Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model

Abstract

Doubly-truncated data arise in many fields, including economics, engineering, medicine, and astronomy. This article develops likelihood-based inference methods for lifetime distributions under the log-location-scale model and the accelerated failure time model based on doubly-truncated data. These parametric models are practically useful, but the methodologies to fit these models to doubly-truncated data are missing. We develop algorithms for obtaining the maximum likelihood estimator under both models, and propose several types of interval estimation methods. Furthermore, we show that the confidence band for the cumulative distribution function has closed-form expressions. We conduct simulations to examine the accuracy of the proposed methods. We illustrate our proposed methods by real data from a field reliability study, called the Equipment-S data.

Appendices

Appendix A: Derivatives of the log-likelihood function

To obtain he MLE numerically, we need the first and second derivatives of the log-likelihood $\ell (\varvec{\theta })$. Define notations $Z_{s}(\varvec{\theta }) = (s - \mu )/\sigma , s \in \{u_i, y_i, v_i\}$ and

$$\begin{aligned} H_{v_i}(\varvec{\theta })&= \frac{\phi \{Z_{v_i}(\varvec{\theta })\}}{\varPhi \{Z_{v_i}(\varvec{\theta })\}-\varPhi \{Z_{u_i}(\varvec{\theta })\}}, \ H_{u_i}(\varvec{\theta }) = \frac{\phi \{Z_{u_i}(\varvec{\theta })\}}{\varPhi \{Z_{v_i}(\varvec{\theta })\}-\varPhi \{Z_{u_i}(\varvec{\theta })\}} \end{aligned}$$

(1)

$$\begin{aligned} K_{v_i}(\varvec{\theta })&= \frac{\phi '\{Z_{v_i}(\varvec{\theta })\}}{\varPhi \{Z_{v_i}(\varvec{\theta })\}-\varPhi \{Z_{u_i}(\varvec{\theta })\}}, \ K_{u_i}(\varvec{\theta }) = \frac{\phi '\{Z_{u_i}(\varvec{\theta })\}}{\varPhi \{Z_{v_i}(\varvec{\theta })\}-\varPhi \{Z_{u_i}(\varvec{\theta })\}} \end{aligned}$$

(2)

Note that $H_{v_i}(\varvec{\theta })$ is the backward hazard rate at time $v_i$ and $H_{u_i}(\varvec{\theta })$ is the forward hazard rate at time $u_i$ under double-truncation (Sankaran and Sunoj 2004).

The first-order derivatives of the log-likelihood are

$$\begin{aligned} \frac{\partial }{\partial \mu } \ell (\varvec{\theta })&= -\frac{1}{\sigma } \sum _{i=1}^n \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{1}{\sigma } \sum _{i=1}^n \{H_{v_i}(\varvec{\theta }) - H_{u_i}(\varvec{\theta })\}, \\ \frac{\partial }{\partial \sigma } \ell (\varvec{\theta })&= -\frac{n}{\sigma } - \frac{1}{\sigma } \sum _{i=1}^n \frac{Z_{y_i}(\varvec{\theta }) \cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}}\\&\ \ \ + \frac{1}{\sigma } \sum _{i=1}^n \{Z_{v_i}(\varvec{\theta }) \cdot H_{v_i}(\varvec{\theta }) - Z_{u_i}(\varvec{\theta }) \cdot H_{u_i}(\varvec{\theta })\}. \end{aligned}$$

