Exact likelihood inference for the two-parameter exponential distribution under Type-II progressively hybrid censoring

Abstract

Hybrid censoring schemes are commonly used in life-testing experiments to reduce the experimental time and the cost. A Type-II progressive hybrid censoring scheme (PHCS) was introduced by Kundu and Joarder (Comput Stat Data Anal 50:2509–2528, 2006) that combines progressive Type-II censoring and Type-I censoring. In this paper, we consider the statistical inference of a two-parameter exponential distribution under the Type-II PHCS. The conditional maximum likelihood estimates (MLEs) of the model parameters and their joint and marginal conditional moment generating functions are derived. Based on these exact conditional moments, bias-reduced estimators are proposed and their distributions are discussed. Confidence intervals of the model parameters based on exact and asymptotic distributions of the MLEs and bias-reduced estimators are developed. The performances of the point and interval estimation procedures are evaluated and compared through exact calculations and Monte Carlo simulations. Recommendations are made based on these results and an illustrative example is presented.

Appendix: Proof of Theorem 1

The explicit expression of the conditional mgf of ${\hat{\theta }}$ and ${\hat{\mu }}$ can be derived by using the following lemmas.

Lemma 1

$$\begin{aligned} \int \limits _{0<w_0<\cdots <w_{m}<T} \text{ e }^{-\sum \limits _{i=1}^ma_iw_i}dw_1\cdots dw_{m} =\sum \limits _{l=0}^m c_l^{(m)}\text{ e }^{-\sum \limits _{i=m-l+1}^ma_iT}. \end{aligned}$$

and

$$\begin{aligned} c_0^{(m)}&= \prod \limits _{j=1}^{m}\frac{1}{\sum \limits _{i=j}^{m}a_i}\\ c_l^{(m)}&= \frac{(-1)^l}{\prod \nolimits _{j=0}^{l-1}\sum \nolimits _{i=m-l+1}^{m-j}a_i}\prod \limits _{j=1}^{m-l}\frac{1}{\sum \nolimits _{i=j}^{m-l}a_i}, \quad \text{ for } 1\le l\le m. \end{aligned}$$

Proof

See Lemma 1 in Balakrishnan et al. (2002). $\square$

Lemma 2

$$\begin{aligned}&c_m\theta ^{-m}\text{ e }^{s\mu } \nonumber \\&\times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{m}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{m}\right) \sum \limits _{i=2}^m(1+R_i)x_i- \left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s \right) x_1}dx_1\cdots dx_{m}\nonumber \\&= p_0\left( 1-\frac{\theta t}{m}\right) ^{-m+1}\text{ e }^{\mu s}\left( 1- \frac{\theta s}{n} \right) ^{-1}\nonumber \\&+\sum \limits _{l=1}^{m-1} p_l\left( 1-\frac{\theta t}{m}\right) ^{-m}\left[ 1+\frac{\theta (R_{m-l}^*t-ms)}{m(n-R_{m-l}^*)}\right] ^{-1}\text{ e }^{T_l^*t+\mu s}\nonumber \\&+p_m\text{ e }^{Ts}\prod \limits _{j=1}^{m}\left[ \frac{1}{\theta }+\frac{R_{m-j}^*t-ms}{m(n-R_{m-j}^*)}\right] ^{-1}, \end{aligned}$$

$p_{l}$ and $T^*_{l}$ are defined by (7). $\square$

Proof

By using Lemma 1, we have

$$\begin{aligned}&c_m\theta ^{-m}\text{ e }^{s\mu } \\&\times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{m}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{m}\right) \sum \nolimits _{i=2}^m(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s\right) x_1}dx_1\cdots dx_{m}\\&= c_m\theta ^{-m}\text{ e }^{s\mu } \sum \limits _{l=0}^m c_l^{(m)}\text{ e }^{-\sum \limits _{i=m-l+1}^ma_i(T-\mu )}, \end{aligned}$$

where $a_1 = \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s, a_i = \left( \frac{1}{\theta }-\frac{t}{m}\right) (1+R_i), \text{ for } 2\le i\le m.$ For $j=1$, we have

$$\begin{aligned} \sum \limits _{i=1}^{m}a_i&= {\left( \frac{1}{\theta }-\frac{t}{m} \right) }\sum \limits _{i=2}^{m}(1+R_i)+\frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s = \frac{n}{\theta }-s \end{aligned}$$

