Exact Likelihood-Based Point and Interval Estimation for Lifetime Characteristics of Laplace Distribution Based on a Time-Constrained Life-Testing Experiment

Abstract

In this paper, we first derive explicit expressions for the MLEs of the location and scale parameters of the Laplace distribution based on a Type-I right-censored sample arising from a time-constrained life-testing experiment by considering different cases. We derive the conditional joint MGF of these MLEs and use them to derive the bias and MSEs of the MLEs for all the cases. We then derive the exact conditional marginal and joint density functions of the MLEs and utilize them to develop exact conditional CIs for the parameters. We also briefly discuss the MLEs of reliability and cumulative hazard functions and the construction of exact CIs for these functions. Next, a Monte Carlo simulation study is carried out to evaluate the performance of the developed inferential results. Finally, some examples are presented to illustrate the point and interval estimation methods developed here under a time-constrained life-testing experiment.

Appendix

1.1 List of notation

$$\begin{aligned} p_0= & {} \frac{1}{2^n}e^{-\frac{n(T-\mu )}{\sigma }},\\ q_0= & {} \left( 1-\frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^n,\\ p_1= & {} \frac{(-1)^ln!e^{-\frac{(T-\mu )(n+l-d)}{\sigma }}}{(1-p_0)2^n(n-d)!j!(d-j-l)!l!},\\ p_2= & {} \frac{(-1)^ln!e^{-\frac{(T-\mu )m}{\sigma }}}{2^nj!(l+1)!(m-j-l-1)!(n-m)!(1-p_0)},\\ p_3= & {} -p_2e^{-\frac{(T-\mu )(l+1)}{\sigma }},\\ p_4= & {} \frac{n!e^{-\frac{(T-\mu )m}{\sigma }}}{2^nm!m!(1-p_0)},\\ p_5= & {} \frac{(-1)^{l_1+d-m-1-l_2}n!e^{-\frac{(T-\mu )(m-1-l_2)}{\sigma }}}{2^nj!(m-1-j-l_1)!(d-m-1-l_2)!(l_2+1)!(n-d)!(l_1+1)!(1-p_0)},\\ p_6= & {} -p_5e^{-\frac{(T-\mu )(l_2+1)}{\sigma }},\\ p_{7,e}= & {} -\frac{p_5}{l_1+l_2+2},\\ p_{7,o}= & {} \frac{(-1)^{l_1+d-m-1-l_2}n!e^{-\frac{(T-\mu )(m-l_2)}{\sigma }}}{ (1-p_0)2^n(l_1+l_2+1)l_1!l_2!j!(n-d)!(m-j-l_1)!(d-m-1-l_2)!},\\ p_{8,e}= & {} -p_{7,e}e^{-\frac{(T-\mu )(l_1+l_2+2)}{\sigma }},\\ p_{8,o}= & {} -p_{7,o}e^{-\frac{(T-\mu )(l_1+l_2+1)}{\sigma }} \end{aligned}$$

$$\begin{aligned} p_9&=\frac{(-1)^{d-m-l-1}n!e^{-\frac{(T-\mu )(m-l-1)}{\sigma }}}{2^nm!(d-m-l-1)!(l+1)!(n-d)!(1-p_0)},\\ p_{10}&=-p_9e^{-\frac{(T-\mu )(l+1)}{\sigma }},\\ p_{11}&=\frac{(-1)^{l_1+l_2}n!e^{-\frac{(T-\mu )(n-d+l_2)}{\sigma }}}{2^nm!(j-m-l_1-1)!l_1!(d-j-l_2)!l_2!(n-d)!(m+l_1+1)(1-p_0)},\\ p_{7\sigma }&= \frac{(-1)^{l_1+d-m-1-l_2}n!e^{-\frac{(T-\mu )(m-1-l_2)}{\sigma }}}{ (1-p_0)2^n(l_1+l_2+1)l_1!l_2!j!(n-d)!(m-j-l_1)!(d-m-1-l_2)!},\\ p_{8\sigma }&=-p_{7\sigma }e^{-\frac{(T-\mu )(l_1+l_2+1)}{\sigma }},\\ p_{2\mu }&=\frac{(-1)^ln!e^{-\frac{(T-\mu )m}{\sigma }}}{(m-l-1)!(l+1)!m!2^{m+l+1}(1-p_0)},\\ p_{3\mu }&=-p_{2\mu }e^{-\frac{(T-\mu )(l+1)}{\sigma }},\\ p_{5\mu }&=\frac{n!(-1)^{l_1+d-m-1-l_2}e^{-\frac{(T-\mu )(m-l_2-1)}{\sigma }}}{2^{m+l_1+1}(m-l_1-1)!(l_1+1)!(d-m-l_2-1)!(l_2+1)!(n-d)!(1-p_0)},\\ p_{6\mu }&=-p_{5\mu }e^{-\frac{(T-\mu )(l_2+1)}{\sigma }},\\ p_{7\mu ,e}&=-\frac{l_2+1}{l_1+l_2+2}p_{5\mu },\\ p_{7\mu ,o}&=\frac{n!(-1)^{l_1+d-m-1-l_2}e^{-\frac{(T-\mu )(m-l_2)}{\sigma }}}{2^{m+l_1+1}(m-l_1)!l_1!(d-m-1-l_2)!l_2!(n-d)!(l_1+l_2+1)(1-p_0)},\\ p_{8\mu ,e}&=-p_{7\mu ,e}e^{-\frac{(T-\mu )(l_1+l_2+2)}{\sigma }},\\ p_{8\mu ,o}&=-p_{7\mu ,o}e^{-\frac{(T-\mu )(l_1+l_2+1)}{\sigma }},\\ p_{11\mu }&=\frac{(-1)^{l_1}n!e^{-\frac{(T-\mu )(n-d)}{\sigma }} \left[ 1-\frac{1}{2}e^{-\frac{T-\mu }{\sigma }}\right] ^{d-m-1-l_1}}{2^{m+l_1+1+n-d}m!(d-m-l_1-1)!l_1!(n-d)!(m+l_1+1)(1-p_0)},\\ q_1&=\frac{n!}{d!(n-d)!(1-q_0)} \left( \frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^{d} \left( 1-\frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^{n-d},\\ q_2&=\frac{n!}{m!m!}\left( \frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^m\left[ 1-\frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right] ^m,\\ q_3&=\frac{(-1)^l\left( 1-\frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^{n-d} \left( \frac{1}{2}e^{-\frac{(\mu -T)}{\sigma }}\right) ^d n!}{(l+m+1)m!l!(d-m-1-l)!(n-d)!(1-q_0)},\\ Z_{\sigma }^{(7)}&\overset{d}{=}\varGamma \left( j+d-m-1,\frac{\sigma }{d}\right) +N\varGamma \left( m-j,\frac{\sigma }{d}\right) +E\left( \frac{(l_2-l_1+1)\sigma }{(l_2+l_1+1)d}\right) \\&+\frac{(T-\mu )(m-1-l_2)}{d},\\ Z_{\sigma }^{(8)}&\overset{d}{=}\varGamma \left( j+d-m-1,\frac{\sigma }{d}\right) +N\varGamma \left( m-j,\frac{\sigma }{d}\right) +E\left( \frac{(l_2-l_1+1)\sigma }{(l_2+l_1+1)d}\right) \\&+\frac{(T-\mu )(m-l_1)}{d}, \end{aligned}$$