The second-order derivatives of the log-likelihood are

$$\begin{aligned} \frac{\partial ^2}{\partial \mu ^2} \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \frac{\phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} - \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ - \frac{1}{\sigma ^2} \sum _{i=1}^n \{K_{v_i}(\varvec{\theta })- K_{u_i}(\varvec{\theta })\} + \frac{1}{\sigma ^2} \sum _{i=1}^n \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}^2 \\ \frac{\partial ^2}{\partial \sigma ^2} \ell (\varvec{\theta })&= \frac{n}{\sigma ^2} + \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{2 Z_{y_i}(\varvec{\theta }) \cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{Z^2_{y_i}(\theta ) \cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \\&\ \ \ - \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{Z_{y_i}(\varvec{\theta }) \cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{2}{\sigma ^2} \sum _{i=1}^n \{ Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })\} \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \{ Z^2_{v_i}(\varvec{\theta }) \cdot K_{v_i}(\varvec{\theta })- Z^2_{u_i}(\varvec{\theta }) \cdot K_{u_i}(\varvec{\theta })\} \\&\ \ \ + \frac{1}{\sigma ^2} \sum _{i=1}^n \{Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })\}^2, \\ \frac{\partial }{\partial \mu \partial \sigma } \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{Z_{y_i}(\varvec{\theta }) \cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \left[ Z_{y_i}(\varvec{\theta }) \left\{ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right\} ^2 \right] \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \{ Z_{v_i}(\varvec{\theta })\cdot K_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot K_{u_i}(\varvec{\theta })\} \\&\ \ \ + \frac{1}{\sigma ^2} \sum _{i=1}^n \{ H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}\{Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })- 1\}. \end{aligned}$$

For the AFT model, we define $Z_s(\varvec{\theta }) = (s - \beta _0 - {\varvec{x}}_i^{\mathrm T} \varvec{\beta })/\sigma , s \in \{u_i, y_i, v_i\}$ and define $H_{v_i}(\varvec{\theta }), \ H_{u_i}(\varvec{\theta }), \ K_{v_i}(\varvec{\theta })$ and $K_{u_i}(\varvec{\theta })$ by Equations (1) and (2).

The first-order derivatives of the log-likelihood are

$$\begin{aligned} \frac{\partial }{\partial \beta _0} \ell (\varvec{\theta })&= -\frac{1}{\sigma } \sum _{i=1}^n \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{1}{\sigma } \sum _{i=1}^n \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}, \\ \frac{\partial }{\partial \varvec{\beta }} \ell (\varvec{\theta })&= -\frac{1}{\sigma } \sum _{i=1}^n \frac{{\varvec{x}}_i \cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{1}{\sigma } \sum _{i=1}^n {\varvec{x}}_i \cdot \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\},\\ \frac{\partial }{\partial \sigma } \ell (\varvec{\theta })&= -\frac{n}{\sigma } - \frac{1}{\sigma } \sum _{i=1}^n \frac{Z_{y_i}(\varvec{\theta })\cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \\&\ \ \ + \frac{1}{\sigma } \sum _{i=1}^n \{Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })\}. \end{aligned}$$

The second-order derivatives of the log-likelihood are

$$\begin{aligned} \frac{\partial ^2}{\partial \beta _0^2} \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \frac{\phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} - \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \{K_{v_i}(\varvec{\theta })- K_{u_i}(\varvec{\theta })\} + \frac{1}{\sigma ^2} \sum _{i=1}^n \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}^2, \\ \frac{\partial ^2}{\partial \varvec{\beta }\partial \varvec{\beta }^{\mathrm {T}}} \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \frac{{\varvec{x}}_i {\varvec{x}}_i^{\mathrm T} \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} - \left[ \frac{{\varvec{x}}_i \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \left[ \frac{{\varvec{x}}_i \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^{\mathrm T} \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i {\varvec{x}}_i^{\mathrm T} \{K_{v_i}(\varvec{\theta })- K_{u_i}(\varvec{\theta })\} + \frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i {\varvec{x}}_i^{\mathrm T} \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}^2, \\ \frac{\partial ^2}{\partial \sigma ^2} \ell (\varvec{\theta })&= \frac{n}{\sigma ^2} + \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{2 Z_{y_i}(\varvec{\theta })\cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{Z^2_{y_i}(\varvec{\theta }) \cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{Z_{y_i}(\varvec{\theta })\cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{2}{\sigma ^2} \sum _{i=1}^n \{ Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })\} \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \{ Z^2_{v_i}(\varvec{\theta }) \cdot K_{v_i}(\varvec{\theta })- Z^2_{u_i}(\varvec{\theta }) \cdot K_{u_i}(\varvec{\theta })\} \\&\ \ \ +\frac{1}{\sigma ^2} \sum _{i=1}^n \{Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })\}^2,\\ \frac{\partial ^2}{\partial \beta _0 \partial \varvec{\beta }} \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \frac{{\varvec{x}}_i \cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} - \frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \cdot \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \cdot \{K_{v_i}(\varvec{\theta })- K_{u_i}(\varvec{\theta })\} + \frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \cdot \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\}^2, \\ \frac{\partial ^2}{\partial \beta _0 \partial \sigma } \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{Z_{y_i}(\varvec{\theta })\cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n Z_{y_i}(\varvec{\theta })\cdot \left[ \frac{\phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n \{Z_{v_i}(\varvec{\theta })\cdot K_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot K_{u_i}(\varvec{\theta })\} \\&\ \ \ +\frac{1}{\sigma ^2} \sum _{i=1}^n \{H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\} \{ Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta })- 1\}, \\ \frac{\partial ^2}{\partial \varvec{\beta }\partial \sigma } \ell (\varvec{\theta })&= \frac{1}{\sigma ^2} \sum _{i=1}^n \left[ \frac{{\varvec{x}}_i \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} + \frac{{\varvec{x}}_i Z_{y_i}(\varvec{\theta })\cdot \phi ''\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \left[ \frac{Z_{y_i}(\varvec{\theta })\cdot \phi '\{Z_{y_i}(\varvec{\theta })\}}{\phi \{Z_{y_i}(\varvec{\theta })\}} \right] ^2 \\&\ \ \ -\frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \{Z_{v_i}(\varvec{\theta })\cdot K_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot K_{u_i}(\varvec{\theta })\} \\&\ \ \ + \frac{1}{\sigma ^2} \sum _{i=1}^n {\varvec{x}}_i \{ H_{v_i}(\varvec{\theta })- H_{u_i}(\varvec{\theta })\} \{Z_{v_i}(\varvec{\theta })\cdot H_{v_i}(\varvec{\theta })- Z_{u_i}(\varvec{\theta })\cdot H_{u_i}(\varvec{\theta }){-} 1\}. \end{aligned}$$

