Exact Inference and Prediction for Exponential Models Under General Progressive Censoring with Application to Tire Wear Data

Abstract

General progressive Type-II censoring is widely applied in life-testing experiments to enhance efficiency by allowing early removal of surviving units, thereby reducing experimental time and cost. This paper develops exact inference and prediction procedures for one- and two-parameter exponential models based on multiple independent general progressively Type-II censored samples. Using the recursive algorithm repeatedly, exact confidence intervals for model parameters and exact prediction intervals for unobserved failure times are constructed. The proposed methods are illustrated with simulated and real (tire wear) data, demonstrating their practical applicability to partially censored reliability experiments.

1. Introduction

General progressive Type-II censoring schemes are widely employed in clinical trials and life-testing experiments because they permit early removal of live units, improving efficiency and conserving both time and resources (see Sen [1], Balakrishnan et al. [2]). In clinical trials, such censoring accommodates partially observed outcomes. For instance, Bhattacharya [3] illustrates a study in which patients with squamous carcinoma of the oropharynx are grouped by lymph node deterioration, with greater deterioration generally leading to shorter survival times. Failures may remain unobserved at the study onset, and patients can be censored as the trial progresses because of improvement, relocation, or other study-specific considerations. Similarly, in industrial experiments and accelerated life tests, such as tire wear studies (Bain and Engelhardt [4]), multiple-sample (K-sample) general progressive Type II censoring naturally arises when comparing products from different production batches or assessing performance under several stress levels, leading to a K-sample general progressive Type II censoring setup. This setting allows flexible termination of tests while efficiently utilizing limited testing resources and retaining informative data.

Extensive research on general progressive Type-II censoring has been conducted for a wide range of lifetime distributions, including the exponential, Gompertz, Pareto, generalized Pareto, Weibull, inverse Weibull, Rayleigh, and scal-family models (Balakrishnan and Sandhu [5], Balakrishnan et al. [2], Balakrishnan and Lin  [6], Fernandez [7], Bhattacharya [3], Soliman [8], Kim and Han [9], Soliman et al. [10], Wang [11], Peng and Yan [12], Rashad [13], Yan et al. [14], and Wang and Gui [15]). Comprehensive reviews of progressive censoring methodologies and their diverse applications are provided by Balakrishnan and Aggarwala [16], Balakrishnan [17], and Balakrishnan and Cramer [18].

Despite these advances, only a limited number of studies, among them Balakrishnan and Sandhu [5], Balakrishnan et al. [2], Bhattacharya [3], and Yan et al. [14], have addressed the K-sample general progressive Type-II censoring framework. These works have concentrated mainly on estimation, developing best linear unbiased estimators (BLUE) and maximum likelihood estimators (MLE) for exponential lifetime models, as well as MLE and Bayesian estimators for Weibull regression models in multi-level stress testing. By comparison, because the joint distribution of pivotal quantities across samples is intractable, prediction problems under such censoring schemes have received limited attention, and exact distributional results for predictive inference remain unavailable. This motivates the present study, which aims to develop exact, unified, and computationally efficient procedures for inference and prediction in exponential models under complex multi-sample censoring structures.

The exponential distribution is a fundamental model in life-testing and reliability analysis due to its analytical tractability and desirable stochastic properties (Nelson [19]; Lawless [20]; Bain and Engelhardt [4]; Johnson et al. [21]; Balakrishnan and Basu [22]; Meeker and Escobar [23]). A notable feature is that the normalized spacings from an exponential sample are independent and identically distributed exponential random variables (David [24]; Arnold et al. [25]), which facilitates the development of exact inference procedures. Specifically, exact chi-square confidence intervals for the scale parameter, exact F-based intervals for the location parameter in the two-parameter exponential model, and exact prediction intervals for future failure times can all be derived under Type-II right censoring (Lawless [26,27]; Likeš [28]; Lingappaiah [29]). Comprehensive reviews of prediction methods for exponential models under the one-sample Type-II right censoring framework are provided by Patel [30], Kaminsky and Nelson [31], and Nagaraja [32].

For more complex censoring schemes, such as doubly Type-II, K-sample doubly Type-II, or general progressive Type-II censoring, Lin and Balakrishnan [33], Balakrishnan et al. [34], and Balakrishnan and Lin [6] extended exact inference for exponential models by employing the recursive algorithm of Huffer and Lin [35] to obtain exact distributions of pivotal quantities and confidence intervals for one- and two-parameter settings. They also developed exact prediction intervals for units censored at the final failure time. However, for the K-sample general progressive Type-II censoring scheme, no exact distributional results for prediction problems have yet been established. Moreover, the core computational codes from earlier studies, which were based on different formulas and derivation structures, cannot be directly adapted to the present censoring framework, leaving a significant methodological gap.

Building upon the BLUE framework proposed by Balakrishnan et al. [2] and the recursive algorithms of Huffer and Lin [35], the present study addresses this gap by developing exact inference and prediction procedures for both one- and two-parameter exponential models under multi-sample general progressive Type-II censoring. The new MAPLE implementation not only unifies but also generalizes the computational procedures of prior studies while correcting minor inaccuracies in earlier one-sample implementations. By making use of the recursive structure of Huffer and Lin [35] and the properties of normalized spacings, the exact distributions of pivotal statistics can be efficiently obtained, providing a rigorous basis for constructing exact confidence and prediction intervals in complex multi-sample censoring frameworks.

To demonstrate the practical utility of the proposed methodology, we further apply it to both simulated data and a real dataset from tire wear studies. Through these two examples, we illustrate how the developed procedures can yield exact confidence and prediction intervals under multiple-sample general progressive Type-II censoring, thereby confirming their effectiveness and potential for applications in reliability and life-testing experiments.

The remainder of this paper is organized as follows. Section 2 introduces the general framework of progressive Type-II censoring. Section 3 reviews the computational algorithm proposed by Huffer and Lin [35]. Section 4 summarizes the results of Balakrishnan et al. [2] on the BLUEs for one- and two-parameter exponential distributions based on K independent progressively Type-II censored samples. Section 5 presents our new developments, building upon the work of Balakrishnan et al. [2] and utilizing the properties of normalized spacings to construct exact confidence intervals for the BLUEs and exact prediction intervals for future failure times. Section 6 illustrates the derivation of the exact distribution of the BLUE for the scale parameter discussed in Section 5, employing the recursive algorithm of Huffer and Lin [35]. Finally, Section 7 demonstrates the proposed methods using a simulated example and a real dataset from tire wear studies, while Appendix A provides the implementation of MAPLE codes for computing the exact distributional results presented in Section 6.