$$\begin{aligned} Z_1^{(1)}&\overset{d}{=}\varGamma \left( j,\frac{\sigma }{d}\right) +N\varGamma \left( d-j,\frac{\sigma }{d}\right) +\frac{(T-\mu )(d-l)}{d},\\ Z_2^{(1)}&\overset{d}{=}\log \left( \frac{n}{2d}\right) Z_1^{(1)}+T,\\ Z_{1,e}^{(2)}&\overset{d}{=}\varGamma \left( j,\frac{\sigma }{m}\right) + N\varGamma \left( m-1-j,\frac{\sigma }{m}\right) -\frac{\sigma }{m} E_2 +(T-\mu ),\\ Z_{2,e}^{(2)}&\overset{d}{=}\frac{\sigma }{2(l+1)}E_2+\frac{T+\mu }{2},\\ Z_1^{(3)}&\overset{d}{=}\varGamma \left( j,\frac{\sigma }{m}\right) + N\varGamma \left( m-1-j,\frac{\sigma }{m}\right) -\frac{\sigma }{m} E_3+\frac{(T-\mu )(m-l-1)}{m},\\ Z_2^{(3)}&\overset{d}{=}\frac{\sigma }{2(l+1)}E_3+T,\\ Z_1^{(4)}&\overset{d}{=}\varGamma \left( m-1,\frac{\sigma }{m}\right) +\frac{\sigma }{m}E_4+(T-\mu ),\\ Z_2^{(4)}&\overset{d}{=}-\frac{\sigma }{n}E_4+\frac{T+\mu }{2},\\ Z_{1,e}^{(5)}&\overset{d}{=}\varGamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\varGamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_{5A}-\frac{\sigma }{d}E_{5B}\\ {}&+\frac{(T-\mu )(m-1-l_2)}{d},\\ Z_{2,e}^{(5)}&\overset{d}{=}\frac{\sigma }{2(l_2+1)}E_{5A}+\frac{\sigma }{2(l_1+1)}E_{5B}+\mu ,\\ Z_{1,e}^{(6)}&\overset{d}{=}\varGamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\varGamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_{6A}-\frac{\sigma }{d}E_{6B} \\ {}&+\frac{(T-\mu )m}{d},\\ Z_{2,e}^{(6)}&\overset{d}{=}\frac{\sigma }{2(l_2+1)}E_{6A}+\frac{\sigma }{2(l_1+1)}E_{6B}+\frac{T+\mu }{2},\\ Z_{1,e}^{(7)}&\overset{d}{=}\varGamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\varGamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_1+l_2+2)d}E_{7A}\\ {}&-\frac{\sigma }{d}E_{7B}+\frac{(T-\mu )(m-l_2-1)}{d},\\ Z_{2,e}^{(7)}&\overset{d}{=}\frac{\sigma }{l_1+l_2+2}E_{7A}+\frac{\sigma }{2(l_1+1)}E_{7B}+\mu ,\\ Z_{1,o}^{(7)}&\overset{d}{=}\varGamma \left( j+d-m-1,\frac{\sigma }{d}\right) + N\varGamma \left( m-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1+1)\sigma }{(l_2+l_1+1)d} E_{7} \\ {}&+\frac{(T-\mu )(m-l_2)}{d},\\ Z_{2,o}^{(7)}&\overset{d}{=}\frac{\sigma }{l_2+l_1+1}E_{7}+\mu ,\\ Z_{1,e}^{(8)}&\overset{d}{=}\varGamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\varGamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_1+l_2+2)d}E_{8A}\\ {}&-\frac{\sigma }{d}E_{8B}+\frac{(T-\mu )(m-l_1-1)}{d},\\ Z_{2,e}^{(8)}&\overset{d}{=}\frac{\sigma }{l_1+l_2+2}E_{8A}+\frac{\sigma }{2(l_1+1)}E_{8B}+T,\\ Z_{1,o}^{(8)}&\overset{d}{=}\varGamma \left( j+d-m-1,\frac{\sigma }{d}\right) + N\varGamma \left( m-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1+1)\sigma }{(l_2+l_1+1)d} E_{8}\\ {}&+\frac{(T-\mu )(m-l_1)}{d},\\ Z_{2,o}^{(8)}&\overset{d}{=}\frac{\sigma }{l_2+l_1+1}E_{8}+T,\\ Z_{1}^{(9)}&\overset{d}{=}\varGamma \left( d-2,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_{9A}+\frac{\sigma }{d}E_{9B}+\frac{(T-\mu )(m-l-1)}{d},\\ Z_{2}^{(9)}&\overset{d}{=}-\frac{\sigma }{n}E_{9A}+\frac{\sigma }{2(l+1)}E_{9B}+\mu ,\\ Z_{1}^{(10)}&\overset{d}{=}\varGamma \left( d-2,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_{10A}+\frac{\sigma }{d}E_{10B}+\frac{(T-\mu )m}{d},\\ Z_{2}^{(10)}&\overset{d}{=}-\frac{\sigma }{n}E_{10A}+\frac{\sigma }{2(l+1)}E_{10B}+\frac{T+\mu }{2},\\ \end{aligned}$$