Appendix B: Regularity conditions

We impose the following conditions to derive the asymptotic distribution of the MLEs.

Assumption (A) There exists a positive definite matrix ${\varvec{I}}(\varvec{\theta })$ such that, as $n \rightarrow \infty$,

$$\begin{aligned} \sum _{i=1}^n {\varvec{I}}_i(\varvec{\theta }) / n \rightarrow {\varvec{I}}(\varvec{\theta }), \quad \forall \varvec{\theta }. \end{aligned}$$

Assumption (B) The partial derivatives and integration can be exchangeable such that

$$\begin{aligned} \int \frac{\partial }{\partial \varvec{\theta }} \log f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \cdot f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \ dy = \frac{\partial }{\partial \varvec{\theta }} \int f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \ dy = {\varvec{0}}, \\ {\varvec{I}}_i(\varvec{\theta }) = -\int \frac{\partial ^2}{\partial \varvec{\theta }\partial \varvec{\theta }^{\mathrm T}} \log f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \cdot f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \ dy. \end{aligned}$$

Assumption (C) There is a measurable function $M_{jsl}(\cdot )$ such that

$$\begin{aligned} \left| \frac{\partial ^3}{\partial \varvec{\theta }_j \partial \varvec{\theta }_s \partial \varvec{\theta }_l} \log f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \right| \le M_{jsl}(y), \quad \forall y, \quad \forall \varvec{\theta }, \quad \forall i, \end{aligned}$$

with $m_{i,jsl} \equiv \mathrm {E}_{\varvec{\theta }^0} \{ M_{jsl}(Y_i) \} < \infty$ and $m_{i,jsl}^2 \equiv \mathrm {E}_{\varvec{\theta }^0} \{ M_{jsl}(Y_i)^2 \} < \infty$. For some $m_{jsl}$ and $m_{jsl}^2$, it holds that $\sum _{i=1}^n m_{i,jsl}/n \rightarrow m_{jsl}$ and $\sum _{i=1}^n m_{i,jsl}^2 /n \rightarrow m_{jsl}^2$ as $n \rightarrow \infty$.