2. General Progressive Type-II Censoring

We first formalize the general progressive Type-II censoring scheme, which serves as the foundation for the inference and prediction procedures developed in later sections. Suppose n units are randomly selected from a lifetime distribution $F\left(x\right)$ and placed on test. The failure times of the first r units to fail are not observed. At the time of the $(r+1)$th failure, $R_{r+1}$ surviving units are randomly withdrawn (censored) from the test. Similarly, at the $(r+i)$th failure $(1\le i\le m-r)$, $R_{r+i}$ surviving units are randomly removed. Finally, at the time of the mth failure, the remaining units are withdrawn, where

$$R_m=n-m-R_{r+1}-R_{r+2}-\cdots -R_{m-1}.$$

The completely observed lifetimes, denoted by

$$X_{r+1:m:n}^{(R_{r+1},\dots ,R_m)}\le X_{r+2:m:n}^{(R_{r+1},\dots ,R_m)}\le \cdots \le X_{m:m:n}^{(R_{r+1},\dots ,R_m)},$$

are called the general progressively Type-II censored order statistics from $F\left(x\right)$ corresponding to a sample of size n under the censoring scheme $(R_{r+1},\dots ,R_m)$. This formulation includes the standard progressively. Type-II censoring scheme as the special case $r=0$.

3. Algorithm

This work is largely motivated by the algorithm proposed by Huffer and Lin [35]. For clarity, we briefly describe their algorithm, as it forms the foundation of our subsequent developments.

Suppose that $Z_1, Z_2, \dots , Z_{n+1}$ are independent and identically distributed (i.i.d.) standard exponential random variables, and define ${\mathit{Z}}^{\left(n\right)}={(Z_1,Z_2,\dots ,Z_{n+1})}^T$. Huffer and Lin [35] developed an algorithm to evaluate probabilities involving linear combinations of i.i.d. exponential random variables with arbitrary rational coefficients, of the form

$$P({\mathit{A}\mathit{Z}}^{\left(n\right)}>t\mathit{b}),$$

(1)

where A is a matrix with rational entries, b is a vector with rational elements, and $t>0$ is a scalar.

The algorithm is based on the systematic use of two recursions (given in Equations (3) and (4)) involving a function Q, defined as

$$Q(\mathit{A},\mathit{b},\lambda ,p)=p!R(p,\lambda )P\left((1-\lambda t)\mathit{A}{\mathit{Z}}^{(n-p)}>t\mathit{b}\right),$$

where

$$R(p,\lambda )={\displaystyle \frac{t^p{e}^{-\lambda t}}{p!}},p\ge 0,\lambda \ge 0.$$

The dependence of Q on n and t, and of R on t, is left implicit. In particular,

$$Q(Ø,Ø,\lambda ,p)=p!R(p,\lambda ).$$

(2)

Result 1.

Let A be an arbitrary matrix with r rows and q columns. For any $r\times 1$ vector x, denote by $A_{i,x}$ the matrix obtained by replacing the ith column of A with x. Let $\mathit{c}={(c_1,\dots ,c_q)}^T$ be any $q\times 1$ vector satisfying ${\sum }_{i=1}^q{c}_i=1$, and define $\mathit{\xi }=\mathit{A}\mathit{c}$. Then

$$Q(\mathit{A},\mathit{b},\lambda ,p)=\sum _{i=1}^q{c}_{i}Q({\mathit{A}}_{i,\xi },\mathit{b},\lambda ,p).$$

(3)

This recursion is an immediate consequence of the more general recursion given by Huffer [36]. See also Huffer and Lin [35,37], as well as the references therein, for various applications of this general recursion.

Result 2.

Let $\mathit{A}=\left(a_{ij}\right)$ and $\mathit{b}=\left(b_j\right)$ satisfy the following conditions for some $k\ge 1$:

(R1) 

$a_{1j}=0$ for $j>k$,

(R2) 

$a_{ij}=a_{i1}$ for $j\le k$ (i.e., the first k columns of A are identical),

(R3) 

$a_{11}>0$ and $b_1>0$.

Then

$$Q(\mathit{A},\mathit{b},\lambda ,p)=\sum _{i=0}^{k-1}{\displaystyle \frac{{\delta }^i}{i!}}Q({\mathit{A}}_{(-i)}^{*},{\mathit{b}}^{*}-\delta a^{*},\lambda +\delta ,p+i),$$

(4)

where $\delta =b_1/a_{11}$, ${\mathit{A}}^{*}$ is the matrix obtained by deleting the first row of A, ${\mathit{A}}_{(-i)}^{*}$ is obtained by deleting the first i columns of ${\mathit{A}}^{*}$, ${\mathit{b}}^{*}$ is the vector obtained by deleting the first entry of b, and ${\mathit{a}}^{*}$ is the vector obtained by taking the first column of A and deleting the first entry.

See Huffer and Lin [35] for a proof of this recursion. Details on examples of the tools and their applications can be found in Huffer and Lin [35,37] and Lin et al. [38]. The survival functions of these test statistics in terms of probabilities of linear combinations of spacings, which are the successive differences between the order statistics, can be found in Lin et al. [38].

In our specific problem, Equation (4) can be further simplified by applying Equation (2):

$$Q(\mathit{A},1,0,0)=\sum _{i=0}^{k-1}{\displaystyle \frac{{\delta }^i}{i!}}Q(\varnothing ,\varnothing ,\delta ,i)=\sum _{i=0}^{k-1}{\displaystyle \frac{{\delta }^i}{i!}}i!R(i,\delta )=\sum _{i=0}^{k-1}{\delta }^{i}R(i,\delta ),$$

where A is a $1\times m$ matrix of rational values and $\delta =1/a_{11}$.

Each recursion re-expresses a probability of the form in Equation (1) as a sum of similar, but simpler, components. The recursions are applied iteratively until all components reduce to closed-form expressions.