$$\begin{aligned} Z_{1,e}^{(11)}&\overset{d}{=}\varGamma \left( d-j+m-1,\frac{\sigma }{d}\right) +N\varGamma \left( j-m-1,\frac{\sigma }{d}\right) \\&+\frac{\sigma }{d}E_{11A}+\frac{(m-l_1-1)\sigma }{(m+l_1+1)d}E_{11B} +\frac{(T-\mu )(n-d+l_2)}{d},\\ Z_{2,e}^{(11)}&\overset{d}{=}-\frac{\sigma }{n}E_{11A}-\frac{\sigma }{m+l_1+1}E_{11B}+\mu ,\\ Z_{1,o}^{(11)}&\overset{d}{=}\varGamma \left( d-j+m,\frac{\sigma }{d}\right) \\&+ N\varGamma \left( j-m-1,\frac{\sigma }{d}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)d} E_{11} +\frac{(T-\mu )(n-d+l_2)}{d},\\ Z_{2,o}^{(11)}&\overset{d}{=}-\frac{\sigma }{m+l_1+1}E_{11}+\mu ,\\ Z_{1}^{(12)}&\overset{d}{=}\varGamma \left( d,\frac{\sigma }{d}\right) ,\\ Z_{2}^{(12)}&\overset{d}{=}\log \left( \frac{n}{2d}\right) Z_1^{(12)}+T,\\ Z_{1}^{(13)}&\overset{d}{=}\varGamma \left( m-1,\frac{\sigma }{m}\right) +\frac{\sigma }{m}E_{13},\\ Z_{2}^{(13)}&\overset{d}{=}-\frac{\sigma }{n}E_{13}+T,\\ Z_{1,e}^{(14)}&\overset{d}{=}\varGamma \left( m-1,\frac{\sigma }{d}\right) +N\varGamma \left( d-m-1,\frac{\sigma }{d}\right) + \frac{\sigma }{d}E_{14A}+\frac{(m-l-1)\sigma }{(m+l+1)d}E_{14B},\\ Z_{2,e}^{(14)}&\overset{d}{=}-\frac{\sigma }{n}E_{14A}-\frac{\sigma }{m+l+1}E_{14B}+T,\\ Z_{1,o}^{(14)}&\overset{d}{=}\varGamma \left( m,\frac{\sigma }{d}\right) + N\varGamma \left( d-m-1,\frac{\sigma }{d}\right) +\frac{(m-l)\sigma }{(m+l+1)d} E_{14},\\ Z_{2,o}^{(14)}&\overset{d}{=}-\frac{\sigma }{m+l+1}E_{14}+T,\\ E.&\overset{d}{=}E(1).\\ ZQ_1&\overset{d}{=}\frac{dT+(T-\mu )(d-l)\left( \log \left( \frac{n}{2d}\right) +q\right) }{d} +\varGamma ^{*}\left( j,\frac{q+\log \left( \frac{n}{2d}\right) }{d}\sigma \right) \\ {}&+\varGamma ^{*}\left( d-j,-\frac{q+\log \left( \frac{n}{2d}\right) }{d}\sigma \right) ,\\ ZQ_2&\overset{d}{=}\frac{(2q+1)T+(1-2q)\mu }{2}+\varGamma ^*\left( j,\frac{q\sigma }{m}\right) +\varGamma ^*\left( m-1-j,-\frac{q\sigma }{m}\right) \\ {}&+E^*\left( \frac{m-2q(l+1)}{n(l+1)}\sigma \right) ,\\ ZQ_3&\overset{d}{=}\frac{(n-2l-2)qT+nT+2(l+1-m)q\mu }{n}+\varGamma ^*\left( j,\frac{q\sigma }{m}\right) \\ {}&+\varGamma ^*\left( m-1-j,-\frac{q\sigma }{m}\right) +E^*\left( \frac{m-2q(l+1)}{n(l+1)}\sigma \right) ,\\ ZQ_4&\overset{d}{=}\frac{(2q+1)T+(1-2q)\mu }{2}+\varGamma ^*\left( m-1,\frac{q\sigma }{m}\right) +E^*\left( \frac{2q-1}{n}\sigma \right) ,\\ ZQ_5&\overset{d}{=}\frac{(T-\mu )(m-1-l_2)q+d\mu }{d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-1-j,-\frac{q\sigma }{d}\right) \\&+E^*\left( \frac{2(l_2+1)q+d}{2d(l_2+1)}\sigma \right) +E^*\left( \frac{d-2(l_1+1)q}{2d(l_1+1)}\sigma \right) ,\\ ZQ_6&\overset{d}{=}\frac{(T-\mu )nq+(T+\mu )d}{2d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-1-j,-\frac{q\sigma }{d}\right) \\&+E^*\left( \frac{2(l_2+1)q+d}{2d(l_2+1)}\sigma \right) +E^*\left( \frac{d-2(l_1+1)q}{2d(l_1+1)}\sigma \right) ,\\ \end{aligned}$$

