Inference and optimal design of accelerated life test using the geometric process for power rayleigh distribution under time-censored data

Abstract

In this paper, a new extension of the standard Rayleigh distribution called the Power Rayleigh distribution (PRD) is investigated for the accelerated life test (ALT) using the geometric process (GP) under Type-I censored data. Point estimates of the formulated model parameters are obtained via the likelihood estimation approach. In addition, interval estimates are obtained based on the asymptotic normality of the derived estimators. To evaluate the performance of the obtained estimates, a simulation study of 4, 5 and 6 levels of stress is conducted for ALT in different combinations of sample sizes and censored times. Simulation results indicated that point estimates are very close to their initial true values, have small relative errors, are robust and are efficient for estimating the model parameters. Similarly, the interval estimates have small lengths and their coverage probabilities are almost converging to their 95% nominated significance level. The estimation procedure is also improved by the approach of finding optimum values of the acceleration factor to have optimum values for the reliability function at the specified design stress level. This work confirms that PRD has the superiority to model the lifetimes in ALT using GP under any censoring scheme and can be effectively used in reliability and survival analysis.

Keywords

Accelerated life test geometric process power ryleigh distribution maximum likelihood estimation optimum test plan

1 Introduction

Due to the rapid development in industrial operations. Under normally used conditions, manufacturing products have long life spans and therefore, it is difficult or even impossible to measure their reliability using standard life testing methods. To overcome this problem, accelerated life tests (ALTs) are utilized by exposing the test units to successively higher stress levels, including the effect of temperature, voltage, pressure, etc. to collect the failure times data that contain sufficient information requested for inference about their life model.

Generally, ALTs are either time-dependent, like the step-stress and the linearly increasing stress ALTs, or independent of time, as in the case of constant stress ALTs. In the constant ALT, the test unit is exposed to only a single level of stress until either it stops working or the life test is terminated. Practically, most of the produced units are normally exposed to a constant level of stress, which makes the constant ALTs more preferable for reliability engineers than the time-dependent ALTs. In addition, constant ALTs are easy to run and quantify, describe the actual use of units, are less costly and quicker to be performed. For these reasons, a lot of authors have adopted ALTs in their research for estimating the parameters of different life models. Optimum test plans and designs of ALTs are introduced by [16 , 36]. Recent studies are also explored using different estimation methods as in [10] for the modified Weibull distribution, [13] for the exponentiated Rayleigh distribution and [26] for the exponentiated Lindley distribution.

In practical life testing experiments, for many reasons such as time constraints and cost reduction, ALTs are almost conducted under censoring schemes. In reliability engineering and medical survival analysis, the Type-I censoring scheme is frequently used. In this scheme, the experiment continues up to a pre-fixed censoring time T > 0 and the number of failures during the test is considered a random variable. As it provides an efficient estimation of the resulting model parameters, ALTs based on Type-I censored data have the greatest interest in the analysis of some common life models. For the Rayleigh distribution, parameter estimation and optimum test plans are considered by [15]. Islam and Ahmad [3] have also addressed the estimation of the Weibull distribution parameters. Li and Zheng [33] have investigated the estimation procedures and optimum test plans for the Gompertez distribution. Basu et al. [25] explored the parameter estimation of the inverse Lindley distribution. Updated research is included in [14] for the generalized half normal distribution and, [2] for the Frechet distribution. Inferences, applications and designs of ALTs under different censoring schemes are also explored by [17 , 24].

The geometric process (GP) is first used by [34] in connection with maintenance problems for the reliability analysis of repairable systems. In the reliability and survival analysis framework, GP is shown to be a good and simple model for data that has a single trend or multiple trends. Particularly, it is implemented with ALTs for estimating the reliability of produced units having different life models, as in [27] for the case of Exponential distribution. Anwar et al. [20, 23] have also investigated estimation in Marshal-Olkin parameters using both complete and censored data. Recent research is explored the optimum ALT design under progressive Type-II censoring for the half-Logistic distribution by [7, 4] for the Burr Type x distribution as in the study of [19] for the generalized Exponential distribution. Reliability analysis of a hybrid system using GP with multiple levels of constant ALT is also implemented under Type-II progressive censored and masked data through a simulation study by [12].

Despite the fact that the standard Rayleigh distribution has significant applications for modeling a variety of data applied in engineering and medical sciences, it lacks the superiority to model some lifetime data sets. Using the power transformation method, [1] have introduced a generalization of the one-parameter Rayleigh distribution called the Power Rayleigh distribution (PRD). This new distribution has two parameters and is expected to model more lifetime data.

