ ($\varpi$ is given in Table 3 for some distributions). Consequently, $\hbox {d}\varpi /\hbox {d}z$ appears in all the second derivatives of $\ell$. Note that $\partial \ell /\partial \sigma$ and $\partial \ell /\partial \lambda$ involve $\xi$, which in turn depends on the particular distribution in the BCS class and the truncation set. The first derivatives of $\xi$ appear in $\partial ^2 \ell /\partial \sigma ^2$, $\partial ^2 \ell /\partial \lambda ^2$ and $\partial ^2 \ell /\partial \sigma \partial \lambda$. The stability of the terms that involve $\xi$ and its first derivatives around $\lambda =0$ may vary according to different distributions. For instance, they may be unstable for the Box–Cox t distribution with small degrees of freedom parameter. Yet, a simulation study of the type I error probability of the likelihood ratio test of $\mathrm{H}_0: \lambda =0$ in the Box–Cox t model for different values of the degrees of freedom parameter performed well; see Sect. 4.