$$\begin{aligned} ZQ_{7,e}&\overset{d}{=}\frac{(T-\mu )(m-1-l_2)q+d\mu }{d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-1-j,-\frac{q\sigma }{d}\right) \\&+E^*\left( \frac{(l_2-l_1)q+d}{d(l_1+l_2+2)}\sigma \right) +E^*\left( \frac{d-2(l_1+1)q}{2d(l_1+1)}\sigma \right) ,\\ ZQ_{7,o}&\overset{d}{=}\frac{(T-\mu )(m-l_2)q+d\mu }{d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-j,-\frac{q\sigma }{d}\right) +E^*\left( \frac{(l_2-l_1)q+d}{d(l_1+l_2+1)}\sigma \right) ,\\ ZQ_{8,e}&\overset{d}{=}\frac{(T-\mu )(m-l_1-1)q+dT}{d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-1-j,-\frac{q\sigma }{d}\right) \\&+E^*\left( \frac{(l_2-l_1)q+d}{d(l_1+l_2+2)}\sigma \right) +E^*\left( \frac{d-2(l_1+1)q}{2d(l_1+1)}\sigma \right) ,\\ ZQ_{8,o}&\overset{d}{=}\frac{(T-\mu )(m-l_1)q+dT}{d} +\varGamma ^*\left( d-m-1+j,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( m-j,-\frac{q\sigma }{d}\right) +E^*\left( \frac{(l_2-l_1)q+d}{d(l_1+l_2+1)}\sigma \right) ,\\ ZQ_9&\overset{d}{=} \frac{(T-\mu )(m-1-l)q+d\mu }{d} +\varGamma ^*\left( d-2,\frac{q\sigma }{d}\right) +E^*\left( \frac{nq-d}{nd}\sigma \right) \\ {}&+E^*\left( \frac{2(l+1)q+d}{2d(l+1)}\sigma \right) ,\\ ZQ_{10}&\overset{d}{=} \frac{(T-\mu )nq+d(T+\mu )}{2d} +\varGamma ^*\left( d-2,\frac{q\sigma }{d}\right) +E^*\left( \frac{nq-d}{nd}\sigma \right) \\ {}&+E^*\left( \frac{2(l+1)q+d}{2d(l+1)}\sigma \right) ,\\ ZQ_{11,e}&\overset{d}{=}\frac{(T-\mu )(n-d+l_2)q+\mu d}{d} +\varGamma ^*\left( d-j+m-1,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( j-m-1,-\frac{q\sigma }{d}\right) \\&+E^*\left( \frac{nq-d}{nd}\sigma \right) +E^*\left( \frac{(m-l_1-1)q-d}{d(m+l_1+1)}\sigma \right) ,\\ ZQ_{11,o}&\overset{d}{=}\frac{(T-\mu )(n-d+l_2)q+\mu d}{d} +\varGamma ^*\left( d-j+m,\frac{q\sigma }{d}\right) \\ {}&+\varGamma ^*\left( j-m-1,-\frac{q\sigma }{d}\right) +E^*\left( \frac{(m-l_1)q-d}{d(m+l_1+1)}\sigma \right) ,\\ ZQ_{12}&\overset{d}{=} \varGamma ^*\left( d,\frac{q+\log \left( \frac{n}{2d}\right) }{d}\sigma \right) ,\\ ZQ_{13}&\overset{d}{=} \varGamma ^*\left( m-1,\frac{q}{m}\sigma \right) \\ {}&+E^*\left( \frac{2q-1}{n}\sigma \right) ,\\ ZQ_{14,e}&\overset{d}{=} \varGamma ^*\left( m-1,\frac{q}{d}\sigma \right) +\varGamma ^*\left( d-m-1,-\frac{q}{d}\sigma \right) +E^*\left( \frac{nq-d}{nd}\sigma \right) \\ {}&+E^*\left( \frac{(m-l-1)q-d}{d(m+l+1)}\sigma \right) ,\\ ZQ_{14,o}&\overset{d}{=} \varGamma ^*\left( m,\frac{q}{d}\sigma \right) +\varGamma ^*\left( d-m-1,-\frac{q}{d}\sigma \right) +E^*\left( \frac{(m-l)q-d}{d(m+l+1)}\sigma \right) . \end{aligned}$$

1.2 Proof of Result 16.1

Here, we will present the detailed proof for the derivation of the MLEs for the case when $d<\frac{n}{2}$ and abstain from presenting the proofs for all other cases for the sake of brevity since their derivations are quite similar. Even though it will not be known whether T is greater than $\mu$ or not, it is clear that the density in (16.3) will take on two different forms and so we will first find the MLE of $\mu$ based on the two cases $T\ge \mu$ and $T\le \mu$ separately. Then, finally the MLE of $\mu$ is determined by comparing the likelihood values under these two cases.