Assumption (D) There is a measurable function $W_{js}(\cdot )$ such that

$$\begin{aligned} \left| \frac{\partial ^2}{\partial \varvec{\theta }_j \partial \varvec{\theta }_s} \log f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \right| \le W_{js}(y), \quad \forall y, \quad \forall \varvec{\theta }, \quad \forall i, \end{aligned}$$

with $w_{i,js} \equiv \mathrm {E}_{\varvec{\theta }^0} \{ W_{js}(Y_i) \} < \infty$ and $w_{i,js}^2 \equiv \mathrm {E}_{\varvec{\theta }^0} \{ W_{js}(Y_i)^2 \} < \infty$. For some $w_{js}$ and $w_{js}^2$, it holds that $\sum _{i=1}^n w_{i,js}/n \rightarrow w_{js}$ and $\sum _{i=1}^n w_{i,js}^2 /n \rightarrow w_{js}^2$ as $n \rightarrow \infty$.

Assumption (E) There is a measurable function $A_j(\cdot )$ such that

$$\begin{aligned} \left| \frac{\partial }{\partial \varvec{\theta }_j} \log f_{\varvec{\theta }}(y | Y \in [u_i, v_i]) \right| \le A_{j}(y), \quad \forall y, \quad \forall \varvec{\theta }, \quad \forall i, \end{aligned}$$

with $\sup _y A_j^2(y) < \infty$.

Assumption (A) requires the Fisher information matrix to be stable for large samples. Here, the matrix ${\varvec{I}}(\varvec{\theta })$ is regarded as the asymptotic information matrix. Assumptions (B)–(C) impose the smoothness and boundedness of $f_{\varvec{\theta }}(y | Y \in [u_i, v_i])$, which are similar to those imposed for the i.i.d. models (pp. 462–463 of Lehmann and Casella 1998). Assumptions (D)–(E) require the boundedness of the score functions and Hessian matrix, as employed by Emura et al. (2017) for the i.n.i.d. models. While these assumptions are too strong without truncation (Hoadley 1971), they can be satisfied under double-truncation since the density is truncated from below and above. See Lemma 4 of Emura et al. (2017).

Appendix C: The derivation of the transformed CI for $F_{\varvec{\theta }}(t)$

The derivation is based on inverting the CI for the quantile. Define the pth quantile function $g(\varvec{\theta }) = y_p = \log (t_p) = \mu + \sigma w_p$, where $w_p = \varPhi ^{-1}(p)$ for $0 \le p \le 1$. The SE of $\widehat{y}_p = \widehat{\mu }+ \widehat{\sigma }w_p$ is

$$\begin{aligned} \mathrm {SE}(\widehat{y}_p) = \sqrt{\begin{bmatrix} 1&w_p \end{bmatrix} \begin{bmatrix} \lambda ^*_{11} &{} \lambda ^*_{12} \\ \lambda ^*_{12} &{} \lambda ^*_{22} \end{bmatrix} \begin{bmatrix} 1 \\ w_p \end{bmatrix}} = \sqrt{\lambda ^*_{11} + 2w_p \lambda ^*_{12} + w_p^2 \lambda ^*_{22}}. \end{aligned}$$

The $(1-\alpha ) 100\%$ CI for the pth quantile is $[\widehat{y}_p(\min ), \widehat{y}_p(\max )]$, where

$$\begin{aligned} \widehat{y}_p(\min )&= \widehat{y}_p - Z_{\alpha /2} \cdot \sqrt{\lambda ^*_{11} + 2 w_p \lambda ^*_{12} + w_p^2 \lambda ^*_{22}},\\ \widehat{y}_p(\max )&= \widehat{y}_p + Z_{\alpha /2} \cdot \sqrt{\lambda ^*_{11} + 2 w_p \lambda ^*_{12} + w_p^2 \lambda ^*_{22}}. \end{aligned}$$

Let $w_p = a, \varOmega = Z_{\alpha /2}/\widehat{\sigma }$ and $\widehat{\xi }= \{\log (t) - \widehat{\mu }\}/\widehat{\sigma }$. Inverting the CI for the quantile means that (1) the solution to $\widehat{y}_p(\min ) = \log (t)$ with respect to p gives the upper CI for $F_{\varvec{\theta }}(t)$ and (2) the solution to $\widehat{y}_p(\max ) = \log (t)$ with respect to p gives the lower CI for $F_{\varvec{\theta }}(t)$. Thus we need to solve

$$\begin{aligned} \widehat{\xi }= w_p - \varOmega \sqrt{\lambda ^*_{11} + 2 w_p \lambda ^*_{12} + w_p^2 \lambda ^*_{22}}, \quad \widehat{\xi }= w_p + \varOmega \sqrt{\lambda ^*_{11} + 2 w_p \lambda ^*_{12} + w_p^2 \lambda ^*_{22}}. \end{aligned}$$

