Planning step-stress test plans under Type-I hybrid censoring for the log-location-scale distribution

Abstract

The optimal design of a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps under Type-I hybrid censoring scheme for a general log-location-scale lifetime distribution is discussed here. Censoring is allowed only at the change-stress point in the final stage. Based on the cumulative exposure model, the determination of the optimal choice for Weibull, lognormal and log-logistic lifetime distributions are considered by minimization of the asymptotic variance of the maximum likelihood estimate of the pth percentile of the lifetime at the normal operating condition. Numerical results show that for these lifetime distributions, the optimal k-step-stress ALT design with unequal duration steps under Type-I hybrid censoring scheme reduces just to a 2-step-stress ALT design.

Appendix

For deriving the entries of the Fisher information matrix, we need the following properties concerning the count variables and order statistics.

Some important properties

• (a) The random variable $n_1$ has a binomial distribution with parameters $(n, \Phi (\eta _1))$. For $i=2, 3, \ldots , k-1$, given $n_1, n_2, \ldots , n_{i-1}$, the random variable $n_i$ has a binomial distribution with parameters $\displaystyle \left( n-\sum _{j=1}^{i-1} n_j, \frac{\Phi (\eta _i)-\Phi (\eta _{i-1})}{1- \Phi (\eta _{i-1})} \right)$; here,

  $$\begin{aligned} \eta _i=\frac{\ln (\tau _i+ s_{i-1}- \tau _{i-1})- (\beta _0+ \beta _1 x_i)}{\sigma }, \quad i=1, 2, \ldots , k, \ \eta _0 \equiv -\infty ; \end{aligned}$$

  (4)

• (b) Given $n_1, n_2, \ldots , n_{k-1}$, the random variable $n_k^*$ has probability mass function

  

• (c) Given $n_1, n_2, \ldots , n_i$, the random variables $y_{i,j}$, $j=1, 2, \ldots , n_i$, are distributed jointly as order statistics from a random sample of size $n_i$ from a doubly truncated population with PDF $\displaystyle \frac{g_i(y)}{\Phi (\eta _{i})-\Phi (\eta _{i-1})}$ for $\tau _{i-1} \le y \le \tau _i$ and $i=1, 2, \ldots , k-1$, where $g_i(\cdot )$ is as defined earlier in (2);

• (d) Given $n_1, n_2, \ldots , n_{k-1}$, the random variables $y_{k,j}$, $j=1, 2, \ldots , n_k^*$, are distributed jointly as order statistics from a random sample of size $n_k^*$ from a doubly truncated population with PDF $\displaystyle \frac{g_k(y)}{\Phi (\eta _{k})-\Phi (\eta _{k-1})}$ for $\tau _{k-1} < y \le \tau _k$.

Using the above key distributional properties, we obtain

$$\begin{aligned}&E(n_i)= n [ \Phi (\eta _i)- \Phi (\eta _{i-1})], \\&E\left\{ \frac{Z^{\ell }_i e^{-q \sigma Z_i}}{\phi (Z_i)} \left[ \frac{1}{\phi (Z_i)} \left( \frac{d \phi (Z_i)}{d Z_i}\right) ^2- \frac{d^2 \phi (Z_i)}{d Z_i^2} \right] \right\} = \frac{A_{i, q,{\ell }}}{\Phi (\eta _{i})-\Phi (\eta _{i-1})}, \\&E\left( \frac{Z^{\ell }_i e^{-q \sigma Z_i}}{\phi (Z_i)} \frac{d \phi (Z_i)}{d Z_i} \right) = \frac{B_{i, q, {\ell }}}{\Phi (\eta _{i})-\Phi (\eta _{i-1})}, \\&E\left( e^{-q \sigma Z_i} \right) = \frac{C_{i,q}}{\Phi (\eta _{i})-\Phi (\eta _{i-1})} , \end{aligned}$$