Case I: $T\ge \mu$

In this case, we readily obtain the MLE of $\mu$ as $\hat{\mu }_1=T$, and the corresponding likelihood to be

$$\begin{aligned} L_1(T, \sigma )= & {} \frac{C_d}{2^n\sigma ^d}e^{-\frac{dT-\sum _{i=1}^dx_{i:n}}{\sigma }}. \end{aligned}$$

(16.30)

Case II: $T\le \mu$

In this case, from the likelihood function $L_2$, we obtain the MLE of $\mu$ as

$$\begin{aligned} \hat{\mu }_2=T+\sigma \log \left( \frac{n}{2d}\right) . \end{aligned}$$

(16.31)

Since for any $\sigma$, $L_1(T, \sigma )=L_2(T, \sigma )<L_2(\hat{\mu }_2, \sigma )$, we now obtain

$$\begin{aligned} \hat{\mu }=T+\hat{\sigma }\log \left( \frac{n}{2d}\right) , \end{aligned}$$

(16.32)

as given in (16.4). Then, the MLE of $\sigma$ can be obtained from the profile likelihood function to be

$$\begin{aligned} \hat{\sigma }=\frac{1}{d}\sum _{i=1}^d\left( T-X_{i:n}\right) , \end{aligned}$$

(16.33)

as given in (16.5).

Proof of Result 16.2.

We shall consider the two cases $\mu <T$ and $\mu \ge T$ separately.

Case I: $\mu < T.$ In this case, we have

$$\begin{aligned} E[e^{t\hat{\sigma }}|D>0]= & {} \sum _{d=1}^m\sum _{j=0}^d E\left[ e^{t\hat{\sigma }}|D=d, J=j\right] P(D=d, J=j|D>0) \nonumber \\&+\sum _{d=m+1}^n\sum _{j=0}^d E\left[ e^{t\hat{\sigma }}|D=d, J=j\right] P(D=d, J=j|D>0). \end{aligned}$$

(16.34)

The joint distribution of J and D, conditional on $D>0$, is

$$\begin{aligned} \nonumber P(D=d, J=j|D>0) \nonumber= & {} P(J=j|D=d)\frac{P(D=d,D>0)}{P(D>0)}\\= & {} (1-p_0)^{-1} 2^{-n} {d\atopwithdelims ()j} {n\atopwithdelims ()d} \left( 1-e^{-\frac{T-\mu }{\sigma }}\right) ^{d-j} e^{-\frac{(T-\mu )(n-d)}{\sigma }}. \end{aligned}$$

(16.35)

Now, the joint distribution of $X_{1:d},\cdots ,X_{d:d}$, conditional on $D=d$ and $J=j$, is given by

$$\begin{aligned} \nonumber&f(X_{1:d},\cdots ,X_{d:d}|D=d,J=j)=j!(d-j)!\prod _{i=1}^j\frac{1}{\sigma }e^{-\frac{\mu -x_{i:d}}{\sigma }} \prod _{i=j+1}^d\frac{\frac{1}{\sigma }e^{-\frac{x_{i:d}-\mu }{\sigma }}}{1-e^{-\frac{T-\mu }{\sigma }}},\\&\qquad \qquad \qquad \qquad \qquad \qquad x_{1:d}<\cdots<x_{j:d}<\mu<x_{j+1:d}<\cdots<x_{d:d}<T. \end{aligned}$$

(16.36)

For the case when $D\le \frac{n}{2}$, and given $J=j$, we find

$$\begin{aligned} \hat{\sigma }|_{D=d,J=j}\overset{d}{=}\sum _{i=1}^jE\left( \frac{\sigma }{d}\right) +\sum _{i=j+1}^d E_R^*\left( \frac{\sigma }{d},T-\mu \right) +T-\mu , \end{aligned}$$

(16.37)

where $E^*(\theta )$ denotes the negative exponential distribution with scale parameter $\theta$, that is, if X follows the exponential distribution, then $-X$ is negative exponentially distributed, i.e., $X\sim E(\theta )$, $-X\sim E^*(\theta )$, and $E_R^*\left( \sigma ,T\right)$ denotes a negative exponential distribution right truncated at time T. Then, we have the conditional MGF of $\hat{\sigma }$ as

$$\begin{aligned} \nonumber E[e^{t\hat{\sigma }}|D=d,J=j]= & {} e^{t(T-\mu )} \left( 1-\frac{t\sigma }{d}\right) ^{-j}\left( 1+\frac{t\sigma }{d}\right) ^{-(d-j)} \left\{ \frac{1-e^{-(T-\mu )(\frac{1}{\sigma }+\frac{t}{d})}}{1-e^{-\frac{T-\mu }{\sigma }}}\right\} ^{d-j}\\= & {} \nonumber \sum _{l=0}^{d-j} {d-j \atopwithdelims ()l} (-1)^l \left( 1-e^{-\frac{T-\mu }{\sigma }}\right) ^{-(d-j)} e^{-\frac{(T-\mu )l}{\sigma }} e^{(T-\mu )\frac{(d-l)t}{d}}\\&\times \left( 1-\frac{t\sigma }{d}\right) ^{-j}\left( 1+\frac{t\sigma }{d}\right) ^{-(d-j)}. \end{aligned}$$

(16.38)

For the case when $D>m$, we need to consider $J\le m$ and $J>m$ separately. When $J\le m$, similarly, we will have the first j failures as i.i.d. exponential random variables, denoted by $X_1,\cdots ,X_j.$ For the remaining $d-j$ failures, we may consider them as $m-j$ i.i.d. failures before $X_{m+1:d}$ and $d-m-1$ i.i.d. failures after $X_{m+1:d}$ with $\mu<X_{m+1:d}<T$, denoted by $X_{j+1},\cdots ,X_{m}$ and $X_{m+2},\cdots ,X_{d}$, respectively.