In this paper, PRD is investigated in the constant ALT using GP under Type-I censored data. Point and interval estimates of the parameters of the formulated model are obtained via the likelihood estimation approach. The performance of these estimates is evaluated through a simulation study of 4, 5 and 6 levels of stress. Unlike other studies that are mainly focused on the optimum censoring time. This study aims to obtain optimum values for the acceleration factor of the constant ALT to improve the estimation procedures.

Application of this work may include a diversity of real-life applications like evaluating the reliability of systems under the maintenance processes, analysis of stress and strength data, testing the evaluability of the equipment in the aircraft, analysis of software system failures data and many other applications that propose PRD as a proper underlying life model. For more details (see [9, 35]).

The rest of this paper is organized as follows: Basic concepts of GP, the PRD life model and assumptions about the ALT procedure are introduced in Section 2. Section 3 is devoted to obtaining point estimates of the model parameters. Using an asymptotic approximation of the derived Fisher information matrix, Section 4 is focused on constructing the confidence intervals for the model parameters. A simulation study is conducted in Section 5 to evaluate the statistical properties of the obtained estimates and to improve the efficiency of the estimation procedure, optimum values for the ratio of the GP are obtained based on the trace optimality criterion in Section 6. Finally, Section 7 includes a conclusion about this work.

2 The model and test procedure

2.1 The geometric process

A stochastic process {X_n, n = 1, 2, … } is said to be a geometric process (GP) if there exists a real value λ > 0 such that {λ^n-1X_n, n = 1, 2, … } forms a renewal process. This implies that if X₁ has a probability density function (pdf) given by f(x) with mean μ and variance σ², then X_n has the pdf λ^n-1f (λ^n-1 x) with mean $μ^{*} = \frac{μ}{λ^{n - 1}}$ and variance $σ^{* 2} = \frac{σ^{2}}{λ^{2 (n - 1)}}$ . Thus μ^∗, σ^∗2, and λ are the important parameters of GP.

2.2 The power rayleigh distribution

The PRD introduced by [1] is a new extension of the standard Rayleigh distribution. It has the probability density (pdf), the cumulative distribution function (cdf) and reliability functions given, respectively, by $f (x; θ, β) = \frac{β}{θ^{2}} x^{2 β - 1} e^{\frac{- x^{2 β}}{2 θ^{2}}}; x > 0, θ, β > 0$ (1) $F (x; θ, β) = 1 - e^{\frac{- x^{2 β}}{2 θ^{2}}}; x > 0, θ, β > 0$ (2) $R (x; θ, β) = e^{\frac{- x^{2 β}}{2 θ^{2}}}; x > 0, θ, β > 0$ (3) As a new generalization of the Rayleigh distribution, PRD can be used for many life applications, particularly in the analysis of reliability and survival data. PRD can effectively model the lifetimes of many pieces of equipment like resistors, transformers and capacitors, for more details; see [5, 9]. The hazard rate function of the PRD is given by $h (x; θ, β) = \frac{β}{θ^{2}} x^{2 β - 1}; x > 0, θ, β > 0$ (4) which is constant when β = 0.5, increasing when β < 0.5, and decreasing when β > 0.5. For this flexibility, as shown by [1], PRD has the superiority to model the stress and strength data.

2.3 Assumptions and test procedure:

An accelerated life test is conducted by exposing a random sample of size n identical units to m increasing levels of stress at the same time.

At each stress level, the life test is terminated at a pre-specified time T and during the test, if the unit fails, it will be removed and the observed failure times x_ij of the sample units i = 1, 2, …, n at the stress level j = 1, 2, …, m are recorded such that x_ij ≤ T.

The life of each unit at each stress level follows the Power Rayleigh distribution PRD (θ, β) and at each stress level, the scale parameter θ is a log-linear function of stress level given by log(θ) = a + bS_j, j = 1, 2, …, m where a and b are unknown parameters depending on the nature of the test units and test method.

The lifetimes of units under each stress level are denoted by the random variables X_θ, X₁, X₂, …, X_m, where X_θ is the lifetime of the units under the design stress at which the units are normally operated and sequence (X_j, j = 1, 2, …, m) forms a geometric process with the ratio λ > 0.

Based on assumption 3, assumption 4 is confirmed by the following theorem (see [30]):

Theorem. In ALT, if the stress level is increasing with a constant difference, then the lifetimes of units form a GP under each stress level. That is, If S_j+1 - S_j is constant for j = 1, 2, …, m - 1, then (X_j, j = 1, 2, …, m) form a GP.

