Appendix
List of notation used throughout
$$\begin{aligned} p_1= & {} \frac{(-1)^ln!e^{-\frac{(T-\mu )(n+l-d)}{\sigma }}}{2^n(n-d)!j!(d-j-l)!l!(1-p_0)},\\ p_{2,a}= & {} \frac{(-1)^ln!}{2^n(n-k)!j!(k-j-1-l)!l!(n-k)!(n-k+l+1)(1-p_0)},\\ p_{2,b}= & {} -p_{2,a}e^{-\frac{(T-\mu )(n-k+l+1)}{\sigma }},\\ p_{3}= & {} \frac{(-1)^l n!}{2^{k+l}(k-1)!(n-k-l)!l!(k+l)(1-p_0)},\\ p_{4,a,e}= & {} \frac{(-1)^l n!e^{-\frac{(T-\mu )m}{\sigma }}}{2^n j!(m-1-j-l)!(l+1)!m!(1-p_0)},\\ p_{4,b,e}= & {} -p_{4,a,e}e^{-\frac{(T-\mu )(l+1)}{\sigma }},\\ p_{5,e}= & {} \frac{n!e^{-\frac{(T-\mu )m}{\sigma }}}{2^n (n-m)! m!(1-p_0)},\\ p_{6,a,e}= & {} \frac{(-1)^ln!}{2^nj!(m-1-j-l)!m!(l+1)!(1-p_0)},\\ p_{6,b,e}= & {} -p_{6,a,e}e^{-\frac{(T-\mu )m}{\sigma }},\\ p_{6,c,e}= & {} \frac{(-1)^{l+1}n!}{2^nj!(m-1-j-l)!(m-1)!(l+1)!(m+l+1)(1-p_0)},\\ p_{6,d,e}= & {} -p_{6,c,e}e^{-\frac{(T-\mu )(m+l+1)}{\sigma }},\\ p_{7,a,e}= & {} \frac{n!}{2^nm!m!(1-p_0)},\\ p_{7,b,e}= & {} -p_{7,a,e}e^{-\frac{m(T-\mu )}{\sigma }},\\ p_{9,e}= & {} \frac{(-1)^l n!}{m!(n-k-l)!l!2^{k+l}(m+k+l)(1-p_0)},\\ p_{10,a,e}= & {} \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_{10,b,e}= & {} -p_{10,a,e}e^{-\frac{(T-\mu )(l_2+1)}{\sigma }},\\ p_{10,c,e}= & {} -\frac{p_{10,a,e}}{l_1+l_2+2},\\ p_{10,d,e}= & {} -p_{10,c,e}e^{-\frac{(T-\mu )(l_1+l_2+2)}{\sigma }},\\ p_{11,a,e}= & {} \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_{11,b,e}= & {} -p_{11,a,e}e^{-\frac{(T-\mu )(l+1)}{\sigma }},\\ p_{12,e}= & {} \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_{13,a}= & {} \frac{(-1)^{l_1+l_2}n!}{2^nj!(m-1-j-l_1)!(l_1+1)!(k-m-2-l_2)!l_2!(n-k)!(n-k+l_2+1)m(1-p_0)},\\ p_{13,b}= & {} -p_{13,a}e^{-\frac{(T-\mu )m}{\sigma }},\\ p_{13,c}= & {} \frac{(-1)^{l_1+l_2+1}n!e^{-\frac{(T-\mu )(n-k+l_2+1)}{\sigma }}}{2^nj!(m-1-j-l_1)!(l_1+1)!(k-m-l_2-1)!l_2!(n-k)!(n-k+l_2+1)(1-p_0)},\\ p_{13,d}= & {} -p_{13,c}e^{-\frac{(T-\mu )(k-m-l_2-1)}{\sigma }},\\ p_{13,e}= & {} \frac{(-1)^{l_1+l_2+1}n!}{2^nj!(m-1-j-l_1)!(l_1+1)!(k-m-2-l_2)!l_2!(n-k)!(n-k+l_2+1)(m+l_1+1)(1-p_0)},\\ p_{13,f}= & {} -p_{13,e}e^{-\frac{(T-\mu )(m+l_1+1)}{\sigma }},\\ p_{13,g}= & {} \frac{(-1)^{l_1+l_2}n!e^{-\frac{n-k+l_2+1}{\sigma }}}{2^nj!(m-1-j-l_1)!(l_1+1)!(k-m-2-l_2)!l_2!(n-k)!(n-k+l_2+1)(k-m-l_2+l_1)(1-p_0)},\\ p_{13,h}= & {} -p_{13,g}e^{-\frac{(T-\mu )(k-m-l_2+l_1)}{\sigma }},\\ p_{13,a}^*= & {} \frac{(-1)^{l_1+k-m-l_2}n!}{2^n j!(m-j-l_1)!l_1!(k-m-2-l_2)!l_2!(n-k)!(m-l_2-1)(m+l_1)},\\ p_{13,b}^*= & {} -p_{13,a}^*e^{-\frac{(T-\mu )(m+l_1)}{\sigma }},\\ p_{13,c}^*= & {} \frac{(-1)^{l_1+k-m-1-l_2}n!e^{-\frac{(T-\mu )(m-l_2-1)}{\sigma }}}{2^n j!(m-j-l_1)!l_1!(k-m-2-l_2)!l_2!(n-k)!(m-l_2-1)(l_1+l_2+1)},\\ p_{13,d}^*= & {} -p_{13,c}^*e^{-\frac{(T-\mu )(l_1+l_2+1)}{\sigma }},\\ p_{14,a}= & {} \frac{(-1)^ln!}{2^nm!(k-m-2-l)!l!(n-k+l+1)m(1-p_0)},\\ p_{14,b}= & {} -p_{14,a}e^{-\frac{(T-\mu )m}{\sigma }},\\ p_{14,c}= & {} \frac{(-1)^{l+1}n!e^{-\frac{(T-\mu )(n-k+l+1)}{\sigma }}}{2^nm!(k-m-1-l)!l!(n-k+l+1)(1-p_0)},\\ p_{14,d}= & {} -p_{14,c}e^{-\frac{(T-\mu )(k-m-1-l)}{\sigma }},\\ p_{15,a}= & {} \frac{n!(-1)^{l_1+l_2}(m+l_1+1)^{-1}}{2^nm!(j-m-1-l_1)!l_1!(k-1-j-l_2)!l_2!(n-k)!(n-k+l_2+1)(1-p_0)},\\ p_{15,b}= & {} -p_{15,a}e^{-\frac{(T-\mu )(n-k+l_2+1)}{\sigma }},\\ p_{16,a}= & {} \frac{(-1)^{l_1+l_2}n!}{2^{l_2+d}m!(k-m-2-l_1)!l_1!(n-k-l_2)!l_2!(k-m-1-l_1+l_2)(m+l_1+1)(1-p_0)},\\ p_{16,b}= & {} \frac{(-1)^{l_1+l_2+1}n!}{2^{l_2+d}m!(k-m-2-l_1)!l_1!(n-k-l_2)!l_2!(k-m-1-l_1+l_2)(k+l_2)(1-p_0)},\\ q_1= & {} P(D=d),\\ q_2= & {} \frac{(-1)^le^{\frac{(T-\mu )(k+l)}{\sigma }}n!