The joint pdf $\mathbf{X}=(X_1,\cdots ,X_m, X_{m+1:d},X_{m+2},\cdots ,X_d)$ is given by

$$\begin{aligned} \nonumber f(\mathbf{X}|D=d,J=j)= & {} \frac{(d-j)!}{(m-j)!(d-m-1)!}\left( \prod _{i=1}^j\frac{1}{\sigma }e^{-\frac{\mu -x_i}{\sigma }}\right) \left( \prod _{i=j}^m\frac{\frac{1}{\sigma }e^{-\frac{x_i-\mu }{\sigma }}}{1-e^{-\frac{T-\mu }{\sigma }}}\right) \\&\times \left( \frac{\frac{1}{\sigma }e^{-\frac{x_{m+1:d}-\mu }{\sigma }}}{1-e^{-\frac{T-\mu }{\sigma }}}\right) \left( \prod _{i=m+2}^d\frac{\frac{1}{\sigma }e^{-\frac{x_i-\mu }{\sigma }}}{1-e^{-\frac{T-\mu }{\sigma }}}\right) , \nonumber \\&\nonumber x_{1},\cdots ,x_{j}<\mu<x_{j+1},\cdots ,x_m<x_{m+1:d}<x_{m+2},\cdots ,x_d<T.\\ \end{aligned}$$

(16.39)

Then, the conditional MGF can be derived as follows:

$$\begin{aligned} \nonumber&E[e^{t\hat{\sigma }}|J=j,D=d]\\ \nonumber= & {} \frac{(d-j)!}{(m-j)!(d-m-1)!} \left( {1-e^{-\frac{T-\mu }{\sigma }}}\right) ^{-(d-j)} e^{\frac{(n-d)(T-\mu )t}{d}} \left[ \int _{-\infty }^\mu \frac{1}{\sigma } e^{-(\mu -x)(\frac{1}{\sigma }-\frac{t}{d})}dx\right] ^j\\&\nonumber \times \int _{\mu }^T \left[ \int _{\mu }^{x_{m+1:d}}\frac{1}{\sigma } e^{-(x-\mu )(\frac{1}{\sigma }+\frac{t}{d})}dx\right] ^{m-j} \left[ \int _{x_{m+1:d}}^T\frac{1}{\sigma } e^{-(x-\mu )(\frac{1}{\sigma }-\frac{t}{d})}dx\right] ^{d-m-1}\\ \nonumber&\times \frac{1}{\sigma } e^{-(x_{m+1:d}-\mu )(\frac{1}{\sigma }-\frac{t}{d})}dx_{m+1:d}\\ \nonumber= & {} \sum _{l_1=0}^{m-j} \sum _{l_2=0}^{d-m-1} (-1)^{l_1+d-m-1-l_2} {m-j \atopwithdelims ()l_1} {d-m-1\atopwithdelims ()l_2} \frac{(d-j)!}{(m-j)!(d-m-1)!}\\&\nonumber \times (l_1+l_2+1)^{-1} \left( {1-e^{-\frac{T-\mu }{\sigma }}}\right) ^{-(d-j)} e^{(T-\mu )\left( \frac{(m-1-l_2)t}{d}-\frac{d-m-1-l_2}{\sigma }\right) } \left( 1+\frac{t\sigma }{d}\right) ^{-(m-j)} \\ \nonumber&\times \left( 1-\frac{t\sigma }{d}\right) ^{-(j+d-m-1)} \left\{ 1-\frac{t(l_2+1-l_1)\sigma }{d(l_1+l_2+1)}\right\} ^{-1} \left\{ 1- e^{-(T-\mu )\left[ \frac{l_1+l_2+1}{\sigma }-\frac{t(l_2+1-l_1)}{d}\right] }\right\} . \end{aligned}$$

We can similarly derive the conditional MGF for the case when $J>m.$

Case II: $\mu >T.$ In this case, we have

$$\begin{aligned} \nonumber P(D=d|D>0)= & {} (1-q_0)^{-1} {n \atopwithdelims ()d} [F(T)]^{d}[1-F(T)]^{n-d}\\= & {} (1-q_0)^{-1} {n \atopwithdelims ()d} \left( \frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^{d} \left( 1-\frac{1}{2}e^{-\frac{\mu -T}{\sigma }}\right) ^{n-d}. \end{aligned}$$

(16.40)

Now, by adopting a similar procedure, we can obtain the corresponding conditional MGF. Finally, upon combining these expressions, we obtain the conditional MGF presented in Result 16.2.

1.3 Marginal and Joint CDF of $W_1$ and $W_2$

To derive the marginal and joint CDF of $W_1$ and $W_2$ as described in Lemma 16.1, we first need the following lemma.

Lemma 16.2

Let $Y_1\sim \varGamma (\alpha _1,\beta _1)$ and $Y_2\sim \varGamma (\alpha _2,\beta _2)$ be independent variables with integer shape parameters $\alpha _1$ and $\alpha _2$ and $\beta _1,\beta _2>0.$ Let $Y=Y_1-Y_2.$ Then, the CDF of Y, denoted by $\varGamma (y,\alpha _1,\alpha _2,\beta _1,\beta _2)$, is given by

$$\begin{aligned} \varGamma (y,\alpha _1,\alpha _2,\beta _1,\beta _2)= & {} P(Y_1\le y+y_2)=1-P(Y_1>y+y_2)\\= & {} 1-\sum \limits _{i=0}^{\alpha _1-1} \int _0^\infty \frac{1}{i!}\left( \frac{y+y_2}{\beta _1}\right) ^i e^{-\frac{y+y_2}{\beta _1}}\frac{y_2^{\alpha _2-1}e^{-\frac{y_2}{\beta _2}}}{\varGamma (\alpha _2)\beta _2^{\alpha _2}}dy_2\\= & {} 1-\frac{\beta _1^{\alpha _2}}{\varGamma (\alpha _2)\left( \beta _1+\beta _2\right) ^{\alpha _2}} \sum \limits _{i=0}^{\alpha _1-1}\sum \limits _{j=0}^{i} \frac{\varGamma (\alpha _2+j)\beta _1^{j-i}\beta _2^jy^{i-j}e^{-\frac{y}{\beta _1}}}{(i-j)!j!\left( \beta _1+\beta _2\right) ^j}, \qquad y\ge 0; \end{aligned}$$

$$\begin{aligned} \varGamma (y,\alpha _1,\alpha _2,\beta _1,\beta _2)= & {} P(Y_2\ge y_1-y)\\= & {} \frac{\beta _2^{\alpha _1}}{\varGamma (\alpha _1)\left( \beta _1+\beta _2\right) ^{\alpha _1}} \sum \limits _{i=0}^{\alpha _2-1}\sum \limits _{j=0}^{i} \\&\times \frac{\varGamma (\alpha _1+j)\beta _2^{j-i}\beta _1^j(-y)^{i-j}e^{\frac{y}{\beta _2}}}{(i-j)!j!\left( \beta _1+\beta _2\right) ^j}, \quad y<0. \end{aligned}$$