4. The Best Linear Unbiased Estimators

In the previous section, we summarized the recursive algorithm of Huffer and Lin [35], which provides a powerful computational tool for evaluating probabilities associated with linear combinations of exponential random variables under general progressive censoring. Building on this framework, we now review the BLUEs and MLEs for the parameters of one- and two-parameter exponential distributions when multiple progressively Type-II censored samples are observed, as originally derived by Balakrishnan et al. [2]. These BLUEs form the foundation for subsequent inference and prediction in reliability studies.

We assume that we have observed data from K independent experiments with corresponding general progressive censoring schemes ${\mathit{R}}_i=(R_{i,r_i+1},R_{i,r_i+2},\dots ,R_{i,m_i})$ with sample sizes $n_i$ for $i=1, 2, \dots , K$. Denote the general progressively Type-II censored sample from the ith sample by

$${\mathit{Y}}_{\mathit{i}}=\{Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i},Y_{i;r_i+2:m_i:n_i}^{{\mathit{R}}_i},\dots ,Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}\}.$$

4.1. One-Parameter Exponential Case

For a one-parameter exponential distribution with probability density function (PDF)

$$f(y;\sigma )={\displaystyle \frac{1}{\sigma }}{e}^{-y/\sigma },y\ge 0,$$

with scale parameter $\sigma >0$, the BLUE of $\sigma$ is given by

$$\begin{array}{ccc}\hfill {\sigma }^{*}& =& {\displaystyle \frac{1}{{\sum }_{i=1}^K\left(m_i-r_i-1+\frac{{\alpha }_{r_i+1:n_i}^2}{{\beta }_{r_i+1:n_i}}\right)}}\left[\sum _{i=1}^K{\displaystyle \frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}}{Y}_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}\right.\hfill \\ & & \left.+\sum _{i=1}^K\sum _{j=r_i+2}^{m_i}(R_{i,j}+1)\left(Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}-Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}\right)\right]\hfill \end{array}$$

(5)

with variance

$$Var\left({\sigma }^{*}\right)={\displaystyle \frac{{\sigma }^2}{{\sum }_{i=1}^K\left(m_i-r_i-1+\frac{{\alpha }_{r_i+1:n_i}^2}{{\beta }_{r_i+1:n_i}}\right)}}$$

(6)

where

$${\alpha }_{r_i+1:n_i}=\sum _{j=1}^{r_i+1}{\displaystyle \frac{1}{n_i-j+1}} \mathrm{and} {\beta }_{r_i+1:n_i}=\sum _{j=1}^{r_i+1}{\displaystyle \frac{1}{{(n_i-j+1)}^2}},1\le i\le K.$$

The MLE of $\sigma$, denoted $\widehat{\sigma }$, is the solution to

$$\sum _{i=1}^K{\displaystyle \frac{r_i{Y}_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}}{exp\left(Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}/\widehat{\sigma }\right)-1}}+\widehat{\sigma }\sum _{i=1}^K(m_i-r_i)=\sum _{i=1}^K\sum _{j=r_i+1}^{m_i}(R_{i,j}+1)Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}$$

(7)

4.2. Two-Parameter Exponential Case

For a two-parameter exponential distribution with PDF

$$f(y;\mu ,\sigma )={\displaystyle \frac{1}{\sigma }}{e}^{-(y-\mu )/\sigma },y\ge \mu ,$$

with location parameter $\mu \in \mathbb{R}$ and scale parameter $\sigma >0$, the BLUEs of $\mu$ and $\sigma$ (for $m_i\ge r_i+2$) are

$$\begin{array}{ccc}\hfill {\mu }^{*}& =& {\displaystyle \frac{1}{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}}\left[\sum _{i=1}^K{\displaystyle \frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}{Y}_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}\right.\hfill \\ & & \left.-\sum _{i=1}^K{\displaystyle \frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}}\sum _{j=r_i+2}^{m_i}(R_{i,j}+1)\left(Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}-Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}\right)\right]\hfill \end{array}$$

(8)

and

$$\begin{array}{ccc}\hfill {\sigma }^{*}& =& {\displaystyle \frac{1}{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}}\sum _{i=1}^K{\displaystyle \frac{1}{{\beta }_{r_i+1:n_i}}}\sum _{j=r_i+2}^{m_i}(R_{i,j}+1)\left(Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}-Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}\right).\hfill \end{array}$$

(9)

For $K=1$, the MLEs reduce to the formulas in Balakrishnan and Sandhu [5]:

$$\widehat{\mu }=Y_{1;r_1+1:m_1:n_1}^{{\mathit{R}}_1}+\widehat{\sigma }ln\left(1-{\displaystyle \frac{r_1}{n_1}}\right),\widehat{\sigma }={\displaystyle \frac{1}{m_1-r_1}}\sum _{j=r_1+2}^{m_1}(R_{1,j}+1)\left(Y_{1;j:m_1:n_1}^{{\mathit{R}}_1}-Y_{1;r_1+1:m_1:n_1}^{{\mathit{R}}_1}\right).$$

For $K\ge 2$, the MLEs generally have no explicit closed forms and require numerical solutions.

The formulas of these BLUEs provide the necessary groundwork for the development of exact confidence and prediction procedures under multiple-sample general progressive Type-II censoring, which are presented in the next section.

5. The Exact Inference and Prediction

Based on the BLUEs reviewed in the previous section for the scale parameter in the one-parameter exponential model and for the location and scale parameters in the two-parameter exponential model, we now develop procedures for exact statistical inference and prediction, applicable to K independent progressively Type-II censored samples. The following subsections detail the one-parameter and two-parameter cases, where exact distributions are obtained using properties of normalized exponential spacings.