for $q=0, 1, 2, \ell =0, 1, 2$, and $i=1, \ldots , k$, where

$$\begin{aligned} A_{i, q, {\ell }}= & {} \int _{\eta _{i-1}}^{\eta _{i}} z^{\ell } e^{-q \sigma z} \left[ \frac{1}{\phi (z)} \left( \frac{d \phi (z)}{d z}\right) ^2- \frac{d^2 \phi (z)}{d z^2}\right] dz,\\ B_{i, q, {\ell }}= & {} \int _{\eta _{i-1}}^{\eta _{i}} z^{\ell } e^{-q \sigma z} \frac{d \phi (z)}{d z} dz,\\ C_{i,q}= & {} \int _{\eta _{i-1}}^{\eta _{i}} e^{-q \sigma z} \phi (z) dz. \end{aligned}$$

For $q=0, 1, 2$, $\ell =0, 1, 2$, and $h=0,1$, we define

$$\begin{aligned} A^*_{q, {\ell }}= & {} \frac{1}{n} E\left\{ \sum _{j=1}^{n^*_k} \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k}\right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} ,\\ B^*_{q, {\ell }}= & {} \frac{1}{n} E\left( \sum _{j=1}^{n^*_k} \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \frac{d \phi (Z_k)}{d Z_k} \right) ,\\ C^*_{q}= & {} \frac{1}{n} E\left( \sum _{j=1}^{n^*_k} e^{-q \sigma Z_k} \right) ,\\ D^*_{q, {\ell }, h}= & {} \frac{1}{n} E \left[ \left( n- \sum _{i=1}^{k-1} n_i -n^*_k\right) \frac{{\eta _k^*}^{\ell } \phi ^h (\eta _k^*) e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \right] ,\\ H^*_{q,{\ell }}= & {} \frac{1}{n} E \left\{ \left( n- \sum _{i=1}^{k-1} n_i -n^*_k\right) \frac{{\eta _k^*}^{\ell } e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \left[ \frac{d \phi (\eta _k^*)}{d \eta _k^*}+ \frac{\phi ^2 (\eta _k^*)}{1-\Phi (\eta _k^*)} \right] \right\} . \end{aligned}$$