and for $j \ge 2$, we have $\sum \nolimits _{i=j}^{m}a_i = \left( \frac{1}{\theta }-\frac{t}{m} \right) \sum \nolimits _{i=j}^{m}(1+R_i) =\left( \frac{1}{\theta }-\frac{t}{m} \right) R_{j-1}^*.$ Thus,

$$\begin{aligned} c_0^{(m)}&= \prod \limits _{j=1}^{m}\frac{1}{\sum \nolimits _{i=j}^{m}a_i} ={\left\{ \prod \limits _{j=0}^{m-1}R_{j}^*\right\} ^{-1}}{\left( \frac{1}{\theta }-\frac{t}{m}\right) ^{-m+1}}{\left( \frac{1}{\theta }- \frac{s}{n}\right) ^{-1}, }\\ c_l^{(m)}&= \frac{(-1)^l}{\prod \limits _{j=0}^{l-1}\sum \nolimits _{i=m-l+1}^{m-j}a_i}\prod \limits _{j=1}^{m-l}\frac{1}{\sum \nolimits _{i=j}^{m-l}a_i}\\&= \frac{(-1)^l}{\prod \nolimits _{j=0}^{l-1}(\frac{1}{\theta }-\frac{t}{m})(R_{m-l}^*-R_{m-j}^*)} \prod \limits _{j=2}^{m-l}\frac{1}{(\frac{1}{\theta }-\frac{t}{m})(R_{j-1}^*-R_{m-l}^*)}\\&\times \frac{1}{\frac{n-R_{m-l}^*}{\theta }+\frac{R_{m-l}^*t}{m}-s}\\&= {\left\{ \prod \limits _{\mathop {j\ne m-l}\limits ^{j=0}}^{m}(R_{j}^*-R_{m-l}^*)\right\} ^{-1}} \\&\times {\left( \frac{1}{\theta }-\frac{t}{m}\right) ^{-m+1}}{\left[ \frac{1}{\theta }+\frac{R_{m-l}^*t-ms}{m(n-R_{m-l}^*)}\right] ^{-1}}, { \text{ for } } 1\le l< m,\\ c_m^{(m)}&= \frac{(-1)^m}{\prod \nolimits _{j=0}^{m-1}\sum \nolimits _{i=1}^{m-j}a_i}\\&= \frac{(-1)^m}{\prod \nolimits _{j=0}^{m-1}\left( \frac{n-R_{m-j}^*}{\theta }+\frac{R_{m-j}^*t}{m}-s \right) }\\&= {\left\{ \prod \limits _{j=1}^{m}(R_{j}^*-n)\right\} ^{-1} \prod \limits _{j=0}^{m-1}\left\{ \frac{1}{\theta }+\frac{R_{m-j}^*t-ms}{m(n-R_{m-j}^*)}\right\} ^{-1}.} \end{aligned}$$

Then, we can write

$$\begin{aligned}&c_m\theta ^{-m}\text{ e }^{s\mu } \\&\times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{m}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{m}\right) \sum \nolimits _{i=2}^m(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s \right) x_1}dx_1\cdots dx_{m}\\&= c_m\theta ^{-m}\text{ e }^{s\mu } \sum \limits _{l=0}^{m}c_l^{(m)}\text{ e }^{-\sum \nolimits _{i=m-l+1}^ma_i(T-\mu )}\\&= c_m\theta ^{-m}\text{ e }^{s\mu } c_0^{(m)}+\sum \limits _{l=1}^{m-1} c_m\theta ^{-m} c_l^{(m)}\text{ e }^{-(\frac{1}{\theta }-\frac{t}{m})R_{m-l}^*(T-\mu )+\mu s}\\&+c_m\theta ^{-m}c_m^{(m)}\text{ e }^{-\frac{n}{\theta }(T-\mu )+Ts} \\&= p_0\left( 1-\frac{\theta t}{m}\right) ^{-m+1}\text{ e }^{\mu s}\left( 1- \frac{\theta s}{n}\right) ^{-1}\\&+\sum \limits _{l=1}^{m-1} p_l\left( 1-\frac{\theta t}{m}\right) ^{-m}\left( 1+\frac{\theta (R_{m-l}^*t-ms)}{m(n-R_{m-l}^*)}\right) ^{-1}\text{ e }^{T_l^*t+\mu s}\\&+p_m\text{ e }^{Ts}\prod \limits _{j=0}^{m-1}\left\{ \frac{1}{\theta }+\frac{R_{m-j}^*t-ms}{m(n-R_{m-j}^*)}\right\} ^{-1}. \end{aligned}$$