5.1. One-Parameter Exponential Case

For $i=1, 2, \dots , K$, let us denote $Y_{i;0:m_i:n_i}=0$ and define

$$\begin{array}{ccc}\hfill D_{i,j}& =& (n_i-j+1)\left(Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}-Y_{i;j-1:m_i:n_i}^{{\mathit{R}}_i}\right),j=1,2,\dots ,r_i+1,\hfill \\ \hfill W_{i,j}& =& \left(n_i-\sum _{\ell =r_i+1}^{j-1}{R}_{i,\ell }-j+1\right)\left(Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}-Y_{i;j-1:m_i:n_i}^{{\mathit{R}}_i}\right),j=r_i+2,\dots ,m_i.\hfill \end{array}$$

For $j=1, 2, \dots , r_i+1$, the $Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}$ denote the first $r_i+1$ order statistics from a sample of size $n_i$, with $D_{i,j}$ denoting the corresponding normalized exponential spacings; see [6,24,25]. For $j=r_i+2, \dots , m_i$, $W_{i,j}$ denote the normalized spacings from progressively Type-II censored samples; see  [5,6,39]. It follows that $D_{i,j}/\sigma$ ($j=1,\dots ,r_i+1$) and $W_{i,j}/\sigma$ ($j=r_i+2,\dots ,m_i$) are independent standard exponential random variables. Thus,

$$Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}=\sum _{j=1}^{r_i+1}{\displaystyle \frac{D_{i,j}}{n_i-j+1}},$$

and

$$Y_{i;j:m_i:n_i}^{{\mathit{R}}_i}=Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}+\sum _{h=r_i+2}^j{\displaystyle \frac{W_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}},j=r_i+2,\dots ,m_i.$$

Using these normalized spacings, the BLUE of the scale parameter $\sigma$ in Equation (5) can be expressed as

$${\sigma }^{*}={\displaystyle \frac{{\sum }_{i=1}^K\frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}{\sum }_{j=1}^{r_i+1}\frac{D_{i,j}}{n_i-j+1}+{\sum }_{i=1}^K{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{W_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\left(m_i-r_i-1+\frac{{\alpha }_{r_i+1:n_i}^2}{{\beta }_{r_i+1:n_i}}\right)}}.$$

The exact distribution of ${\sigma }^{*}$ can then be determined using the recursive algorithm described in Section 3. Specifically, for a given significance level $\alpha$, we determine t such that

$$\begin{array}{ccc}\hfill \alpha & =& P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>t\right)\hfill \\ & =& P\left({\displaystyle \frac{{\sum }_{i=1}^K\frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}{\sum }_{j=1}^{r_i+1}\frac{Z_{i,j}}{n_i-j+1}+{\sum }_{i=1}^K{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\left(m_i-r_i-1+\frac{{\alpha }_{r_i+1:n_i}^2}{{\beta }_{r_i+1:n_i}}\right)}}>t\right),\hfill \end{array}$$

(10)

where $Z_{i,j}$’s are independent standard exponential random variables for $i=1, \dots , K$ and $j=1, \dots , m_i$. Equivalently, for a given significance level $\alpha$, we can determine $t_1$ and $t_2$ such that

$$P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>t_1\right)={\displaystyle \frac{\alpha }{2}} \mathrm{and} P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>t_2\right)=1-{\displaystyle \frac{\alpha }{2}};$$

then, an exact $100(1-\alpha )\%$ confidence interval for $\sigma$ is $({\sigma }^{*}/t_1,{\sigma }^{*}/t_2)$.

Following Balakrishnan and Aggarwala [16], exact prediction intervals can also be constructed for the failure times of the last $R_{i,m_i}$ items censored at $Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}$ in the $i^{\mathrm{th}}$ sample. Specifically, for $s=1, \dots , R_{i,m_i}$, these unobserved failure times correspond to $Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}$, where

$${\mathit{R}}_i^{*}=(R_{i,r_i+1},\dots ,R_{i,m_i-1},\underset{s\mathrm{zeros}}{\underbrace{0,\dots ,0}},R_{i,m_i}-s),$$

and the previously observed data $Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}, \dots , Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}$ are treated as $Y_{i;r_i+1:m_i+s:n_i}^{{\mathit{R}}_i^{*}}, \dots$, $Y_{i;m_i:m_i+s:n_i}^{{\mathit{R}}_i^{*}}$. The exact value of t is determined from

$$\begin{array}{ccc}\hfill \alpha & =& P\left({\displaystyle \frac{Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}-Y_{i;m_i:m_i+s:n_i}^{{\mathit{R}}_i^{*}}}{{\sigma }^{*}}}>t\right)\hfill \\ & =& P\left({\displaystyle \frac{Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}-Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}}{{\sigma }^{*}}}>t\right)\hfill \\ & =& P(\sum _{j=m_i+1}^{m_i+s}{\displaystyle \frac{Z_{i,j}}{n_i-{\sum }_{\ell =r_i+1}^{m_i-1}{R}_{i,\ell }-h+1}}\hfill \\ & & -{\displaystyle \frac{{\sum }_{k=1}^K\frac{{\alpha }_{r_k+1:n_k}}{{\beta }_{r_k+1:n_k}}{\sum }_{j=1}^{r_k+1}\frac{Z_{k,j}}{n_k-j+1}+{\sum }_{k=1}^K{\sum }_{j=r_k+2}^{m_k}(R_{k,j}+1){\sum }_{h=r_k+2}^j\frac{Z_{k,h}}{n_k-{\sum }_{\ell =r_k+1}^{h-1}{R}_{k,\ell }-h+1}}{{\sum }_{k=1}^K\left(m_k-r_k-1+\frac{{\alpha }_{r_k+1:n_k}^2}{{\beta }_{r_k+1:n_k}}\right)}}t>0).\hfill \end{array}$$

(11)

Then, for a given significance level $\alpha$, values $t_1$ and $t_2$ are obtained such that

$$P\left({\displaystyle \frac{Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}-Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}}{{\sigma }^{*}}}>t_1\right)={\displaystyle \frac{\alpha }{2}},P\left({\displaystyle \frac{Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}-Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}}{{\sigma }^{*}}}>t_2\right)=1-{\displaystyle \frac{\alpha }{2}}.$$

An exact $100(1-\alpha )\%$ prediction interval for $Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}$ is then

$$\left(Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}+t_2{\sigma }^{*},Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}+t_1{\sigma }^{*}\right),$$

for $s=1, \dots , R_{i,m_i}$ and $i=1, \dots , K$. This provides exact prediction intervals for unobserved failure times under the general progressive Type-II censoring scheme, completing the inference framework for the one-parameter exponential model.