We then have

$$\begin{aligned} A^*_{q, {\ell }}= & {} \frac{1}{n} E\left\{ \sum _{j=1}^{n^*_k} \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k} \right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} \\= & {} \frac{1}{n} E\left\{ E(n^*_k) E \left\{ \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k} \right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} \bigg | \tau _k^* \right\} \\= & {} \frac{1}{n} E\left\{ E(n^*_k) E\left\{ \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k} \right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} \bigg | Y_{r:n}> \tau _k \right\} \\&\times \, P(Y_{r:n}> \tau _k)\\&+\ \frac{1}{n} E\left\{ E(n^*_k) E\left\{ \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k} \right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} \bigg | \tau _{k-1}< Y_{r:n}< \tau _k \right\} \\&\times P(\tau _{k-1}< Y_{r:n}< \tau _k)\\= & {} \frac{1}{n} E(n_k) E\left\{ \frac{Z_k^{\ell } e^{-q \sigma Z_k}}{\phi (Z_k)} \left[ \frac{1}{\phi (Z_k)} \left( \frac{d \phi (Z_k)}{d Z_k} \right) ^2- \frac{d^2 \phi (Z_k)}{d Z_k^2} \right] \right\} P(Y_{r:n}> \tau _k)\\&+\ \frac{1}{n} E\left\{ n \left[ \Phi \left( Z_{k, r-\sum _{i=1}^{k-1} n_i}\right) - \Phi (\eta _{k-1})\right] \right. \\&\left. \times \int _{\eta _{k-1}}^{Z_{k, r-\sum _{i=1}^{k-1} n_i}} \frac{x^{\ell } e^{-q \sigma x}}{\phi (x)} \left[ \frac{1}{\phi (x)} \left( \frac{d \phi (x)}{d x} \right) ^2- \frac{d^2 \phi (x)}{d x^2} \right] \right. \\&\times \left. \frac{\phi (x)}{\Phi \left( Z_{k, r-\sum _{i=1}^{k-1} n_i}\right) - \Phi (\eta _{k-1})} \ dx \bigg | \tau _{k-1}< Y_{r:n}< \tau _k \right\} P(\tau _{k-1}< Y_{r:n}< \tau _k)\\= & {} A_{k, q, \ell } P(Y_{r:n}> \tau _k)\\&+\ E \left\{ \int _{\eta _{k-1}}^{Z_{k, r-\sum _{i=1}^{k-1} n_i}} x^{\ell } e^{-q \sigma x} \left[ \frac{1}{\phi (x)} \left( \frac{d \phi (x)}{d x} \right) ^2- \frac{d^2 \phi (x)}{d x^2} \right] \ dx \bigg | \tau _{k-1}< Y_{r:n}< \tau _k \right\} \\&\times P(\tau _{k-1}< Y_{r:n} < \tau _k)\\= & {} A_{k, q, \ell } P(Y_{r:n}> \tau _k) +\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)}\\&\times \int ^{\tau _k}_{\tau _{k-1}} \left\{ \int _{\eta _{k-1}}^{z_{k, r-\sum _{i=1}^{k-1} n_i}} x^{\ell } e^{-q \sigma x} \left[ \frac{1}{\phi (x)} \left( \frac{d \phi (x)}{d x} \right) ^2- \frac{d^2 \phi (x)}{d x^2} \right] \ dx \right\} \\&\times \left[ \frac{G_k (y)-G_{k-1} (\tau _{k-1})}{1-G_{k-1} (\tau _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-G_k (y)}{1-G_{k-1} (\tau _{k-1})}\right] ^{n-r} \frac{g_k (y)}{1-G_{k-1} (\tau _{k-1})} \ dy\\= & {} A_{k, q, \ell } P(Y_{r:n}> \tau _k) +\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)} \\&\times \int ^{\eta _k}_{\eta _{k-1}} \left\{ \int _{\eta _{k-1}}^{w} x^{\ell } e^{-q \sigma x} \left[ \frac{1}{\phi (x)} \left( \frac{d \phi (x)}{d x} \right) ^2- \frac{d^2 \phi (x)}{d x^2} \right] \ dx \right\} \\&\times \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw,\\ B^*_{q, {\ell }}= & {} B_{k, q, \ell } P(Y_{r:n}> \tau _k)\\&+\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)} \int ^{\eta _k}_{\eta _{k-1}} \left\{ \int _{\eta _{k-1}}^{w} x^{\ell } e^{-q \sigma x} \frac{d \phi (x)}{d x} \ dx \right\} \\&\times \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw,\\ C^*_{q}= & {} C_{k, q} P(Y_{r:n} > \tau _k)\\&+\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)} \int ^{\eta _k}_{\eta _{k-1}} \left\{ \int _{\eta _{k-1}}^{w} e^{-q \sigma x} \phi (x) \ dx \right\} \\&\times \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw, \end{aligned}$$