Manipulating these equations, we have

$$\begin{aligned} (1 - \varOmega ^2 \lambda ^*_{22}) a^2 - 2 (\widehat{\xi }+ \varOmega ^2 \lambda ^*_{12})a + (\widehat{\xi }^2 - \varOmega ^2 \lambda ^*_{11}) = 0. \end{aligned}$$

The solutions to the above equation are

$$\begin{aligned} a_{\min } = \widehat{\xi }+ G_1(\varLambda ^*) - G_2(\varLambda ^*), \quad a_{\max } = \widehat{\xi }+ G_1(\varLambda ^*) + G_2(\varLambda ^*), \end{aligned}$$

where

$$\begin{aligned} G_1(\varLambda ^*)&= \frac{\varOmega ^2 (\lambda ^*_{12} + \widehat{\xi }\lambda ^*_{22})}{1 - \varOmega ^2 \lambda ^*_{22}}, \\ G_2(\varLambda ^*)&= \frac{\sqrt{\varOmega ^2 (\lambda ^*_{11} + 2 \widehat{\xi }\lambda ^*_{12} + \widehat{\xi }^2 \lambda ^*_{22}) - \varOmega ^4 (\lambda ^*_{11} \lambda ^*_{22} - \lambda ^*_{12})}}{1 - \varOmega ^2 \lambda ^*_{22}}. \end{aligned}$$

Therefore, the transformed CI for $F_{\varvec{\theta }}(t)$ is $[\varPhi (a_{\min }), \varPhi (a_{\max })]$ with the restriction $1 - \varOmega ^2 \lambda ^*_{22} > 0$.

Appendix D: The derivation of the CB for $F_{\varvec{\theta }}(t)$

Here we assume the regularity conditions given in “Appendix C”. For the location-scale model, the observed information matrix can be written as

$$\begin{aligned} {\varvec{i}}(\widehat{\varvec{\theta }}) = - \frac{\partial \ell (\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^{\mathrm T}} \Bigg |_{\varvec{\theta }= \widehat{\varvec{\theta }}} = \begin{bmatrix} -\frac{\partial ^2}{\partial \mu ^2} \ell (\varvec{\theta }) &{} -\frac{\partial ^2}{\partial \mu \partial \sigma } \ell (\varvec{\theta }) \\ -\frac{\partial ^2}{\partial \mu \partial \sigma } \ell (\varvec{\theta }) &{} -\frac{\partial ^2}{\partial \sigma ^2} \ell (\varvec{\theta }) \end{bmatrix} \Bigg |_{\varvec{\theta }= \widehat{\varvec{\theta }}} = \frac{n}{\widehat{\sigma }^2} \begin{bmatrix} i_{11} &{} i_{12} \\ i_{12} &{} i_{22}\end{bmatrix}. \end{aligned}$$

Define the scaled information matrix as ${\varvec{i}}_S(\widehat{\varvec{\theta }}) = \begin{bmatrix} i_{11} &{} i_{12} \\ i_{12} &{} i_{22} \end{bmatrix}$, and the inverse of the scaled information matrix $\varLambda = \begin{bmatrix} i_{11} &{} i_{12} \\ i_{12} &{} i_{22} \end{bmatrix}^{-1} = \begin{bmatrix} \lambda _{11} &{} \lambda _{12} \\ \lambda _{12} &{} \lambda _{22} \end{bmatrix}$. Letting $\varvec{\delta }^{\mathrm T} = (0, 1)$, we find $\min \varvec{\delta }^{\mathrm T} \varvec{\theta }$ for the constrained region $(\widehat{\varvec{\theta }}- \varvec{\theta }) {\varvec{i}}(\widehat{\varvec{\theta }}) (\widehat{\varvec{\theta }}- \varvec{\theta }) \le \gamma$ such that