5.2. Two-Parameter Exponential Case

Analogous to the one-parameter case, let $Y_{i;0:m_i:n_i}=\mu$ and

$$Y_{i;r_i+1:m_i:n_i}^{{\mathit{R}}_i}=\mu +\sum _{j=1}^{r_i+1}{\displaystyle \frac{D_{i,j}}{n_i-j+1}}.$$

Then, using the property of normalized spacings, the pivot quantity $({\mu }^{*}-\mu )/{\sigma }^{*}$ can be expressed as

$$\begin{array}{ccc}& & {\displaystyle \frac{{\mu }^{*}-\mu }{{\sigma }^{*}}}\hfill \\ & =& {\displaystyle \frac{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=1}^{r_i+1}\frac{Z_{i,j}}{n_i-j+1}-{\sum }_{i=1}^K\frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\frac{1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}}\hfill \end{array}$$

and

$${\displaystyle \frac{{\sigma }^{*}}{\sigma }}={\displaystyle \frac{{\sum }_{i=1}^K\frac{1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}},$$

where $Z_{i,j}$’s are i.i.d. standard exponential random variables for $i=1, \dots , K$ and $j=1, \dots , m_i$.

Using these representations, the exact probabilities for inference on $\mu$ and $\sigma$ are obtained by finding t such that

$$\begin{array}{ccc}\hfill \alpha & =& P\left({\displaystyle \frac{{\mu }^{*}-\mu }{{\sigma }^{*}}}>t\right)\hfill \\ & =& P\left({\displaystyle \frac{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=1}^{r_i+1}\frac{Z_{i,j}}{n_i-j+1}-{\sum }_{i=1}^K\frac{{\alpha }_{r_i+1:n_i}}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\frac{1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}}>t\right)\hfill \\ & =& P\left(\sum _{i=1}^K{\displaystyle \frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}\sum _{j=1}^{r_i+1}{\displaystyle \frac{Z_{i,j}}{n_i-j+1}}-\sum _{i=1}^K{\displaystyle \frac{{\alpha }_{r_i+1:n_i}+t}{{\beta }_{r_i+1:n_i}}}\sum _{j=r_i+2}^{m_i}(R_{i,j}+1)\sum _{h=r_i+2}^j{\displaystyle \frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}>0\right)\hfill \end{array}$$

(12)

and

$$\alpha =P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>t\right)=P\left({\displaystyle \frac{{\sum }_{i=1}^K\frac{1}{{\beta }_{r_i+1:n_i}}{\sum }_{j=r_i+2}^{m_i}(R_{i,j}+1){\sum }_{h=r_i+2}^j\frac{Z_{i,h}}{n_i-{\sum }_{\ell =r_i+1}^{h-1}{R}_{i,\ell }-h+1}}{{\sum }_{i=1}^K\frac{m_i-r_i+1}{{\beta }_{r_i+1:n_i}}}}>t\right)$$

(13)

hold for a given significance level $\alpha$. This allows the construction of exact confidence intervals for both parameters under the general progressive Type-II censoring scheme.

Similarly, exact prediction intervals for the unobserved failure times among the last $R_{i,m_i}$ censored items in the i-th sample, $Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}$ for $s=1, \dots , R_{i,m_i}$, can be derived by determining t that satisfies

$$\begin{array}{ccc}\hfill \alpha & =& P\left({\displaystyle \frac{Y_{i;m_i+s:m_i+s:n_i}^{{\mathit{R}}_i^{*}}-Y_{i;m_i:m_i:n_i}^{{\mathit{R}}_i}}{{\sigma }^{*}}}>t\right)\hfill \\ & =& P(\sum _{j=m_i+1}^{m_i+s}{\displaystyle \frac{Z_{i,j}}{n_i-{\sum }_{\ell =r_i+1}^{m_i-1}{R}_{i,\ell }-h+1}}\hfill \\ & & -{\displaystyle \frac{t}{{\sum }_{k=1}^K\frac{m_k-r_k+1}{{\beta }_{r_k+1:n_k}}}}\sum _{k=1}^K{\displaystyle \frac{1}{{\beta }_{r_k+1:n_k}}}\sum _{j=r_k+2}^{m_k}(R_{k,j}+1)\sum _{h=r_k+2}^j{\displaystyle \frac{Z_{k,h}}{n_k-{\sum }_{\ell =r_k+1}^{h-1}{R}_{k,\ell }-h+1}}>0).\hfill \end{array}$$

(14)

These intervals provide rigorous probabilistic statements regarding the unobserved failure times, thereby completing the exact inference and prediction framework for the two-parameter exponential model.

6. Illustrative Example of Computational Implementation

We evaluate $P(\mathbf{A}{\mathbf{Z}}^{\left(n\right)}>t)=Q(\mathbf{A},1,0,0)$, where $\mathbf{A}$ is a $1\times m$ matrix with rational elements. The MAPLE implementation proceeds through a series of computational steps. For illustration, we consider a one-sample general progressive Type-II censored dataset with $n=19$, $m=8$, $r=1$, and $R=(-,0,3,0,3,0,0,5)$ in (1).

Under this censoring scheme, the detailed elements of $\mathbf{A}$ in Equation (3), together with the exact expression for evaluating the probabilities of the BLUE of $\sigma$ in Equation (10), are provided in Appendix A. The complete MAPLE implementation for the corresponding numerical evaluation is also included therein.

Using the procedure R(j,w), the resulting expression can be evaluated for any real value $t>0$, yielding an explicit function of t. By solving this function for t corresponding to a given significance level $\alpha$, the desired quantiles can be directly obtained. For example, using the commands tval(0.025) and tval(0.975), we obtain

$$P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>1.65521\right)=0.025,\mathrm{and}P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>0.53719\right)=0.975.$$

The computational times required to run the commands tval(0.025) and tval(0.975) are 0.280 and 1.310 s, respectively. To ensure high numerical precision, the command Digits is incorporated into the program; in our implementation, we set Digits: = 40.

7. Simulation Study and Application

To illustrate the practical applicability of our procedures, we analyze simulated data and a real dataset from a tire wear study, demonstrating how the proposed inference and prediction intervals can be implemented under multiple-sample progressive censoring.