}{2^{k+l} (k-1)!l!(n-k-l)!(k+l)(1-q_0)},\\ q_{3,e}= & {} P(D=m),\\ q_{4,e}= & {} \frac{(-1)^l n!e^{\frac{(T-\mu )(m+l+1)}{\sigma }}}{2^{k+l} m! l!(m-1-l)!(m+l+1)(1-q_0)},\\ q_5= & {} \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)}\\ q_6= & {} \frac{(-1)^{l_1+l_2}e^{\frac{(T-\mu )(k+l_2)}{\sigma }} n!}{2^{k+l_2}(l_1+m+1)(k+l_2)m!l_1!(k-m-2-l_1)!l_2!(n-k-l_2)!(1-q_0)}\\ p_{4,a,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_{4,b,o}= & {} -p_{4,a,o}e^{-\frac{(T-\mu )(l_1+l_2+1)}{\sigma }},\\ p_{5,o}= & {} \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_{6,a,o}= & {} \frac{(-1)^{l_1+l_2}n!}{ 2^nj!(m-j-l_1)!l_1!(k-m-2-l_2)!l_2!(n-k)!(n-k+l_2+1)(m+l_1+1)(1-p_0)},\\ p_{6,b,o}= & {} -p_{6,a,o}e^{-\frac{(T-\mu )(m+l_1+1)}{\sigma }},\\ p_{6,c,o}= & {} \frac{(-1)^{l_1+l_2+1}n!e^{-\frac{(T-\mu )(n-d+l_2+1)}{\sigma }}}{ (1-p_0)2^nj!(m-j-l_1)!l_1!(k-m-2-l_2)!l_2!(n-k)!(n-k+l_2+1)(k-m-l_2+l_1-1)},\\ p_{6,d,o}= & {} -p_{6,c,o}e^{-\frac{(T-\mu )(d-m+l_1-l_2-1)}{\sigma }},\\ p_{7,a,o}= & {} \frac{(-1)^{l_1+l_2}n!}{ 2^nm!(j-m-1-l_1)!l_1!(k-1-j-l_2)!l_2!(n-k)!(m+l_1+1)(n-k+l_2+1)(1-p_0)},\\ p_{7,b,o}= & {} -p_{7,a,o}e^{-\frac{(T-\mu )(n-k+l_2+1)}{\sigma }},\\ p_{8,a,o}= & {} \frac{(-1)^{k-m-2-l_1+l_2}n!}{ 2^{k+l_2}m!(k-m-2-l_1)!l_1!(n-k-l_2)!l_2!(l_1+l_2+1)(k-l_1-1)(1-p_0)},\\ p_{8,b,o}= & {} \frac{(-1)^{k-m-1-l_1+l_2}n!}{ 2^{k+l_2}m!(k-m-2-l_1)!l_1!(n-k-l_2)!l_2!(l_1+l_2+1)(k+l_2)(1-p_0)},\\ \end{aligned}$$
$$\begin{aligned}&q_{3,o}=P(D=m+1),\\&p_{16}^*=\frac{(-1)^{l_1+l_2}n!}{m!(k-m-2-l_1)!l_1!(n-k-l_2)!l_2!(m+1+l_1)(k+l_2)(1-p_0)},\\&Z_{p1}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{d}\right) +N\Gamma \left( d-j,\frac{\sigma }{d}\right) +\frac{(T-\mu )(d-l)}{d},\\&Z_{p1}^{(2)}\overset{d}{=}\log \left( \frac{n}{2d}\right) Z_{p1}^{(1)}+T,\\&Z_{p2,a}^{(1)}\overset{d}{=}\Gamma _{A}\left( j,\frac{\sigma }{k}\right) +N\Gamma _{A}\left( k-1-j,\frac{\sigma }{k}\right) +\frac{(k-l-1)\sigma }{(n-k+l+1)k}E_1,\\&Z_{p2,a}^{(2)}\overset{d}{=}\log \left( \frac{n}{2k}\right) \left[ \Gamma _{A}\left( j,\frac{\sigma }{k}\right) +N\Gamma _{A}\left( k-1-j,\frac{\sigma }{k}\right) \right] +\left[ \log \left( \frac{n}{2k}\right) \frac{k-l-1}{n-k+l+1}+\frac{k}{n-k+l+1}\right] \frac{\sigma }{k} E_1+\mu ,\\&Z_{p2,a}^{(3)}\overset{d}{=}\frac{\sigma }{n-k+l+1}E_1+\mu ,\\&Z_{p2,b}^{(1)}\overset{d}{=}\Gamma _{A}\left( j,\frac{\sigma }{k}\right) +N\Gamma _{A}\left( k-1-j,\frac{\sigma }{k}\right) +\frac{(k-l-1)\sigma }{(n-k+l+1)k}E_2+\frac{(T-\mu )(k-l-1)}{k},\\&Z_{p2,b}^{(2)}\overset{d}{=}\log \left( \frac{n}{2k}\right) \left[ \Gamma _{A}\left( j,\frac{\sigma }{k}\right) +N\Gamma _{A}\left( k-1-j,\frac{\sigma }{k}\right) +\frac{(T-\mu )(k-l-1)}{k}\right] \\&+\left[ \log \left( \frac{n}{2k}\right) \frac{k-l-1}{n-k+l+1}+\frac{k}{n-k+l+1}\right] \frac{\sigma }{k} E_1+T,\\&Z_{p2,b}^{(3)}\overset{d}{=} \frac{\sigma }{n-k+l+1}E_1+T\\&Z_{p3}^{(1)}\overset{d}{=}\Gamma \left( k-1,\frac{\sigma }{k}\right) ,\\&Z_{p3}^{(2)}\overset{d}{=}\log \left( \frac{n}{2k}\right) Z_{p3}^{(1)}+NE\left( \frac{\sigma }{k+l}\right) +\mu ,\\&Z_{p3}^{(3)}\overset{d}{=}NE\left( \frac{\sigma }{k+l}\right) +\mu ,\\&Z_{p4,a,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{m}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m}\right) -\frac{\sigma }{m}E+(T-\mu ),\\&Z_{p4,a,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E+\frac{T+\mu }{2},\\&Z_{p4,b,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{m}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m}\right) -\frac{\sigma }{m}E+\frac{(T-\mu )(m-l-1)}{m},\\&Z_{p4,b,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E+T,\\&Z_{p5,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{m}\right) +\frac{\sigma }{m}E+(T-\mu ),\\&Z_{p5,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E+\frac{T+\mu }{2},\\&Z_{p6,a,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{m+1}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m+1}\right) -\frac{\sigma }{m+1}E_1+\frac{\sigma }{m+1}E_2,\\&Z_{p6,a,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E_{1}+\frac{\sigma }{n}E_{2}+\mu ,\\&Z_{p6,b,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{m+1}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m+1}\right) -\frac{\sigma }{m+1}E_{1}+\frac{\sigma }{m+1}E_{2}+\frac{m(T-\mu )}{m+1},\\&Z_{p6,b,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E_{1}+\frac{\sigma }{n}E_{2}+\frac{T+\mu }{2},\\&Z_{p6,c,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{m+1}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m+1}\right) -\frac{\sigma }{m+1}E_{1}+\frac{\sigma (m-l-1)}{(m+l+1)(m+1)}E_{2},\\&Z_{p6,c,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E_{1}+\frac{\sigma }{m+l+1}E_{2}+\mu ,\\&Z_{p6,d,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{m+1}\right) -\frac{\sigma }{m+1}E_{1}+\frac{\sigma (m-l-1)}{(m+l+1)(m+1)}E_{2}+\frac{(T-\mu )(m-l-1)}{m+1},\\&Z_{p6,d,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l+1)}E_{1}+\frac{\sigma }{m+l+1}E_{2}+T,\\&Z_{p7,a,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{m+1}\right) +\frac{\sigma }{k}E_{1}+\frac{\sigma }{m+1}E_{2},\\&Z_{p7,a,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_{1}+\frac{\sigma }{n}E_{2}+\mu ,\\&Z_{p7,b,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{m+1}\right) +\frac{\sigma }{m+1}E_1+\frac{\sigma }{m+1}E_2+\frac{m(T-\mu )}{m+1},\\&Z_{p7,b,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{n}E_2+\frac{(T+\mu )}{2},\\&Z_{p9,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E,\\&Z_{p9,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E+NE\left( \frac{\sigma }{m+l+1}\right) +\mu ,\\&Z_{p10,a,e}^{(1)}\overset{d}{=}\Gamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_1-\frac{\sigma }{d}E_2+\frac{(T-\mu )(m-1-l_2)}{d},\\&Z_{p10,a,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_2+1)}E_1+\frac{\sigma }{2(l_1+1)}E_2+\mu ,\\&Z_{p10,b,e}^{(1)}\overset{d}{=}\Gamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_1-\frac{\sigma }{d}E_2 +\frac{(T-\mu )m}{d},\\&Z_{p10,b,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_2+1)}E_1+\frac{\sigma }{2(l_1+1)}E_2+\frac{T+\mu }{2},\\&Z_{p10,c,e}^{(1)}\overset{d}{=}\Gamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_1+l_2+2)d}E_1-\frac{\sigma }{d}E_2+\frac{(T-\mu )(m-l_2-1)}{d},\\&Z_{p10,c,e}^{(2)}\overset{d}{=}\frac{\sigma }{l_1+l_2+2}E_1+\frac{\sigma }{2(l_1+1)}E_2+\mu ,\\&Z_{p10,d,e}^{(1)}\overset{d}{=}\Gamma \left( d-m-1+j,\frac{\sigma }{d}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_1+l_2+2)d}E_1-\frac{\sigma }{d}E_2 +\frac{(T-\mu )(m-l_1-1)}{d},\\&Z_{p10,d,e}^{(2)}\overset{d}{=}\frac{\sigma }{l_1+l_2+2}E_1+\frac{\sigma }{2(l_1+1)}E_2+T,\\&Z_{p11,a,e}^{(1)}\overset{d}{=}\Gamma \left( d-2,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_1+\frac{\sigma }{d}E_2+\frac{(T-\mu )(m-l-1)}{d},\\&Z_{p11,a,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{2(l+1)}E_2+\mu ,\\&Z_{p11,b,e}^{(1)}\overset{d}{=}\Gamma \left( d-2,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_1+\frac{\sigma }{d}E_2+\frac{(T-\mu )m}{d},\\&Z_{p11,b,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{2(l+1)}E_2+\frac{T+\mu }{2},\\&Z_{p12,e}^{(1)}\overset{d}{=}\Gamma \left( d-j+m-1,\frac{\sigma }{d}\right) +N\Gamma \left( j-m-1,\frac{\sigma }{d}\right) +\frac{\sigma }{d}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)d}E_2 +\frac{(T-\mu )(n-d+l_2)}{d}, \end{aligned}$$