$\square$

Lemma 3

$$\begin{aligned}&\sum \limits _{j=1}^{m-1} c_{j}\theta ^{-j} \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) R_j^*(T-\mu )+s\mu }\nonumber \\&\times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{j}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) \sum \nolimits _{i=2}^{j}(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{j}-s\right) x_1} dx_1\cdots dx_{j}\nonumber \\&= \sum \limits _{j=1}^{m-1} p_{j,0} \text{ e }^{T^*_{j,0}t+s\mu }\left( 1-\frac{\theta t}{j}\right) ^{-j+1} \left[ 1+\frac{\theta ({R_{j}^*t}-js)}{j(n-R_{j}^*)}\right] ^{-1}\nonumber \\&+\sum \limits _{j=1}^{m-1} \sum \limits _{l=1}^{j-1} p_{j,l} \text{ e }^{T^*_{j,l}t+s\mu } \left( 1-\frac{\theta t}{j}\right) ^{-j+1} \left[ 1+\frac{\theta ({R_{j-l}^*t}-js)}{j(n-R_{j-l}^*)}\right] ^{-1}\nonumber \\&+\sum \limits _{j=1}^{m-1}p_{j,j}\text{ e }^{Ts}\prod \limits _{j'=1}^{j}\left[ 1+\frac{\theta ({R_{j'}^*t}-js)}{j(n-R_{j'}^*)}\right] ^{-1} \end{aligned}$$

$p_{j,l}$ and $T^*_{j,l}$ are defined by (8). $\square$

Proof

By using Lemma 1, we have

where $a_1 = \frac{1+R_1}{\theta }+\frac{R^*_1t}{j}-s, a_i = \left( \frac{1}{\theta }-\frac{t}{j} \right) (1+R_i), \text{ for } 2\le i\le j.$ For $j'=1$, we have $\sum \nolimits _{i=1}^{j}a_i = \frac{1}{\theta }(n-R_{j}^*)+\frac{R_{j}^*t}{j}-s$, and for $j'\ge 2$, we have $\sum \nolimits _{i=j'}^{j}a_i = \left( \frac{1}{\theta }-\frac{t}{j} \right) (R_{j'-1}^*-R_{j}^*).$ Thus,

$$\begin{aligned} c_0^{(j)}&= \prod \limits _{j'=1}^{j}\frac{1}{\sum \nolimits _{i=j'}^{j}a_i} = {\left\{ \prod \limits _{j'=0}^{j-1}(R_{j'}^*-R_{j}^*)\right\} ^{-1}\left( \frac{1}{\theta }-\frac{t}{j} \right) ^{-j+1} \left( \frac{1}{\theta }+\frac{{R_{j}^*t}-js}{j(n-R_{j}^*)}\right) ^{-1}}, \\ c_l^{(j)}&= \frac{(-1)^l}{\prod \nolimits _{j'=0}^{l-1}\sum \nolimits _{i=j-l+1}^{j-j'}a_i} \prod \limits _{j'=1}^{j-l}\frac{1}{\sum \nolimits _{i=j'}^{j-l}a_i}\\&= {\left\{ \prod \limits _{\mathop {j'\ne j-l}\limits ^{j'=0}}^{j}(R_{j'}^*-R_{j-l}^*)\right\} ^{-1}} \\&\times \left( \frac{1}{\theta }-\frac{t}{j}\right) ^{-j+1}{\left( \frac{1}{\theta }+\frac{{R_{j-l}^*t}-js}{j (n-R_{j-l}^*)}\right) ^{-1}}, \text{ for } 1 \le l \le j-1, \\ c_j^{(j)}&= \frac{(-1)^j}{\prod \nolimits _{j'=0}^{j-1}\sum \nolimits _{i=1}^{j-j'}a_i}\\&= {\left\{ \prod \limits _{j'=1}^{j}(R_{j'}^*-n)\right\} ^{-1} \prod \limits _{j'=1}^{j}\left( \frac{1}{\theta }+\frac{{R_{j'}^*t}-js}{j(n-R_{j'}^*)}\right) ^{-1}.} \end{aligned}$$