7.1. Example 1

A three-sample general progressively Type-II censored dataset from a one-parameter exponential distribution with scale parameter $\sigma =100$ was simulated. The design parameters $(n_i,m_i,r_i)$, $i=1, 2, 3$, are given respectively as $(40,8,2)$, $(40,8,1)$, and $(40,8,1)$, and the corresponding censoring schemes ${\mathit{R}}_i$ are listed below:

j	1	2	3	4	5	6	7	8
Sample 1	–	–	12.89	12.97	19.15	19.50	26.63	27.27
${\mathit{R}}_1$	–	–	4	0	4	0	4	20
Sample 2	–	5.71	6.51	7.44	8.13	14.24	17.61	25.32
${\mathit{R}}_2$	–	3	0	5	0	4	0	20
Sample 3	–	1.57	1.72	2.84	10.40	17.09	18.80	19.16
${\mathit{R}}_3$	–	0	3	3	3	3	0	20

The BLUE of $\sigma$ is calculated as ${\sigma }^{*}=90.34350$, with variance $Var\left({\sigma }^{*}\right)={\sigma }^2/23.99804$, based on Equations (5) and (6). Using the recursions in Equations (3) and (4) in Equation (10), we obtain

$$P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>0.68770\right)=0.975\mathrm{and}P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>1.38133\right)=0.025,$$

so that the exact $95\%$ confidence interval for $\sigma$ is

$$\left[{\displaystyle \frac{{\sigma }^{*}}{1.38133}},{\displaystyle \frac{{\sigma }^{*}}{0.68770}}\right]=[65.40327,131.37051].$$

For comparison, the normal approximation gives

$$\left[{\displaystyle \frac{{\sigma }^{*}}{1+1.96/\sqrt{23.99804}}},{\displaystyle \frac{{\sigma }^{*}}{1-1.96/\sqrt{23.99804}}}\right]=[64.52648,150.59750],$$

which corresponds to an approximate confidence level of $97.5584\%$. It is also observed that the width of the exact confidence interval is $65.96724$, substantially narrower than that of the normal approximation ($86.07102$), indicating that the normal approximation tends to produce a wider interval. Additionally, it is worth noting that the MLE of $\sigma$ from Equation  (7) is $\widehat{\sigma }=90.34422$, which is almost identical to the BLUE.

For further comparison of the performance of the BLUE and MLE, alternative censoring schemes ${\mathit{R}}_1=(-,-,3,3,3,0,3,20)$, ${\mathit{R}}_2=(-,0,4,0,4,0,4,20)$, and ${\mathit{R}}_3=(-,2,2,0,3$, $3,2,20)$ are considered. Under the given censoring conditions, the BLUE and MLE of $\sigma$ are obtained as ${\sigma }^{*}=91.52901$ and $\widehat{\sigma }=91.52982$, respectively. Again, under similar censoring configurations, the MLE is nearly identical to the BLUE. The negligible difference between the two estimates indicates that moderate variations in the censoring pattern have only a minimal impact on estimation efficiency and bias.

Using the recursions in Equations (3) and (4), we obtain

$$P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>0.69202\right)=0.975\mathrm{and}P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>1.37600\right)=0.025.$$

Accordingly, the exact $95\%$ confidence interval for $\sigma$ is

$$[67.31887,133.85563],$$

while the corresponding normal approximation yields

$$[65.37321,152.57368],$$

with an approximate confidence level of $97.7289\%$.

These findings illustrate the importance of employing exact confidence intervals, as the normal approximation tends to yield wider ranges and inflated coverage probabilities. Furthermore, the results indicate that variations in censoring configurations can have a modest impact on both the point estimates and their associated exact confidence intervals.

Next, under the first censoring configuration, we further examine the prediction intervals for the last 20 unobserved failure times. Table 1 presents the exact values of t used to construct the prediction intervals for the censored items $Y_{i;8+j:8+j,40}$, given $Y_{i;8:8:40}$, for $i=1, 2, 3$ and $j=1, \dots , 20$, at significance levels $\alpha =0.005, 0.025, 0.05, 0.95, 0.975$, and $0.995$. These t-values are identical across the three samples because each sample involves the same number of observed failures and final removals.

Table 1. Exact values of t from Equation  (11) for constructing prediction intervals for the last 20 items $Y_{i;8+j:8+j,40}$ (censored at $Y_{i;8:8:40}$), with $i=1, 2, 3$ and $j=1, \dots , 20$, under various choices of $\alpha$ in the one-parameter exponential model.

j	0.995	0.975	0.95	0.05	0.025	0.005
1	0.000251	0.001266	0.002567	0.157009	0.195449	0.287876
2	0.005236	0.012281	0.018050	0.258148	0.307141	0.421632
3	0.017351	0.031913	0.042312	0.355045	0.413379	0.547812
4	0.035141	0.057353	0.072216	0.453086	0.520481	0.674469
5	0.057505	0.087363	0.106602	0.554583	0.631131	0.805004
6	0.083865	0.121433	0.145053	0.661119	0.747138	0.941689
7	0.113978	0.159423	0.187511	0.774111	0.870102	1.086514
8	0.147821	0.201427	0.234144	0.895044	1.001692	1.241538
9	0.185549	0.247719	0.285304	1.025632	1.143817	1.409118
10	0.227476	0.298753	0.341530	1.167968	1.298818	1.592138
11	0.274089	0.355183	0.403574	1.324734	1.469686	1.794303
12	0.326089	0.417917	0.472471	1.499495	1.660411	2.020574
13	0.384462	0.488218	0.549650	1.697185	1.876530	2.277884
14	0.450605	0.567870	0.637130	1.924942	2.126090	2.576391
15	0.526558	0.659477	0.737863	2.193682	2.421458	2.931871
16	0.615426	0.767040	0.856403	2.521315	2.783066	3.370735
17	0.722277	0.897157	1.000310	2.940263	3.248235	3.942058
18	0.856258	1.061943	1.183601	3.518426	3.896097	4.752325
19	1.036896	1.287938	1.437412	4.437388	4.942115	6.100933
20	1.322616	1.658069	1.861503	6.510669	7.378900	9.427787

Applying the recursion in Equation (3) to Equation (11) and using the corresponding values from Table 1 yields

$$P\left({\displaystyle \frac{Y_{1;9:9:40}^{{\mathit{R}}_i^{*}}-Y_{1;8:8:40}^{\mathit{R}}}{{\sigma }^{*}}}>0.001266\right)=0.975$$

and

$$P\left({\displaystyle \frac{Y_{1;9:9:40}^{{\mathit{R}}_i^{*}}-Y_{1;8:8:40}^{\mathit{R}}}{{\sigma }^{*}}}>0.195449\right)=0.025.$$

Accordingly, the exact $95\%$ prediction interval for the first item among the last 20 censored items, with $Y_{1;8:8:40}=27.27$ in the first sample, is

$$\left[27.27+0.001266\times 90.34350,27.27+0.195449\times 90.34350\right]=[27.38437,44.92755].$$

This example demonstrates the effectiveness of the recursive procedure in obtaining exact prediction intervals. Such intervals provide a more reliable inference framework under complex progressive censoring schemes, where the commonly used normal approximations tend to yield wider and less accurate results.