$$\begin{aligned} \mathrm {argmin}_{\varvec{\theta }} \varvec{\delta }^{\mathrm T} \varvec{\theta }= \mathrm {argmin}_{\varvec{\theta }} \varvec{\delta }^{\mathrm T} (\varvec{\theta }- \widehat{\varvec{\theta }}) = \mathrm {argmin}_{\varvec{\theta }} \left[ \left( \sqrt{{\varvec{i}}^{-1}_S(\widehat{\varvec{\theta }})} \varvec{\delta }\right) ^{\mathrm T} \sqrt{{\varvec{i}}_S(\widehat{\varvec{\theta }})} (\varvec{\theta }- \widehat{\varvec{\theta }}) \right] , \end{aligned}$$

subject to the constraint $(\widehat{\varvec{\theta }}- \varvec{\theta })^{\mathrm T} {\varvec{i}}_S(\widehat{\varvec{\theta }}) (\widehat{\varvec{\theta }}- \varvec{\theta }) \le \widehat{\sigma }^2 (\gamma /n) \equiv \widehat{\sigma }^2 \gamma _0$.

Let ${\varvec{d}}^{\mathrm T} = \left( \sqrt{{\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }})} \varvec{\delta }\right) ^{\mathrm T}$ and ${\varvec{k}}= \sqrt{{\varvec{i}}_S(\widehat{\varvec{\theta }})} (\varvec{\theta }- \widehat{\varvec{\theta }})$, we then have ${\varvec{k}}^{\mathrm T} {\varvec{k}}= \gamma _0 \widehat{\sigma }^2$. From the Cauchy-Schwartz inequality, ${\varvec{d}}^{\mathrm T} {\varvec{k}}\le \sqrt{{\varvec{d}}^{\mathrm T} {\varvec{d}}} \sqrt{{\varvec{k}}^{\mathrm T} {\varvec{k}}} \le \gamma _0 \widehat{\sigma }^2$. The equality holds when ${\varvec{k}}= a {\varvec{d}}$, where a is constant. Since we have $(a {\varvec{d}})^{\mathrm T} (a {\varvec{d}}) = \gamma _0 \widehat{\sigma }^2$, the minimum is attained when $a = -\sqrt{\gamma _0 \widehat{\sigma }^2 / {\varvec{d}}^{\mathrm T} {\varvec{d}}}$ at

$$\begin{aligned} {\varvec{k}}&= -\sqrt{\gamma _0 \widehat{\sigma }^2 / {\varvec{d}}^{\mathrm T} {\varvec{d}}} \times {\varvec{d}}= -\sqrt{\gamma _0 \widehat{\sigma }^2 / \varvec{\delta }^{\mathrm T} {\varvec{i}}^{-1}_S(\widehat{\varvec{\theta }}) \varvec{\delta }} \times \sqrt{{\varvec{i}}^{-1}_S(\widehat{\varvec{\theta }})} \varvec{\delta }. \end{aligned}$$

This minimum is attained when

$$\begin{aligned} \varvec{\theta }= \widehat{\varvec{\theta }}- \sqrt{\gamma _0 \widehat{\sigma }^2 / \varvec{\delta }^{\mathrm T} {\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }}) \varvec{\delta }} \times {\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }}) \varvec{\delta }\end{aligned}$$

and

$$\begin{aligned} \sigma _{(\min )} = \min \varvec{\delta }^{\mathrm T} \varvec{\theta }= \widehat{\sigma }- \widehat{\sigma }^2 \sqrt{\gamma _0 \varvec{\delta }^{\mathrm T} {\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }}) \varvec{\delta }} = \widehat{\sigma }(1 - \sqrt{\gamma _0 \lambda _{22}}). \end{aligned}$$

