= \omega s_1^*(y) I(y\leq \theta) + (1-\omega) s_2^*(y) I(y>\theta),\,\,\, y>0,\end{aligned}$$ where $\omega = \dfrac{s_2(\theta)S_1(\theta)}{s_2(\theta)S_1(\theta) + s_1(\theta)[1-S_2(\theta)]}$, $\theta>0$ is a threshold parameter, $s_1^*(y)=\dfrac{s_1(y)}{S_1(\theta)}$, $s_2^*(y)=\dfrac{s_2(y)}{1-S_2(\theta)}$, $s_1$ and $s_2$ are continuous PDFs with support on ${\mathbb R}_+$, and $S_1$ and $S_2$ are the corresponding CDFs. If we fix $s_1(y)=\dfrac{1}{\sigma_1 y}s\left(\dfrac{\log(y)-\mu}{\sigma_1}\right)$, $s_2(y)=\dfrac{1}{\sigma_1 y}s\left(\dfrac{\log(y)-\mu}{\sigma_2}\right)$, for some symmetric density $s$ with support on ${\mathbb R}$, and $\theta=e^{\mu}$, then it follows that (\[CompPDF\]) coincides with (\[LTPPDF\]), up to a reparameterisation.

From this alternative construction, we conclude that the family of LTP distributions represents a subclass of composite models with the appealing properties and interpretability of parameters discussed above. This also allows us to motivate the use of LTP distributions as survival and size distributions.

Models and Maximum Likelihood Estimation {#MaximumLikelihoodEstimation}
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In this section, we present the parameter estimation procedure for time-to-event and accelerated failure time (AFT) models.

Time-to-event model
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Let ${\bf T}=(T_1,\dots,T_n)$ be an independent sample of survival times distributed as in (\[LTPPDF\]). The likelihood function of the parameters $(\mu,\sigma,\gamma,\delta)$ is defined as: $$\begin{aligned}
L(\mu,\sigma,\gamma,\delta) = \prod_{j=1}^n s_l(T_j;\mu,\sigma,\gamma,\delta).\end{aligned}$$ The MLE is defined as the parameter values that maximise the likelihood function. By noting that $$\begin{aligned}
L(\mu,\sigma,\gamma,\delta) \propto \prod_{j=1}^n s_{tp}(\log(T_j);\mu,\sigma,\gamma,\delta),\end{aligned}$$ it follows that the MLEs of 