Now, the corresponding PDF is given by

$$\begin{aligned} f(y)= {\left\{ \begin{array}{ll} \frac{\beta _1^{\alpha _2-\alpha _1}}{\varGamma (\alpha _2)\left( \beta _1+\beta _2\right) ^{\alpha _2}} \sum \limits _{j=0}^{\alpha _1-1} \frac{\varGamma (\alpha _2+j)\beta _1^j\beta _2^jy^{\alpha _1-j-1}e^{-\frac{y}{\beta _1}}}{(\alpha _1-j-1)!j!\left( \beta _1+\beta _2\right) ^j}, &{} y\ge 0,\\ \frac{\beta _2^{\alpha _1-\alpha _2}}{\varGamma (\alpha _1)\left( \beta _1+\beta _2\right) ^{\alpha _1}} \sum \limits _{j=0}^{\alpha _2-1} \frac{\varGamma (\alpha _1+j)\beta _1^j\beta _2^j(-y)^{\alpha _2-j-1}e^{\frac{y}{\beta _2}}}{(\alpha _2-j-1)!j!\left( \beta _1+\beta _2\right) ^j}, &{} y<0. \end{array}\right. } \end{aligned}$$

The marginal distribution of $W_2$ is either exponential or is a linear combination of two exponential variables, and so its CDF is easy to obtain and is therefore omitted for brevity. By using Lemma 16.2, the marginal CDF of $W_1$ can be readily obtained as presented in the following lemma.

Lemma 16.3

Let $Y_1\sim \varGamma (\alpha _1,\beta _1)$, $Y_2\sim N\varGamma (\alpha _2,\beta _2)$, with $\alpha _1$ and $\alpha _2$ being positive integer shape parameters and $\beta _1$ and $\beta _2>0.$ Further, let $Z_1\sim E(1)$ and $Z_2\sim E(1)$, with $a_1\ne a_2\ne \beta _1~(\text {and}~\beta _2)>0.$ If $Y_1$, $Y_2$, $Z_1$, and $Z_2$ are all independent, then the CDF of $W_1=Y_1+Y_2+a_1Z_1-a_2Z_2$ is as follows:

$$\begin{aligned} \nonumber P(W_1\le w_1)= & {} \frac{a_2 e^{\frac{w_1}{a_2}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _1-1}C_jF\left( w_1,\infty ,\alpha _1-j,\frac{1}{\beta _1}+\frac{1}{a_2}\right) +\varGamma (w_1,\alpha _1,\alpha _2,\beta _1,\beta _2)\\ \nonumber&-\frac{a_1 e^{-\frac{w_1}{a_1}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,0,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_1}\right) \\&-\frac{a_1 e^{-\frac{w_1}{a_1}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _1-1}C_jF\left( 0,w_1,\alpha _1-j,\frac{1}{\beta _1}-\frac{1}{a_1}\right) , \quad w_1\ge 0; \end{aligned}$$

(16.41)

$$\begin{aligned} \nonumber P(W_1\le w_1)= & {} \frac{a_2 e^{\frac{w_1}{a_2}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( w_1,0,\alpha _2-j,\frac{1}{a_2}-\frac{1}{\beta _2}\right) \\&\nonumber + \frac{a_2 e^{\frac{w_1}{a_2}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _1-1}C_jF\left( 0,\infty ,\alpha _1-j,\frac{1}{\beta _1}+\frac{1}{a_2}\right) +\varGamma (w_1,\alpha _1,\alpha _2,\beta _1,\beta _2)\\&-\frac{a_1 e^{-\frac{w}{a_1}}}{a_1+a_2} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,w_1,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_1}\right) , \quad w_1<0. \end{aligned}$$

(16.42)

The CDF of $W_1=Y_1+Y_2+a_1Z_1+a_2Z_2$ is as follows:

$$\begin{aligned} P(W_1\le w_1) \nonumber= & {} \varGamma (w_1,\alpha _1,\alpha _2,\beta _1,\beta _2)-\frac{a_2}{a_2-a_1}e^{-\frac{w_1}{a_2}} \sum \limits _{j=0}^{\alpha _1-1}C_jF\left( 0,w_1,\alpha _1-j,\frac{1}{\beta _1}-\frac{1}{a_2}\right) \\ \nonumber&-\frac{a_2 e^{-\frac{w_1}{a_2}}}{a_2-a_1} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,0,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_2}\right) \\ \nonumber&+\frac{a_1}{a_2-a_1}e^{-\frac{w}{\theta _1}} \sum \limits _{j=0}^{\alpha _1-1}C_jF\left( 0,w_1,\alpha _1-j,\frac{1}{\beta _1}-\frac{1}{a_1}\right) \\&+\frac{a_1}{a_2-a_1}e^{-\frac{w_1}{a_1}} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,0,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_1}\right) , \quad w_1\ge 0; \end{aligned}$$