Therefore,

$$\begin{aligned}&\sum \limits _{j=1}^{m-1} c_{j}\theta ^{-j} \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) R_j^*(T-\mu )+s\mu }\nonumber \\&\quad \quad \times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{j}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) \sum \nolimits _{i=2}^{j}(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{j}-s \right) x_1} dx_1\cdots dx_{j}\nonumber \\&\quad =\sum \limits _{j=1}^{m-1} p_{j,0} \text{ e }^{T^*_{j,0}t+s\mu }\left( 1-\frac{\theta t}{j}\right) ^{-j+1} \left( 1+\frac{\theta ({R_{j}^*t}-js)}{j(n-R_{j}^*)}\right) ^{-1}\nonumber \\&\quad \quad +\sum \limits _{j=1}^{m-1} \sum \limits _{l=1}^{j-1} p_{j,l} \text{ e }^{T^*_{j,l}t+s\mu } \left( 1-\frac{\theta t}{j}\right) ^{-j+1} \left[ 1+\frac{\theta ({R_{j-l}^*t}-js)}{j(n-R_{j-l}^*)}\right] ^{-1}\nonumber \\&\quad \quad +\sum \limits _{j=1}^{m-1}p_{j,j}\text{ e }^{Ts}\prod \limits _{j'=1}^{j}\left[ 1+\frac{\theta ({R_{j'}^*t}-js)}{j(n-R_{j'}^*)}\right] ^{-1}. \end{aligned}$$

$\square$

Proof of Theorem 1

Condition on the values of $D$ according to the different situations in Eq. (3), we have

$$\begin{aligned}&E\left[ \text{ e }^{t{\hat{\theta }}+s{\hat{\mu }}}|D>0\right] P\left( D>0\right) \nonumber \\&\quad = E\left[ \text{ e }^{t{\hat{\theta }}+s{\hat{\mu }}}|D\ge m\right] P\left( D\ge m\right) +\sum \limits _{j=1}^{m-1}E\left[ \text{ e }^{t{\hat{\theta }}+s{\hat{\mu }}}|D=j\right] P\left( D=j\right) \nonumber \\&\quad =\frac{c_m}{\theta ^{m}} \nonumber \\&\quad \quad \times \int \limits _{\mu \le x_1\le x_2\le \cdots \le x_{m}<T} \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{m}\right) \sum \limits _{i=2}^m(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s \right) x_1+\frac{n\mu }{\theta }}dx_1\cdots dx_{m}\nonumber \\&\quad \quad +\sum \limits _{j=1}^{m-1} c_{j}\theta ^{-j}\text{ e }^{\frac{n\mu }{\theta }-(\frac{1}{\theta }-\frac{t}{j})R_j^*T}\nonumber \\&\quad \quad \times \int \limits _{\mu \le x_1\le x_2\le \cdots \le x_{j}<T} \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) \sum \limits _{i=2}^{j}(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{j}-s \right) x_1} dx_1\cdots dx_{j}.\nonumber \\&\quad =c_m\theta ^{-m}\text{ e }^{s\mu }\nonumber \\&\quad \quad \times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{m}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{m}\right) \sum \nolimits _{i=2}^m(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{m}-s \right) x_1}dx_1\cdots dx_{m}\nonumber \\&\quad \quad +\sum \limits _{j=1}^{m-1} c_{j}\theta ^{-j} \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) R_j^*(T-\mu )+s\mu }\nonumber \\&\quad \quad \times \int \limits _{0\le x_1\le x_2\le \cdots \le x_{j}<T-\mu } \text{ e }^{-\left( \frac{1}{\theta }-\frac{t}{j}\right) \sum \nolimits _{i=2}^{j}(1+R_i)x_i-\left( \frac{1+R_1}{\theta }+\frac{R^*_1t}{j}-s \right) x_1} dx_1\cdots dx_{j}.\nonumber \\ \end{aligned}$$

(30)

Upon carrying out the integrations in the Eq. (30) by using Lemmas 2 and 3, we obtain the formula in Eq. (6) and hence Theorem 1 is proved. $\square$