$$\begin{aligned}&Z_{p12,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1-\frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{p13,a}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2,\\&Z_{p13,a}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{n}E_2+\mu ,\\&Z_{p13,b}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2+\frac{m(T-\mu )}{k},\\&Z_{p13,b}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{n}E_2+\frac{T+\mu }{2},\\&Z_{p13,c}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2+\frac{(n-k+l_2+1)(T-\mu )}{k},\\&Z_{p13,c}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{2(k-m-l_2-1)}E_2+\mu ,\\&Z_{p13,d}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2+\frac{m(T-\mu )}{k},\\&Z_{p13,d}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{2(k-m-l_2-1)}E_2+\frac{T+\mu }{2},\\&Z_{p13,e}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2,\\&Z_{p13,e}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{p13,f}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2 +\frac{(m-l_1-1)(T-\mu )}{k},\\&Z_{p13,f}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{m+l_1+1}E_2+T,\\&Z_{p13,g}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{(k-m-l_1-l_2-2)\sigma }{(k-m-l_2+l_1)k}E_2\\&\qquad \quad +\frac{(n-k+l_2+1)(T-\mu )}{k},\\&Z_{p13,g}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{k-m-l_2+l_1}E_2+\mu ,\\&Z_{p13,h}^{(1)}\overset{d}{=}\Gamma \left( k-m-1+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) -\frac{\sigma }{k}E_1+\frac{(k-m-l_1-l_2-2)\sigma }{(k-m-l_2+l_1)k}E_2\\&\qquad \quad +\frac{(m-l_1-1)(T-\mu )}{k},\\&Z_{p13,h}^{(2)}\overset{d}{=}\frac{\sigma }{2(l_1+1)}E_1+\frac{\sigma }{k-m-l_2+l_1}E_2+T,\\&Z_{p13,a}^*\overset{d}{=}\Gamma \left( k-1-m+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-j,\frac{\sigma }{k}\right) +E\left( \frac{(m-l_1)\sigma }{(m+l_1)k}\right) ,\\&Z_{p13,b}^*\overset{d}{=}\Gamma \left( k-1-m+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-j,\frac{\sigma }{k}\right) +E\left( \frac{(m-l_1)\sigma }{(m+l_1)k}\right) +\frac{(T-\mu )(m-l_1)}{k},\\&Z_{p13,c}^*\overset{d}{=}\Gamma \left( k-1-m+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-j,\frac{\sigma }{k}\right) +E\left( \frac{(l_2-l_1+1)\sigma }{(l_1+l_2+1)k}\right) +\frac{(T-\mu )(m-l_2-1)}{k},\\&Z_{p13,d}^*\overset{d}{=}\Gamma \left( k-1-m+j,\frac{\sigma }{k}\right) +N\Gamma \left( m-j,\frac{\sigma }{k}\right) +E\left( \frac{(l_2-l_1+1)\sigma }{(l_1+l_2+1)k}\right) +\frac{(T-\mu )(m-l_1)}{k},\\&Z_{p14,a}^{(1)}\overset{d}{=}\Gamma \left( k-2,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1 +\frac{\sigma }{k}E_2,\\&Z_{p14,a}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{n}E_2+\mu ,\\&Z_{p14,b}^{(1)}\overset{d}{=}\Gamma \left( k-2,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1 +\frac{\sigma }{k}E_2+\frac{m(T-\mu )}{k},\\&Z_{p14,b}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{n}E_2+\frac{T+\mu }{2},\\&Z_{p14,c}^{(1)}\overset{d}{=}\Gamma \left( k-2,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1 +\frac{\sigma }{k}E_2+\frac{(T-\mu )(n-k+l+1)}{k},\\&Z_{p14,c}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{2(k-m-l-1)}E_2+\mu ,\\&Z_{p14,d}^{(1)}\overset{d}{=}\Gamma \left( k-2,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1 +\frac{\sigma }{k}E_2+\frac{(T-\mu )m}{k},\\&Z_{p14,d}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{2(k-m-l-1)}E_2+\frac{T+\mu }{2},\\&Z_{p15,a}^{(1)}\overset{d}{=}\Gamma \left( m-1+k-j,\frac{\sigma }{k}\right) + N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1 +\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2,\\&Z_{p15,a}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1 -\frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{p15,b}^{(1)}\overset{d}{=}\Gamma \left( m-1+k-j,\frac{\sigma }{k}\right) + N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2+\frac{(T-\mu )(n-k+l_2+1)}{k},\\&Z_{p15,b}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1 -\frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{p16,a}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) + N\Gamma \left( d-m-2,\frac{\sigma }{k}\right) +NE\left( \frac{(m-l_1-1)\sigma }{(k-m-1-l_1+l_2)k}\right) +\frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2,\\&Z_{p16,a}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1 -\frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{p16,b}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) + N\Gamma \left( d-m-2,\frac{\sigma }{k}\right) +NE\left( \frac{(m-l_1-1)\sigma }{(k-m-1-l_1+l_2)k}\right) +\frac{\sigma }{k}E,\\&Z_{p16,b}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E +NE\left( \frac{\sigma }{k+l_2}\right) +\mu ,\\&Z_{q1}^{(1)}\overset{d}{=}\Gamma \left( d,\frac{\sigma }{d}\right) ,\\&Z_{q1}^{(2)}\overset{d}{=}\log \left( \frac{n}{2d}\right) Z_{q1}^{(1)}+T,\\&Z_{q2,a}^{(1)}\overset{d}{=}\Gamma \left( k-1,\frac{\sigma }{k}\right) ,\\&Z_{q2,a}^{(2)}\overset{d}{=}\log \left( \frac{n}{2k}\right) Z_{q2,a}^{(1)}+NE\left( \frac{\sigma }{k+l}\right) +T,\\&Z_{q2,b}^{(1)}\overset{d}{=}\Gamma \left( k-1,\frac{\sigma }{k}\right) ,\\&Z_{q2,b}^{(2)}\overset{d}{=}NE\left( \frac{\sigma }{k+l}\right) +T,\\&Z_{q3,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{m}\right) +\frac{\sigma }{m}E,\\&Z_{q3,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E+T,\\&Z_{q4,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E,\\&Z_{q4,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E+NE\left( \frac{\sigma }{m+l+1}\right) +T,\\&Z_{q5,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{d}\right) +N\Gamma \left( d-m-1,\frac{\sigma }{d}\right) + \frac{\sigma }{d}E_1+\frac{(m-l-1)\sigma }{(m+l+1)d}E_2,\\&Z_{q5,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1-\frac{\sigma }{m+l+1}E_2+T,\\&Z_{q6,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) +N\Gamma \left( k-m-2,\frac{\sigma }{k}\right) + \frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{(m+l_1+1)k}E_2,\\&Z_{q6,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1-\frac{\sigma }{m+l_1+1}E_2+NE\left( \frac{\sigma }{k+l_2}\right) +T,\\&Z_{p4,a,o}^{(1)}\overset{d}{=}\Gamma \left( j+d-m-1,\frac{\sigma }{d}\right) + N\Gamma \left( m-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_2+l_1+1)d} E +\frac{(T-\mu )(m-l_2)}{d},\\&Z_{p4,a,o}^{(2)}\overset{d}{=}\frac{\sigma }{l_2+l_1+1}E+\mu ,\\&Z_{p4,b,o}^{(1)}\overset{d}{=}\Gamma \left( j+d-m-1,\frac{\sigma }{d}\right) + N\Gamma \left( m-j,\frac{\sigma }{d}\right) +\frac{(l_2-l_1)\sigma }{(l_2+l_1+1)d} E+\frac{(T-\mu )(m-l_1)}{d},\\&Z_{p4,b,o}^{(2)}\overset{d}{=}\frac{\sigma }{l_2+l_1+1}E+T,\\&Z_{p5,o}^{(1)}\overset{d}{=}\Gamma \left( d-j+m,\frac{\sigma }{d}\right) + N\Gamma \left( j-m-1,\frac{\sigma }{d}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)d} E +\frac{(T-\mu )(n-d+l_2)}{d},\\&Z_{p5,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{p6,a,o}^{(1)}\overset{d}{=}\Gamma \left( j+k-m-1,\frac{\sigma }{k}\right) + N\Gamma \left( m-j,\frac{\sigma }{k}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)k} E,\\&Z_{p6,a,o}^{(2)}\overset{d}{=}\frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{p6,b,o}^{(1)}\overset{d}{=}\Gamma \left( j+k-m-1,\frac{\sigma }{k}\right) + N\Gamma \left( m-j,\frac{\sigma }{k}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)k} E+\frac{(m-l_1)(T-\mu )}{k},\\&Z_{p6,b,o}^{(2)}\overset{d}{=}\frac{\sigma }{m+l_1+1}E+T,\\&Z_{p6,c,o}^{(1)}\overset{d}{=}\Gamma \left( j+k-m-1,\frac{\sigma }{k}\right) + N\Gamma \left( m-j,\frac{\sigma }{k}\right) ++\frac{(k-m-l_1-l_2-2)\sigma }{(k-m+l_1-l_2-1)k} E+\frac{(n-k+l_2+1)(T-\mu )}{k},\\&Z_{p6,c,o}^{(2)}\overset{d}{=}\frac{\sigma }{k-m-l_2+l_1-1}E+\mu ,\\&Z_{p6,d,o}^{(1)}\overset{d}{=}\Gamma \left( j+k-m-1,\frac{\sigma }{k}\right) + N\Gamma \left( m-j,\frac{\sigma }{k}\right) +\frac{(k-m-l_1-l_2-2)\sigma }{(k-m+l_1-l_2-1)k} E+\frac{(m-l_1)(T-\mu )}{k},\\&Z_{p6,d,o}^{(2)}\overset{d}{=}\frac{\sigma }{k-m+l_1-l_2-1}E+T,\\&Z_{p7,a,o}^{(1)}\overset{d}{=}\Gamma \left( m+k-j,\frac{\sigma }{k}\right) +N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)k}E,\\&Z_{p7,a,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{p7,b,o}^{(1)}\overset{d}{=}\Gamma \left( m+k-j,\frac{\sigma }{k}\right) +N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{(m-l_1)\sigma }{(m+l_1+1)k}E+\frac{(n-k+1+l_2)(T-\mu )}{k},\\&Z_{p7,b,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{p8,a,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{k}\right) +N\Gamma \left( k-m-2,\frac{\sigma }{k}\right) +NE\left( \frac{n-k+l_1+1}{(l_1+l_2+1)k}\right) +\frac{n+1+l_1-k}{(k-l_1-1)k}E,\\&Z_{p8,a,o}^{(2)}\overset{d}{=}-\frac{\sigma }{k-l_1-1}E+\mu ,\\&Z_{p8,b,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{k}\right) +N\Gamma \left( k-m-2,\frac{\sigma }{k}\right) +NE\left( \frac{n-k+l_1+1}{(l_1+l_2+1)k}\right) ,\\&Z_{p8,b,o}^{(2)}\overset{d}{=}NE\left( \frac{\sigma }{k+l_2}\right) +\mu ,\\&Z_{q3,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{m+1}\right) +\frac{\sigma }{m+1}E,\\&Z_{q3,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+1}E+T,\\&Z_{q5,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{d}\right) +N\Gamma \left( d-m-1,\frac{\sigma }{d}\right) + \frac{(m-l)\sigma }{(m+l+1)d}E,\\&Z_{q5,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l+1}E+T,\\&Z_{q6,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{k}\right) +N\Gamma \left( k-m-2,\frac{\sigma }{k}\right) + \frac{(m-l_1)\sigma }{(m+l_1+1)k}E,\\&Z_{q6,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l_1+1}E+NE\left( \frac{\sigma }{k+l_2}\right) +T,\\&Z_{p16}^*\overset{d}{=}\Gamma \left( m,\frac{\sigma }{k}\right) +N\Gamma \left( k-2-m,\frac{\sigma }{m}\right) +E\left( \frac{(m-l_1-1)\sigma }{k(m+l_1+1)}\right) . \end{aligned}$$
List of notation used in Lemma 7.3