(16.43)

$$\begin{aligned} P(W_1\le w_1) \nonumber= & {} \varGamma (w_1,\alpha _1,\alpha _2,\beta _1,\beta _2) -\frac{a_2 e^{-\frac{w_1}{a_2}}}{a_2-a_1} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,w_1,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_2}\right) \\&+\frac{a_1 e^{-\frac{w_1}{a_1}}}{a_2-a_1} \sum \limits _{j=0}^{\alpha _2-1}C_j^*F\left( -\infty ,w_1,\alpha _2-j,-\frac{1}{\beta _2}-\frac{1}{a_1}\right) , \quad w_1<0. \end{aligned}$$

(16.44)

By using Lemmas 16.2 and 16.3, the joint CDF of $W_1$ and $W_2$ can be derived as presented in Tables 16.10, 16.11, 16.12, 16.13, 16.14, 16.15, 16.16, 16.17, 16.18, 16.19, 16.20, and 16.21. Note in these tables that all the coefficients $a_1, a_2, b_1$, and $b_2$ are positive, all the scale parameters $\beta _1, \beta _2$ are positive, and the shape parameters $\alpha _1$ and $\alpha _2$ are positive integers. Moreover, in these tables, we have used the notation

$$\begin{aligned}&r_1=w_1-\frac{a_1w_2}{b_1},\\&r_2=w_1,\\&r_3=w_1+\frac{a_2w_2}{b_2},\\&r_4=w_1+\frac{a_1w_2}{b_1},\\&r_i^*=\max (r_i,0), \qquad i=1,\cdots ,4,\\&S\varGamma \left( l,r,\alpha _1,\alpha _2,\beta _1,\beta _2\right) =\varGamma \left( r,\alpha _1,\alpha _2,\beta _1,\beta _2\right) -\varGamma \left( l,\alpha _1,\alpha _2,\beta _1,\beta _2\right) ,\\&C_j= \frac{\varGamma (\alpha _2+j)\beta _1^{\alpha _2-\alpha _1+j}\beta _2^j}{\varGamma (\alpha _2)(\alpha _1-j-1)!j!\left( \beta _1+\beta _2\right) ^{\alpha _2+j}},\\&C_j^*=\frac{\varGamma (\alpha _1+j)\beta _1^j\beta _2^{\alpha _1-\alpha _2+j}}{\varGamma (\alpha _1)(\alpha _2-j-1)!j!\left( \beta _1+\beta _2\right) ^{\alpha _1+j}},\\&C_\varGamma =\frac{1}{\varGamma (\alpha )\beta ^\alpha },\\ \end{aligned}$$

and

$$\begin{aligned} F\left( l,r,\theta _1,\theta _2\right)= & {} \int _l^rx^{\theta _1-1}e^{-x\theta _2}dx\\= & {} {\left\{ \begin{array}{ll} \frac{\varGamma (\theta _1)}{\theta _2^{\theta _1}}\left( \varGamma (r,\theta _1,\theta _2^{-1})-\varGamma (l,\theta _1,\theta _2^{-1})\right) , &{} \theta _2>0,l>0,\\ \frac{\varGamma (\theta _1)}{(-\theta _2)^{\theta _1}}\left( \varGamma (-l,\theta _1,\theta _2^{-1})-\varGamma (-r,\theta _1,\theta _2^{-1})\right) , &{} \theta _2>0,r<0,\\ \frac{1}{\theta _1}\left( r^{\theta _1}-l^{\theta _1}\right) , &{} \theta _2=0,\\ -\sum \limits _{j=0}^{\theta _1-1}\frac{(\theta _1-1)!}{j!\theta _2^{\theta _1-j}} \left( r^je^{-r\theta _2}-l^je^{-l\theta _2}\right) , &{} \theta _2<0.\\ \end{array}\right. } \end{aligned}$$

By using these results, we can also obtain a more general result for the joint CDF of $W_1=Y_1+Y_2+a_1Z_1+a_2Z_2$ and $W_2=b_1Z_1+b_2Z_2$ by using the known results on the joint CDF of $-W_1$ and $W_2$ and the CDF of $W_2$ as

$$\begin{aligned} P(W_1\le w_1, W_2\le w_2)= & {} P(-W_1\ge -w_1, W_2\le w_2)\\= & {} P(W_2\le w_2)-P(-W_1\le -w_1, W_2\le w_2). \end{aligned}$$

Table 16.10 Joint CDF of $W_1=Y_1+Y_2-a_1Z_1-a_2Z_2$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.11 Joint CDF of $W_1=Y_1+Y_2+a_1Z_1+a_2Z_2$ and $W_2=b_1Z_1-b_2Z_2.$

Table 16.12 Joint CDF of $W_1=Y_1+Y_2+a_1Z_1-a_2Z_2$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.13 Joint CDF of $W_1=Y_1-a_1Z_1-a_2Z_2$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.14 Joint CDF of $W_1=Y_1+a_1Z_1+a_2Z_2$ and $W_2=b_1Z_1-b_2Z_2$

Table 16.15 Joint CDF of $W_1=Y_1+a_1Z_1-a_2Z_2$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.16 Joint CDF of $W_1=Y_1+Y_2+a_1Z_1$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.17 Joint CDF of $W_1=Y_1+Y_2+a_1Z_1$ and $W_2=b_1Z_1$

Table 16.18 Joint CDF of $W_1=Y_1+a_1Z_1$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.19 Joint CDF of $W_1=Y_1-a_1Z_1$ and $W_2=b_1Z_1+b_2Z_2$

Table 16.20 Joint CDF of $W_1=Y_1+a_1Z_1$ and $W_2=b_1Z_1$

Table 16.21 Joint CDF of $W_1=Y_1-a_1Z_1$ and $W_2=b_1Z_1$