Proof. From assumption 3 $log (\frac{θ_{j + 1}}{θ_{j}}) = b (θ_{j + 1} - θ_{j}) = b Δ S$ (5) The above equation indicates that the increased stress levels form an athematic sequence with a constant difference (ΔS). This implies $\frac{θ_{j + 1}}{θ_{j}} = e^{b Δ S} = \frac{1}{λ} (say)$ (6) Therefore, we can write the scale parameter θ at the stress level j = 1, 2, …, m as: $θ_{j} = \frac{1}{λ} θ_{j - 1} = \frac{1}{λ^{2}} θ_{j - 2} = \dots = \frac{1}{λ^{j}} θ$ (7) Hence, the lifetime pdf and the cdf of a unit at the jth stress level are given respectively as: ${f_{x}}_{j} (x) = \frac{β λ^{2 j}}{θ^{2}} x_{j}^{2 β - 1} exp (- \frac{x_{j}^{2 β} λ^{2 j}}{2 θ^{2}})$ (8) ${F_{x}}_{j} (x) = 1 - exp (- \frac{x_{j}^{2 β} λ^{2 j}}{2 θ^{2}})$ (9) From eq. (8), the lifetime pdf of a unit at the jth stress level can be written in the form: ${f_{x}}_{j} (x) = λ^{j} {f_{x}}_{0} (λ^{j} x)$ (10) It is clear from eq. (10) that, if the pdf of the lifetime of a unit at the design stress level X₀ is f_x₀ (x), then the pdf of the lifetimes at the jth stress level X_j is given by f_x_j (x) = λ^jf_x₀ (λ^jx) j = 1, 2, …, m and thus referring to the definition of the GP. The sequence of lifetimes X₀, X₁, X₂, …, X_m under the arithmetically increasing stress levels S₀, S₁, S₂, …, S_m form a GP with ratio λ.□

Eq. (10) confirms also that, if the life distribution at a design stress level is PRD with scale parameter θ, then the life distribution at the j^th level will also be PRD with scale parameter $(\frac{θ}{λ^{j}})$ .

3 Maximum likelihood estimation

The maximum likelihood estimation method is the most commonly used method for estimating the parameters of ALT since it can be implemented with complex structures, is robust and frequently gives estimates with statistically desired characteristics. On the other hand, interval estimates of the model parameters can also be obtained by the asymptotic normality of the estimators derived by this method.

Let the test at each stress level is terminated at time T and only x_ij ≤ T failure times are observed. Assume that r_j ≤ n failures at the j^th stress level are observed before the test is suspended and (n - r_j) units are still working within the entire test. Then based on ALT from GP under the Type-I censored data from the PRD, the likelihood function of the parameters θ, β, and λ at the stress level j is given by $\begin{matrix} L_{j} (θ, β, λ) = \\ \frac{n!}{(n - r_{j})!} \frac{β}{θ^{2}} (\prod_{i = 1}^{r_{j}} λ^{2 j} x_{ij}^{2 β - 1}) \times \\ exp (- \frac{1}{2 θ^{2}} (\sum_{i = 1}^{r_{j}} x_{ij}^{2 β} λ^{2 j} + (n - r_{j}) T^{2 β} λ^{2 j})) \end{matrix}$ (11) Therefore, the full likelihood function can be formulated as: $L (θ, β, λ) = \prod_{j = 1}^{m} L_{j} (θ, β, λ)$ (12) This implies that the natural log-likelihood function is $\begin{matrix} ln (L (θ, β, λ)) = \sum_{j = 1}^{m} ln (L_{j} (θ, β, λ)) = \\ \sum_{j = 1}^{m} ln (\frac{n!}{(n - r_{j})!}) + mln (β) - 2 mln (θ) + \\ [\sum_{j = 1}^{m} \sum_{i = 1}^{r_{j}} 2 j λ + (2 β - 1) ln (x_{ij})] - \\ \frac{1}{2 θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} x_{ij}^{2 β} λ^{2 j}) + (n - r_{j}) T^{2 β} λ^{2 j} ⌉ \end{matrix}$ (13) The maximum likelihood estimates (MLEs) of the parameters θ, β and λ are attained by solving the system of the following normal equations: $\begin{matrix} \frac{\partial ln (L (θ, β, λ))}{\partial θ} = \frac{- 2 m}{θ} + \frac{1}{θ^{3}} \times \\ ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} x_{ij}^{2 β} λ^{2 j}) + (n - r_{j}) T^{2 β} λ^{2 j} ⌉ \\ = 0 \end{matrix}$ (14)