$$\begin{aligned}&p_{a1}=\frac{(-1)^l n!}{2^nj!(k-1-j-l)!l!(n-k)!(n-k+l+1)},\\&p_{a2}=\frac{(-1)^l n!}{2^{k+l}(k-1)!l!(n-k-l)!(k+l)},\\&p_{b1,e}=\frac{(-1)^l n!}{2^nj!(k-1-j-l)!l!(n-k)!(k+l+1)},\\&p_{b2}=\frac{(-1)^l n!}{2^{k+l}(k-1)!(n-k-l)!l!(k+l)},\\&p_{b1,o}=\frac{(-1)^l n!}{2^nj!(k-1-j-l)!l!(n-k)!(k+l)},\\&p_{c1}=\frac{(-1)^l n!}{2^nj!(m-j-1-l)!l!(n-k+1)!(n-k+l+2)},\\&p_{c2}=\frac{n!}{2^n m!(n-k+1)!},\\&p_{c3}=\frac{(-1)^ln! }{2^{k+l}m!(n-k-l)!l! (l+m+1)},\\&p_{d1,e}=\frac{(-1)^{l_1+l_2}n!}{2^nj!(m-1-j-l_1)!l_1!(k-2-m-l_2)!l_2!(n-k)! m(m+l_1+1)(n-k+l_2+1)},\\&p_{d1,o}=\frac{(-1)^{l_1+l_2}n!}{2^nj!(m-j-l_1)!l_1!(k-2-m-l_2)!l_2!(n-k)! (m+l_1+1)(n-k+l_2+1)},\\&p_{d2}=\frac{(-1)^{l}n!}{2^nm!(k-2-m-l)!l!(n-k)!(n-r+l+1) m},\\&p_{d3}=\frac{(-1)^{l_1+l_2}n!}{2^nm!(j-1-m-l_1)!l_1!(k-1-j-l_2)!l_2!(n-k)! (l_1+m+1)(n-k+1+l_2)},\\&p_{d4}=\frac{(-1)^{l_1+l_2}n!}{2^{k+l_1}m!(k-2-m-l_2)!l_2!(n-k-l_1)!l_1!(k+l_1) (m+l_2+1)},\\&Z_{a1}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{k}\right) +N\Gamma \left( k-1-j,\frac{\sigma }{k}\right) +\frac{k-1-l}{k(n-r+l+1)}E,\\&Z_{a1}^{(2)}\overset{d}{=} \ln \left( \frac{n}{2k}\right) Z_{a1}^{(1)}+\frac{\sigma }{(n-r+l+1)}E+\mu ,\\&Z_{a2}^{(1)}\overset{d}{=}\Gamma \left( k-1,\frac{\sigma }{k}\right) ,\\&Z_{a2}^{(2)}\overset{d}{=}\ln \left( \frac{n}{2k}\right) Z_{a2}^{(1)}+NE\left( \frac{1}{l+k}\right) +\mu ,\\&Z_{b1,e}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{k}\right) +N\Gamma \left( k-1-j,\frac{\sigma }{k}\right) +\frac{(k-l-1)\sigma }{k(k+l+1)}E,\\&Z_{b1,e}^{(2)}\overset{d}{=}\frac{\sigma }{(k+l+1)}E+\mu ,\\&Z_{b1,o}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{k}\right) +N\Gamma \left( k-1-j,\frac{\sigma }{k}\right) +\frac{(k-1-l)\sigma }{k(k+l)}E,\\&Z_{b1,o}^{(2)}\overset{d}{=}\frac{\sigma }{k+l}E+\mu ,\\&Z_{b2}^{(1)}\overset{d}{=}\Gamma \left( k-1,\frac{\sigma }{k}\right) ,\\&Z_{b2}^{(2)}\overset{d}{=}NE\left( \frac{\sigma }{k+l}\right) +\mu ,\\&Z_{c1}^{(1)}\overset{d}{=}\Gamma \left( j,\frac{\sigma }{k}\right) +N\Gamma \left( m-j-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{(n-k-l)\sigma }{k(n-k+l+2)}E_2,\\&Z_{c1}^{(2)}\overset{d}{=}\frac{\sigma }{n}E_1+\frac{1}{n-k+l+2}E_2+\mu ,\\&Z_{c2}^{(1)}\overset{d}{=} \Gamma \left( m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2,\\&Z_{c2}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1+\frac{\sigma }{n}E_2+\mu ,\\&Z_{c3}^{(1)}\overset{d}{=} \Gamma \left( m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E,\\&Z_{c3}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E+NE\left( \frac{\sigma }{m+l+1}\right) +\mu ,\\&Z_{d1,e}^{(1)}\overset{d}{=} \Gamma \left( k+j-m-1,\frac{\sigma }{k}\right) +N\Gamma \left( m-1-j,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+ \frac{(m-l_1-1)\sigma }{k(m+l_1+1)}E_2,\\&Z_{d1,e}^{(2)}\overset{d}{=} \frac{\sigma }{n}E_1+ \frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{d1,o}^{(1)}\overset{d}{=} \Gamma \left( k+j-m-1,\frac{\sigma }{k}\right) +N\Gamma \left( m-j,\frac{\sigma }{k}\right) + \frac{(m-l_1)\sigma }{k(m+l_1+1)}E_2,\\&Z_{d1,o}^{(2)}\overset{d}{=} \frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{d2}^{(1)}\overset{d}{=} \Gamma \left( k-2,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{\sigma }{k}E_2,\\&Z_{d2}^{(2)}\overset{d}{=} -\frac{\sigma }{n}E_1+\frac{\sigma }{n}E_2+\mu ,\\&Z_{d3,e}^{(1)}\overset{d}{=}\Gamma \left( k+m-j-1,\frac{\sigma }{k}\right) +N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{(m-l_1-1)\sigma }{k(m+l_1+1)}E_2,\\&Z_{d3,e}^{(2)}\overset{d}{=}-\frac{\sigma }{n}E_1- \frac{\sigma }{m+l_1+1}E_2+\mu ,\\&Z_{d3,o}^{(1)}\overset{d}{=}\Gamma \left( k+m-j,\frac{\sigma }{k}\right) +N\Gamma \left( j-m-1,\frac{\sigma }{k}\right) +\frac{(m-l_1)\sigma }{k(m+l_1+1)}E,\\&Z_{d3,o}^{(2)}\overset{d}{=}-\frac{\sigma }{m+l_1+1}E+\mu ,\\&Z_{d4,e}^{(1)}\overset{d}{=}\Gamma \left( m-1,\frac{\sigma }{k}\right) +N\Gamma \left( k-2-m,\frac{\sigma }{k}\right) +\frac{\sigma }{k}E_1+\frac{(m-l_2-1)\sigma }{k(m+l_2+1)}E_2,\\&Z_{d4,e}^{(2)}\overset{d}{=} -\frac{\sigma }{n}E_1-\frac{\sigma }{m+l_2+1}E_2 +NE\left( \frac{\sigma }{k+l_1}\right) +\mu ,\\&Z_{d4,o}^{(1)}\overset{d}{=}\Gamma \left( m,\frac{\sigma }{k}\right) +N\Gamma \left( k-2-m,\frac{\sigma }{k}\right) +\frac{(m-l_2)\sigma }{k(m+l_2+1)}E,\\&Z_{d4,o}^{(2)}\overset{d}{=} -\frac{\sigma }{m+l_2+1}E +NE\left( \frac{\sigma }{k+l_1}\right) +\mu . \end{aligned}$$