and

$$\begin{aligned} D^*_{q, {\ell }, h}= & {} \frac{1}{n} E \left[ \left( n- \sum _{i=1}^{k-1} n_i -n^*_k\right) \frac{{\eta _k^*}^{\ell } \phi ^h (\eta _k^*) e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \right] \\= & {} \frac{1}{n} E\left[ \left( n- \sum _{i=1}^{k-1} E(n_i) -E(n^*_k) \right) E \left( \frac{{\eta _k^*}^{\ell } \phi ^h (\eta _k^*) e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \right) \bigg | \tau _k^* \right] \\= & {} \frac{1}{n} E\left[ \left( n- \sum _{i=1}^{k-1} E(n_i) -E(n^*_k) \right) E \left( \frac{{\eta _k^*}^{\ell } \phi ^h (\eta _k^*) e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \right) \bigg | Y_{r:n}> \tau _k \right] P(Y_{r:n}> \tau _k)\\&+\ \frac{1}{n} E\left[ \left( n- \sum _{i=1}^{k-1} E(n_i) -E(n^*_k) \right) E \left( \frac{{\eta _k^*}^{\ell } \phi ^h (\eta _k^*) e^{-q \sigma \eta _k^*}}{1-\Phi (\eta ^*_k)} \right) \bigg | \tau _{k-1}< Y_{r:n}< \tau _k \right] \\&\times P(\tau _{k-1}< Y_{r:n}< \tau _k)\\= & {} \frac{1}{n} \left( n- \sum _{i=1}^{k-1} E(n_i) -E(n_k) \right) \frac{{\eta _k}^{\ell } \phi ^h (\eta _k) e^{-q \sigma \eta _k}}{1-\Phi (\eta _k)} P(Y_{r:n}> \tau _k)\\&+\ \frac{1}{n} E\left\{ \left\{ n- \sum _{i=1}^{k-1} n [ \Phi (\eta _i)- \Phi (\eta _{i-1})] -n \left[ \Phi \left( Z_{k, r-\sum _{i=1}^{k-1} n_i}\right) - \Phi (\eta _{k-1})\right] \right\} \right. \\&\times \frac{Z_{k, r-\sum _{i=1}^{k-1} n_i}^{\ell } \phi ^h (Z_{k, r-\sum _{i=1}^{k-1} n_i}) e^{-q \sigma Z_{k, r-\sum _{i=1}^{k-1} n_i}}}{1-\Phi (Z_{k, r-\sum _{i=1}^{k-1} n_i})} \bigg | \tau _{k-1}< Y_{r:n}< \tau _k \Bigg \} P(\tau _{k-1}< Y_{r:n} < \tau _k)\\= & {} {\eta _k}^{\ell } \phi ^h (\eta _k) e^{-q \sigma \eta _k} P(Y_{r:n}> \tau _k) +\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)}\\&\times \int ^{\tau _k}_{\tau _{k-1}} z_{k, r-\sum _{i=1}^{k-1} n_i}^{\ell } \phi ^h (z_{k, r-\sum _{i=1}^{k-1} n_i}) e^{-q \sigma z_{k, r-\sum _{i=1}^{k-1} n_i}}\\&\times \left[ \frac{G_k (y)-G_{k-1} (\tau _{k-1})}{1-G_{k-1} (\tau _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-G_k (y)}{1-G_{k-1} (\tau _{k-1})}\right] ^{n-r} \frac{g_k (y)}{1-G_{k-1} (\tau _{k-1})} \ dy\\= & {} {\eta _k}^{\ell } \phi ^h (\eta _k) e^{-q \sigma \eta _k} P(Y_{r:n}> \tau _k)\\&+\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)} \int ^{\eta _k}_{\eta _{k-1}} w^{\ell } \phi ^{h} (w) e^{-q \sigma w}\\&\times \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw,\\ H^*_{q, {\ell }}= & {} {\eta _k}^{\ell } e^{-q \sigma \eta _k} \left[ \frac{d \phi (\eta _k)}{d \eta _k}+ \frac{\phi ^2 (\eta _k)}{1-\Phi (\eta _k)} \right] P(Y_{r:n} > \tau _k)\\&+\left[ r-\sum _{i=1}^{k-1} E(n_i) \right] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)} \int ^{\eta _k}_{\eta _{k-1}} w^{\ell } e^{-q \sigma w} \left[ \frac{d \phi (w)}{d w}+ \frac{\phi ^2 (w)}{1-\Phi (w)} \right] \\&\times \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw, \end{aligned}$$

and

$$\begin{aligned} P(Y_{r:n} > \tau _k)= & {} \bigg [ r-\sum _{i=1}^{k-1} E(n_i) \bigg ] {n-\sum _{i=1}^{k-1} E(n_i) \atopwithdelims ()r-\sum _{i=1}^{k-1} E(n_i)}\int ^{\infty }_{\eta _k} \left[ \frac{\Phi (w)-\Phi (\eta _{k-1})}{1-\Phi (\eta _{k-1})}\right] ^{r-\sum _{i=1}^{k-1} E(n_i)-1} \\&\times \left[ \frac{1-\Phi (w)}{1-\Phi (\eta _{k-1})}\right] ^{n-r} \frac{\phi (w)}{1-\Phi (\eta _{k-1})} \ dw. \end{aligned}$$