This gives the restriction $(1 - \sqrt{\gamma _0 \lambda _{22}}) > 0$, that is, $i_{11}(i_{22} - \gamma _0) - i_{12}^2 > 0$. Similarly, under ${\varvec{c}}^{\mathrm T} = (1, w_p)$ for the pth quantile,

$$\begin{aligned} \mathrm {argmin}_{\varvec{\theta }} {\varvec{c}}^{\mathrm T} \varvec{\theta }= \mathrm {argmin}_{\varvec{\theta }} {\varvec{c}}^{\mathrm T} (\varvec{\theta }- \widehat{\varvec{\theta }}) = \mathrm {argmin}_{\varvec{\theta }} \left[ \left( \sqrt{{\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }})} {\varvec{c}}\right) ^{\mathrm T} \sqrt{{\varvec{i}}_S(\widehat{\varvec{\theta }})} (\varvec{\theta }- \widehat{\varvec{\theta }}) \right] . \end{aligned}$$

The maximum and minimum hold when

$$\begin{aligned} \varvec{\theta }_{(\max )}&= \widehat{\varvec{\theta }}+ \sqrt{\gamma _0 \widehat{\sigma }^2 / {\varvec{c}}^{\mathrm T} {\varvec{i}}_S^{-1} (\widehat{\varvec{\theta }}) {\varvec{c}}} \times \sqrt{{\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }})} {\varvec{c}},\\ \varvec{\theta }_{(\min )}&= \widehat{\varvec{\theta }}- \sqrt{\gamma _0 \widehat{\sigma }^2 / {\varvec{c}}^{\mathrm T} {\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }}) {\varvec{c}}} \times \sqrt{{\varvec{i}}_S^{-1}(\widehat{\varvec{\theta }})} {\varvec{c}}. \end{aligned}$$

The CBs for the quantile function are

$$\begin{aligned} \widehat{y}_p(\max ) = \max {\varvec{c}}^{\mathrm T} \varvec{\theta }= {\varvec{c}}^{\mathrm T} \varvec{\theta }_{(\max )} = (\widehat{\mu }+ w_p \widehat{\sigma }) + \widehat{\sigma }\sqrt{\gamma _0(\lambda _{11} + 2 w_p \lambda _{12} + w_p^2 \lambda _{22})}, \\ \widehat{y}_p(\min ) = \min {\varvec{c}}^{\mathrm T} \varvec{\theta }= {\varvec{c}}^{\mathrm T} \varvec{\theta }_{(\min )} = (\widehat{\mu }+ w_p \widehat{\sigma }) - \widehat{\sigma }\sqrt{\gamma _0(\lambda _{11} + 2 w_p \lambda _{12} + w_p^2 \lambda _{22})}. \end{aligned}$$

The CBs for $F_{\varvec{\theta }}(t)$ are derived by inverting the CB for the quantile function, that is,

$$\begin{aligned} \widehat{\xi }= q - \sqrt{\gamma _0(\lambda _{11} + 2q\lambda _{12} + q^2 \lambda _{22})}, \end{aligned}$$

where $\widehat{\xi }= (y - \widehat{\mu })/\widehat{\sigma }= \{\log (t) - \widehat{\mu }\}/\widehat{\sigma }$ and $w_p = q$. This equation yields the solutions

$$\begin{aligned} q_{\min } = \widehat{\xi }+ h_1(\varLambda , \widehat{p}) - h_2(\varLambda , \widehat{p}) \quad \text {and} \quad q_{\max } = \widehat{\xi }+ h_1(\varLambda , \widehat{p}) + h_2(\varLambda , \widehat{p}), \end{aligned}$$

where $\widehat{p} = \varPhi [\{\log (t) - \widehat{\mu }\}/\widehat{\sigma }], w_{\widehat{p}} = \varPhi {-1}(\widehat{p}) = \{\log (t) - \widehat{\mu }\}/\widehat{\sigma }$,

$$\begin{aligned} h_1(\varLambda , \widehat{p})&= \frac{\gamma _0(\lambda _{12} + w_{\widehat{p}} \lambda _{22})}{1 - \gamma _0 \lambda _{22}},\\ h_2(\varLambda , \widehat{p})&= \frac{\sqrt{\gamma _0(\lambda _{11} + 2 w_{\widehat{p}} + w_{\widehat{p}} \lambda _{22}^2) - \gamma _0^2(\lambda _{11} \lambda _{22} - \lambda _{12}^2)}}{1 - \gamma _0 \lambda _{22}} \end{aligned}$$

Therefore, the CB for $F_{\varvec{\theta }}(t)$ is $[\varPhi (q_{\min }), \varPhi (q_{\max })]$.