Proof of Lemma 3.4 Here, we will only give the proof for the case when \(k>m+1\), and all other cases can all be proved similarly. We have
$$\begin{aligned} E\left( e^{t{\hat{\sigma }}+s{\hat{\mu }}}, D=k|D>0\right)= & {} \sum \limits _{j=0}^{m-1}E\left( e^{t{\hat{\sigma }}+s{\hat{\mu }}}, D=k, J=j|D>0\right) \\&+\,E\left( e^{t{\hat{\sigma }}+s{\hat{\mu }}}, D=k, J=m|D>0\right) \\&+\,\sum \limits _{j=m+1}^{k-1}E\left( e^{t{\hat{\sigma }}+s\hat{\mu }}, D=k, J=j|D>0\right) \\&+\,E\left( e^{t{\hat{\sigma }}+s{\hat{\mu }}}, D=k, J=k|D>0\right) , \end{aligned}$$
where J is the number of X’s less than \(\mu \) as defined earlier in Sect. 3. As the derivation of these four expectations are quite similar, we only derive the last one and omit others for the sake of brevity. In this case, we only need to focus on three order statistics \(X_{m:n}\), \(X_{m+1:n}\) and \(X_{k:n}\), and then consider that there exists \(m-1\) i.i.d. observations less than \(X_{m:n}\), and \(k-m-2\) i.i.d. observations are between \(X_{m+1:n}\) and \(X_{k:n}\). We then have
$$\begin{aligned} E\left( e^{t{\hat{\sigma }}+s\hat{\mu }}, D=k, J=k|D>0\right)= & {} \frac{n!}{(m-1)!(k-m-2)!(n-k)!(1-p_0)} \int _{-\infty }^\mu \int _{-\infty }^{X_{m+1:n}}\int _{X_{m+1:n}}^{\mu }\\&\,\times \, \left[ \int _{-\infty }^{X_{m:n}} e^{-tx}\frac{1}{2\sigma }e^{\frac{x-\mu }{\sigma }}\right] ^{m-1} \left[ \int _{X_{m+1:n}}^{X_{k:n}} e^{tx}\frac{1}{2\sigma }e^{\frac{x-\mu }{\sigma }}\right] ^{k-m-2}\\&\,\times \, e^{t(n-k+1)x_{k:n}} \frac{1}{2\sigma }e^{\frac{x_{k:n}-\mu }{\sigma }}\left( 1-\frac{1}{2}e^{\frac{x_{k:n}-\mu }{\sigma }}\right) ^{n-k}dx_{k:n}\\&\,\times \, e^{\left( \frac{s}{2}-t\right) x_{m:n}}\frac{1}{2\sigma }e^{\frac{x_{m:n}-\mu }{\sigma }} dx_{m:n}\\&\,\times \, e^{\left( \frac{s}{2}+t\right) x_{m+1:n}}\frac{1}{2\sigma }e^{\frac{x_{m+1:n}-\mu }{\sigma }}dx_{m+1:n}\\= & {} \sum \limits _{l_1=0}^{k-m-2}\sum \limits _{l_2=0}^{n-k} \left[ p_{16,a}M_{Z_{p16,a}^{(1)},Z_{p16,a}^{(2)}}(t,s)+p_{16,b}M_{Z_{p16,b}^{(1)},Z_{p16,b}^{(2)}}(t,s)\right] . \end{aligned}$$
The last equation is obtained by using the binomial expansion.
In this case, if we are only interested in the marginal distribution of \({\hat{\sigma }}\), we can then focus only on \(X_{m+1:n}\) and \(X_{d:n}\) and consider that there exists m i.i.d. observations less then \(X_{m+1:n}\) and \(k-m-2\) i.i.d. observations between \(X_{m+1:n}\) and \(X_{k:n}\). We then have
$$\begin{aligned} E\left( e^{t{\hat{\sigma }}}, D=k, J=k|D>0\right)= & {} \frac{n!}{m!(k-m-2)!(n-k)!(1-p_0)} \int _{-\infty }^\mu \int _{-\infty }^{X_{k:n}}\\&\times \,\,\left[ \int _{-\infty }^{X_{m+1:n}} e^{-tx}\frac{1}{2\sigma }e^{\frac{x-\mu }{\sigma }}\right] ^{m} \left[ \int _{X_{m+1:n}}^{X_{k:n}} e^{tx}\frac{1}{2\sigma }e^{\frac{x-\mu }{\sigma }}\right] ^{k-m-2}\\&\times \, e^{tx_{m+1:n}}\frac{1}{2\sigma }e^{\frac{x_{m+1:n}-\mu }{\sigma }}dx_{m+1:n}\\&\times \,\, e^{t(n-k+1)x_{k:n}} \frac{1}{2\sigma }e^{\frac{x_{k:n}-\mu }{\sigma }}\left( 1-\frac{1}{2}e^{\frac{x_{k:n}-\mu }{\sigma }}\right) ^{n-k}dx_{k:n}\\= & {} \sum \limits _{l_1=0}^{k-m-2}\sum \limits _{l_2=0}^{n-k} p_{16}^* M_{Z^*_{p16}}(t), \end{aligned}$$
as required. \(\blacksquare \)
