Log-symmetric regression models under the presence of non-informative left- or right-censored observations

Abstract

In this paper, an extension to allow the presence of non-informative left- or right-censored observations in log-symmetric regression models is addressed. Under such models, the log-lifetime distribution belongs to the symmetric class and its location and scale parameters are described by semi-parametric functions of explanatory variables, whose nonparametric components are approximated using natural cubic splines or P-splines. An iterative process of parameter estimation by the maximum penalized likelihood method is presented. The large sample properties of the maximum penalized likelihood estimators are studied analytically and by simulation experiments. Diagnostic methods such as deviance-type residuals and local influence measures are derived. The package ssym, which includes an implementation in the computational environment R of the methodology addressed in this paper, is also discussed. The proposed methodology is illustrated by the analysis of a real data set.

Appendix

Theorem 2

Under Conditions (1)–(4) the maximum penalized likelihood estimator of ${\varvec{\theta }}$ is consistent and

$$\begin{aligned} \Bigl [-\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )\Bigr ]^{ -\frac{1}{2}} \Bigl [-\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )+\mathbf{M}\Bigr ] \Bigl (\hat{{\varvec{\theta }}} - {\varvec{\theta }}^{^{[0]}}\Bigr ) \xrightarrow [n \rightarrow \infty ]{\mathcal {D}} \mathcal {N}(\mathbf{0},\mathbf{I}). \end{aligned}$$

Proof

By a Taylor series expansion of the score function of ${\varvec{\theta }}{ around}{\varvec{\theta }}^{^{[0]}}{}$ it is possible to write

$$\begin{aligned} \dfrac{\partial \mathsf{PL}({{\varvec{\theta }}})}{\partial {\varvec{\theta }}}(\hat{{\varvec{\theta }}})&=\mathbf{U}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}{\varvec{\theta }}^{^{[0]}}+\left[ \mathbf{J}({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}\right] \left( \hat{{\varvec{\theta }}}-{\varvec{\theta }}^{^{[0]}}\right) \\&\quad +\;\frac{1}{2}\sum \limits _{l=1}\sum \limits _{l'=1} \left( \hat{\theta }_l-\theta ^{^{[0]}}_{l}\right) \left( \hat{\theta }_{l'}-\theta ^{^{[0]}}_{l'}\right) \dfrac{\partial {\mathbf{J}_{{ ll'}} ({\varvec{\theta }})}}{{\partial {\varvec{\theta }}^{\top }}}({\varvec{\theta }}^{^*}), \end{aligned}$$

where ${\varvec{\theta }}^{^*}{} { isonthelinesegmentjoining}\hat{{\varvec{\theta }}}{} { and}{\varvec{\theta }}^{^{[0]}},\,{ and}\partial {\mathbf{J}_{{ ll'}} ({\varvec{\theta }})}{} { isthe}(l,l'){ thelementof}{\mathbf{J}({\varvec{\theta }})}{}$. From Conditions 1, 2, 3 (expressions (a) and (b)) and 4, it follows that

• (1) $n^{-1}\left[ \mathbf{U}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}{\varvec{\theta }}^{^{[0]}}\right] \xrightarrow [n \rightarrow \infty ]{\mathcal {P}}\mathbf{0}$ and

• (2) $n^{-1}\left[ \mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}\right] \xrightarrow [n \rightarrow \infty ]{\mathcal {P}}-{\varvec{\varOmega }}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr ).$

Therefore, from 1 and 2 and Condition 3 (expression (e)), and using the argument in Billingsley (1961, page 12), it follows that $\hat{{\varvec{\theta }}}{} { isaconsistentestimatorof}{\varvec{\theta }}^{^{[0]}}.{ Moreover},\,{ byanotherTaylorseriesexpansionofthescorefunctionof}\hat{{\varvec{\theta }}}{} { around}{\varvec{\theta }}^{^{[0]}}{}$, it is possible to write

$$\begin{aligned} \dfrac{\partial \mathsf{PL}({\varvec{\theta }})}{\partial {\varvec{\theta }}}(\hat{{\varvec{\theta }}})=\mathbf{0}=\mathbf{U}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}{\varvec{\theta }}^{^{[0]}}+ \left[ \mathbf{J}({\varvec{\theta }}^*\bigr )-\mathbf{M}\right] \left( \hat{{\varvec{\theta }}}-{\varvec{\theta }}^{^{[0]}}\right) , \end{aligned}$$

where ${\varvec{\theta }}^*{ isonthelinesegmentjoining}\hat{{\varvec{\theta }}}{} { and}{\varvec{\theta }}^{^{[0]}}{}$. By rearranging this expression, we arrive at

$$\begin{aligned} \left[ -\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )\right] ^{-\frac{1}{2}} \left[ -\mathbf{J}\bigl ({\varvec{\theta }}^*\bigr )+\mathbf{M}\right] \left( \hat{{\varvec{\theta }}}-{\varvec{\theta }}^{^{[0]}}\right) =\left[ -\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )\right] ^{-\frac{1}{2}} \left[ \mathbf{U}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}{\varvec{\theta }}^{^{[0]}}\right] . \end{aligned}$$

(5)

From Conditions 1, 2, 3 (expressions (c) and (d)) and 4, it follows that

• (3) $\left[ -\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )\right] ^{-\frac{1}{2}} \left[ \mathbf{U}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )-\mathbf{M}{\varvec{\theta }}^{^{[0]}}\right] \xrightarrow [n \rightarrow \infty ]{\mathcal {D}}\mathcal {N}(\mathbf{0},\mathbf{I})$      and

• (4) $n^{-1}\left[ \mathbf{J}\bigl ({\varvec{\theta }}^*\bigr )-\mathbf{M}\right] \xrightarrow [n \rightarrow \infty ]{\mathcal {P}}-{\varvec{\varOmega }}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr ).$

Therefore, from 3 and 4 and (5), and using the Slutsky theorem, it follows that

$$\begin{aligned} \Bigl [-\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )\Bigr ]^{ -\frac{1}{2}} \Bigl [-\mathbf{J}\bigl ({\varvec{\theta }}^{^{[0]}}\bigr )+\mathbf{M}\Bigr ] \Bigl (\hat{{\varvec{\theta }}} - {\varvec{\theta }}^{^{[0]}}\Bigr ) \xrightarrow [n \rightarrow \infty ]{\mathcal {D}} \mathcal {N}(\mathbf{0},\mathbf{I}). \end{aligned}$$