Hence, the entries of the Fisher information matrix are as follows:

$$\begin{aligned} I_{11}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} A_{i,0,0}+ A^*_{0,0}+ H^*_{0,0} \right\} ,\\ I_{12}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} x_i A_{i,0,0}+ x_k A^*_{0,0} -\sum _{i=2}^{k-1} A_{i,1,0} \sum _{h=2}^{i} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h} \right. \\&\left. -\ A^*_{1,0} \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h} + x_k H^*_{0,0}{-} H^*_{1,0} \sum _{h=2}^{k} (x_{h} - x_{h-1}) s_{h-1} e^{-\mu _h} \right\} \!,\\ I_{13}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} A_{i,0,1}+ A^*_{0,1}+ H^*_{0,1} \right\} ,\\ I_{22}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} x_i^2 A_{i,0,0} +x_k^2 A^*_{0,0} -2 \sum _{i=2}^{k-1} x_i A_{i,1,0} \sum _{h=2}^{i} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h} \right. \\&-\ 2 x_k A^*_{1,0} \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h}\\&+\ \sum _{i=2}^{k-1} A_{i,2,0} \left[ \sum _{h=2}^{i} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h}\right] ^2\\&+A^*_{2,0} \left[ \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h}\right] ^2\\&-\ \sigma \sum _{i=2}^{k-1} (B_{i,1,0}- \sigma C_{i,1}) \sum _{h=2}^{i} \sum _{\ell =2}^{h} (x_{h} -x_{h-1}) (x_{\ell } -x_{\ell -1}) s_{\ell -1} e^{ -\mu _\ell }\\&-\ \sigma (B^*_{1,0}- \sigma C^*_{1}- D^*_{1,0,1}) \sum _{h=2}^{k} \sum _{\ell =2}^{h} (x_{h} -x_{h-1}) (x_{\ell } -x_{\ell -1}) s_{\ell -1} e^{ -\mu _\ell }\\&-\ \sigma \sum _{i=2}^{k-1} (B_{i,1,0}- \sigma C_{i,1}) \sum _{h=2}^{i} (x_{h} -x_{h-1}) (x_{i} -x_{h}) s_{h-1} e^{ -\mu _h}\\&-\ \sigma (B^*_{1,0}- \sigma C^*_{1}- D^*_{1,0,1}) \sum _{h=2}^{k} (x_{h} -x_{h-1}) (x_{k} -x_{h}) s_{h-1} e^{ -\mu _h}\\&+\ \sigma \sum _{i=2}^{k-1} (B_{i,2,0}- \sigma C_{i,2}) \left[ \sum _{h=2}^{i} (x_{h} -x_{h-1}) s_{h-1} e^{ -\mu _h} \right] ^2\\&+\ \sigma (B^*_{2,0}- \sigma C^*_{2}- D^*_{2,0,1}) \left[ \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{ -\mu _h} \right] ^2\\&+\ x_k^2 H^*_{0,0} -2x_k H^*_{1,0} \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _{h}}\\&\left. +H^*_{2,0} \left[ \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _{h}}\right] ^2 \right\} ,\\ I_{23}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} x_i A_{i,0,1} +x_k A^*_{0,1} -\sum _{i=2}^{k-1} (A_{i,1,1}- \sigma C_{i,1}) \sum _{h=2}^{i} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h} \right. \\&-\ (A^*_{1,1}- \sigma C^*_{1}) \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _h} +x_k H^*_{0,1}\\&\left. - H^*_{1,1} \sum _{h=2}^{k} (x_{h} -x_{h-1}) s_{h-1} e^{-\mu _{h}} \right\} ,\\ \text{ and } \\ I_{33}= & {} \frac{n}{\sigma ^2} \left\{ \sum _{i=1}^{k-1} (A_{i,0,2}-B_{i,0,1})+ A^*_{0,2}- B^*_{0,1}+ D^*_{0,1,1}+ H^*_{0,2} \right\} . \end{aligned}$$