7.2. Example 2

The data reported in the exercise of Bain and Engelhardt ([4], p. 205) consist of the first 20 wearout times (in thousands of miles) of tire tread wear from a total sample of 40, obtained under the Present, Additive, and Thickness manufacturing processes, respectively. The exercise assumes two-parameter exponential distributions to study multiple-sample hypothesis testing across these datasets.

Table 2 reports the Kolmogorov-Smirnov (KS) distances between the empirical distribution functions and the fitted distributions for both the one- and two-parameter models, along with the corresponding p-values. The results clearly indicate that the two-parameter model provides a superior fit for all three samples, whereas the one-parameter model yields a comparatively poorer fit. Hence, we will discuss this data set under the two-parameter model.

Table 2. The first 20 wearout times of tire tread wear from a total sample size of 40 under three manufacturing processes, the corresponding KS test statistics, and the p-values of the KS tests for three selected models.

Process	First 20 Wearout Times	Model	Estimated Parameters	KS	p-Value
Present	10.03, 10.47, 10.58, 11.48, 11.60, 12.41, 13.03, 13.51, 14.48, 16.96,	Exp$(\mu ,\sigma )$	$\widehat{\mu }=10.02956$, $\widehat{\sigma }=6.161439$	0.22529	0.2253
	17.08, 17.27, 17.90, 18.21, 19.30, 20.10, 20.51, 21.78, 21.79, 25.34	Exp$\left(\sigma \right)$	$\widehat{\sigma }=16.1915$	0.46177	0.0002007
Additive	10.10, 11.01, 11.20, 12.95, 13.19, 14.81, 16.03, 17.01, 18.96, 24.10,	Exp$(\mu ,\sigma )$	$\widehat{\mu }=10.09932$, $\widehat{\sigma }=12.30943$	0.22935	0.2086
	24.15, 24.52, 26.05, 26.44, 28.59, 30.24, 31.03, 33.51, 33.61, 40.68	Exp$\left(\sigma \right)$	$\widehat{\sigma }=22.409$	0.36283	0.007256
Thickness	19.07, 19.51, 19.62, 20.47, 20.78, 21.37, 22.08, 22.61, 23.47, 26.02,	Exp$(\mu ,\sigma )$	$\widehat{\mu }=19.06955$, $\widehat{\sigma }=6.183934$	0.22501	0.2265
	26.23, 26.47, 27.07, 27.43, 28.28, 29.10, 29.66, 30.67, 30.81, 34.36	Exp$\left(\sigma \right)$	$\widehat{\sigma }=25.254$	0.53005	<0.00001

For illustrative purposes, three-sample progressively Type-II censored data are constructed from these samples, using the first design parameters $(n_i,m_i,r_i)=(40,8,2)$, $(40,8,1)$, and $(40,8,1)$, together with the corresponding censoring schemes ${\mathit{R}}_i$, $i=1, 2, 3$, as described in Example 1 and listed below.

j	1	2	3	4	5	6	7	8
Present	–	–	10.58	11.60	12.41	13.51	16.96	19.30
${\mathit{R}}_1$	–	–	4	0	4	0	4	20
Additive	–	11.01	11.20	12.95	16.03	18.96	28.59	33.51
${\mathit{R}}_2$	–	3	0	5	0	4	0	20
Thickness	–	19.51	19.62	20.47	20.78	26.23	28.28	34.36
${\mathit{R}}_3$	–	0	3	3	3	3	0	20

The BLUEs of $\mu$ and $\sigma$ are calculated as

$${\mu }^{*}=11.44268,{\sigma }^{*}=51.18745$$

from Equations (8) and (9), respectively.

Using the recursions in Equations (3) and (4) to Equation  (13),

$$P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>0.48784\right)=0.975\mathrm{and}P\left({\displaystyle \frac{{\sigma }^{*}}{\sigma }}>1.06612\right)=0.025,$$

so that the exact $95\%$ confidence interval for $\sigma$ is

$$\left[{\displaystyle \frac{{\sigma }^{*}}{1.06612}},{\displaystyle \frac{{\sigma }^{*}}{0.48784}}\right]=[48.01284,104.92672].$$

Similarly, using the recursions in Equations (3) and (4) to Equation  (12),

$$P\left({\displaystyle \frac{{\mu }^{*}-\mu }{{\sigma }^{*}}}>-0.028016\right)=0.975\mathrm{and}P\left({\displaystyle \frac{{\mu }^{*}-\mu }{{\sigma }^{*}}}>0.10804\right)=0.025,$$

yielding the exact $95\%$ confidence interval for $\mu$,

$$\left[{\mu }^{*}-0.10804\times {\sigma }^{*},{\mu }^{*}+0.028016\times {\sigma }^{*}\right]=[5.91239,12.87675].$$

Next, for comparison, we obtain the MLEs of $\mu$ and $\sigma$: $\widehat{\mu }=9.41$ with an approximate standard error of $1.48$, and $\widehat{\sigma }=80.84$ with an approximate standard error of $19.73$. Upon using the asymptotic normality of the MLEs, we find an approximate 95% confidence interval for $\mu$ (based on MLEs) to be

$$\left[\widehat{\mu }-1.96\times 1.48,\widehat{\mu }+1.96\times 1.48\right]=[6.5092,12.3108],$$

and an approximate 95% confidence interval for $\sigma$ (based on MLEs) to be

$$\left[\widehat{\sigma }-1.96\times 19.73,\widehat{\sigma }+1.96\times 19.73\right]=[42.1692,119.5108].$$

It is important to note that the confidence intervals based on the BLUEs are exact, whereas those based on the MLEs are approximate. Numerically, the two sets of intervals are close. Moreover, both confidence intervals for $\mu$ exclude 0, indicating a potential minimum wear-out threshold in the tire data and supporting the appropriateness of the two-parameter exponential model.

We further examine the prediction intervals for the last 20 unobserved failure times. Table 3 presents the exact values of t used for constructing prediction intervals for the last 20 items, $Y_{i;8+j:8+j,40}$, censored at $Y_{i;8:8:40}$, for $i=1, 2, 3$ and $j=1, \dots , 20$, at significance levels $\alpha =0.005, 0.025, 0.05, 0.95, 0.975, 0.995$. As in Table 1, these values are identical across the three samples, since they share the same number of observed failures and the same final removals.

Table 3. Exact values of t from Equation  (14) for constructing prediction intervals for the last 20 items $Y_{i;8+j:8+j,40}$ (censored at $Y_{i;8:8:40}$), with $i=1, 2, 3$ and $j=1, \dots , 20$, under various choices of $\alpha$ in the two-parameter exponential model.

j	0.995	0.975	0.95	0.05	0.025	0.005
1	0.000338	0.001707	0.003459	0.214124	0.267248	0.395960
2	0.007030	0.016499	0.024261	0.353008	0.421275	0.582098
3	0.023226	0.042771	0.056755	0.486419	0.568200	0.758237
4	0.046923	0.076715	0.096707	0.621641	0.716600	0.935394
5	0.076616	0.116661	0.142556	0.761806	0.870121	1.118230
6	0.111521	0.161923	0.193743	0.909067	1.031234	1.309873
7	0.151303	0.212309	0.250190	1.065360	1.202134	1.513073
8	0.195928	0.267941	0.312120	1.232723	1.385115	1.730684
9	0.245592	0.329185	0.380003	1.413510	1.582814	1.965985
10	0.300707	0.396640	0.454554	1.610604	1.798462	2.222988
11	0.361914	0.471173	0.536776	1.827695	2.036192	2.506849
12	0.430135	0.553990	0.628045	2.069693	2.301518	2.824470
13	0.506667	0.646764	0.730262	2.343387	2.602085	3.185489
14	0.593354	0.751861	0.846114	2.658598	2.948997	3.604009
15	0.692885	0.872742	0.979537	3.030326	3.359299	4.101902
16	0.809366	1.014726	1.136602	3.483160	3.861117	4.715720
17	0.949504	1.186598	1.327408	4.061543	4.505735	5.513228
18	1.125438	1.404527	1.570707	4.858405	5.401673	6.641093
19	1.363162	1.704025	1.908269	6.121730	6.843679	8.510531
20	1.740802	2.196520	2.474421	8.959656	10.186063	13.095425

Applying recursion in Equation  (3) to Equation  (14) and using the corresponding values from Table 3 gives

$$P\left({\displaystyle \frac{Y_{1;9:9:40}^{{\mathit{R}}_i^{*}}-Y_{1;8:8:40}^{\mathit{R}}}{{\sigma }^{*}}}>0.001707\right)=0.975,P\left({\displaystyle \frac{Y_{1;9:9:40}^{{\mathit{R}}_i^{*}}-Y_{1;8:8:40}^{\mathit{R}}}{{\sigma }^{*}}}>0.267248\right)=0.025.$$

Hence, the exact 95% prediction interval for the first item among the last 20 items, censored at $Y_{1;8:8:40}=19.30$ in the first sample, is

$$\left[19.30+0.001707\times 51.18745,19.30+0.267248\times 51.18745\right]=[19.387377,32.97974].$$

If the sample sizes $n_i$, observed failures $m_i$, or final removals $R_{i,m_i}$ vary across samples, Table 1 and Table 3 should be generated individually for each sample i.

8. Conclusions

In this paper, we developed exact inference and prediction procedures for one- and two-parameter exponential models under multiple-sample general progressive Type-II censoring. Building upon the BLUEs established by Balakrishnan et al. [2], we expressed these estimators in terms of normalized exponential spacings and employed the recursive algorithm of Huffer and Lin [35] to derive their exact distributions. This approach enabled the construction of exact confidence intervals for both scale and location parameters, as well as prediction intervals for unobserved failure times. The proposed methodology provides exact, unified, and computationally efficient procedures that extend beyond single-sample frameworks, offering practitioners effective tools for analyzing complex censoring schemes commonly encountered in reliability and survival studies.

This work addresses a longstanding gap in the literature, where multiple-sample censoring structures have received limited attention. The proposed framework demonstrates how exact distributional results for pivotal statistics can be systematically obtained. Notably, confidence intervals based on the BLUEs are exact, while those derived from the MLEs are approximate and tend to yield wider and less accurate results. The examples using simulated and tire wear data further illustrate the practical utility and robustness of the proposed procedures in real-world applications.

Future research could extend this work by comparing the BLUE and MLE estimators across various sample sizes and censoring configurations to assess differences in bias and mean squared error (MSE). We also plan to build upon the work of Lin et al. [38] and Wang [40] by developing a Shiny- and R-based web interface that implements our approach for obtaining exact distributional results of pivotal statistics under exponential models. The proposed interface will allow users to upload their own data, specify design parameters $(n_i,m_i,r_i)$ and corresponding censoring schemes ${\mathit{R}}_i$, $i=1, 2, \dots , K$, and automatically compute the critical values required to construct exact confidence and prediction intervals. Although the critical values of t in Equations (10) and (13) can be efficiently obtained using the fsolve command, computations for Equations (11), (12), and (14) are considerably more demanding. Therefore, pre-storing these critical values (as in Table 1 and Table 3) will substantially improve computational efficiency and make the procedure more practical for online implementation. The platform will also feature a library of commonly used censoring schemes, allowing users to perform diverse K-sample prediction analyses conveniently and accurately.

Such an online system will provide researchers, policy-makers, and practitioners with a flexible and efficient tool for constructing exact confidence and prediction intervals, thereby overcoming the limitations of pre-tabulated results and fixed design settings. We are currently developing this platform and will present the corresponding results in future work.