$\begin{matrix} \frac{\partial ln (L (θ, β, λ))}{\partial β} = \frac{m}{β} + 2 \sum_{j = 1}^{m} \sum_{i = 1}^{r_{j}} ln (x_{ij}) - \\ \frac{1}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} ln (x_{ij}) x_{ij}^{2 β} λ^{2 j}) + \\ (n - r_{j}) {ln (T) T}^{2 β} λ^{2 j} ⌉ = 0 \end{matrix}$ (15) $\begin{matrix} \frac{\partial ln (L (θ, β, λ))}{\partial λ} = 2 \sum_{j = 1}^{m} {jr}_{j} - \frac{1}{θ^{2}} \times \\ ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {jx}_{ij}^{2 β} λ^{2 j - 1}) + \\ j (n - r_{j}) T^{2 β} λ^{2 j - 1} ⌉ = 0 \end{matrix}$ (16) The system of the nonlinear eqs. (14), (15) and (16) cannot be solved mathematically. Therefore, Newton-Raphson numerical methods can be efficiently used to simultaneously obtain the MLEs, $\hat{θ}$ , $\hat{β}$ , and $\hat{λ}$ , of the parameters θ, β, and λ respectively. This implies the MLE of the reliability function at time t given the stress level j, j = 1, 2, …, m is given by $\hat{R} (t) = exp (- \frac{t^{2 \hat{β}} {\hat{λ}}^{2 j}}{2 {\hat{θ}}^{2}}), t > 0$ (17)

4 Fisher information matrix and asymptotic confidence intervals

Since the MLEs of the parameters θ, β and λ are not in closed form, it is difficult to construct their exact confidence intervals. Hence, the Fisher information matrix can be used to find approximate confidence intervals of the parameters β and λ. The Fisher information matrix of the parameters (θ, β, λ) is given by $\begin{matrix} I (θ, β, λ) = \\ E (\begin{matrix} - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ^{2}} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial β} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial λ} \\ - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial β} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β^{2}} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β \partial λ} \\ - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial λ} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β \partial λ} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial λ^{2}} \end{matrix}) \end{matrix}$ (18) Exact expressions of the expectations in the above Fisher information matrix are also difficult to find, Hence, the observed Fisher information matrix can be used instead, (see [8]), which is given by

$\begin{matrix} I (\hat{θ}, \hat{β}, \hat{λ}) = \\ (\begin{matrix} - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ^{2}} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial β} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial λ} \\ - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial β} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β^{2}} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β \partial λ} \\ - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial θ \partial λ} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial β \partial λ} & - \frac{\partial^{2} ln (l (θ, β, λ))}{\partial λ^{2}} \end{matrix}) \end{matrix}$ (19) Where: $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial θ^{2}} = \\ \frac{2 m}{θ^{2}} - \frac{3}{θ^{4}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} x_{ij}^{2 β} λ^{2 j}) + \\ (n - r_{j}) T^{2 β} λ^{2 j} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial β^{2}} = \\ \frac{- m}{β^{2}} - \frac{2}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} (ln (x_{ij}))^{2} x_{ij}^{2 β} λ^{2 j}) + \\ (n - r_{j}) {{(ln (T))}^{2} T}^{2 β} λ^{2 j} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial β^{2}} = \\ \frac{- m}{β^{2}} - \frac{2}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} (ln (x_{ij}))^{2} x_{ij}^{2 β} λ^{2 j}) + \\ (n - r_{j}) {{(ln (T))}^{2} T}^{2 β} λ^{2 j} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial λ^{2}} = \\ - \frac{1}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {j (2 j - 1) x}_{ij}^{2 β} λ^{2 j - 2}) + \\ j (2 j - 1) (n - r_{j}) T^{2 β} λ^{2 j - 2} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial θ \partial β} = \\ \frac{2}{θ^{3}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {ln (x_{ij}) x}_{ij}^{2 β} λ^{2 j}) + \\ (n - r_{j}) ln (T) T^{2 β} λ^{2 j} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial θ \partial λ} = \\ \frac{2}{θ^{3}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {jx}_{ij}^{2 β} λ^{2 j - 1}) + \\ j (n - r_{j}) T^{2 β} λ^{2 j - 1} ⌉ \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ln (L (θ, β, λ))}{\partial β \partial λ} = \\ \frac{- 2}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} jln (x_{ij}) x_{ij}^{2 β} λ^{2 j - 1}) + \\ j (n - r_{j}) {ln (T) T}^{2 β} λ^{2 j - 1} ⌉ \end{matrix}$

Table 1

Simulation results using GP with ALT for PRD when m = 4, T = 8

Sample size(n)	Parameter	mean	SE	MSE	RAB	RSE	length	coverage
30	θ	1.7642	0.2448	0.0599	0.1179	0.1223	0.9596	0.97
	β	1.2539	0.1861	0.0605	0.1640	0.1521	0.7295	0.97
	λ	1.0299	0.1737	0.0305	0.1417	0.1165	0.6809	0.97
50	θ	1.8003	0.2025	0.0411	0.0999	0.1014	0.7938	0.97
	β	1.2711	1.1731	0.0501	0.1318	0.1414	0.6785	0.97
	λ	1.0489	0.1527	0.0322	0.1259	0.1016	0.5983	0.96
80	θ	1.8045	0.1986	0.0394	0.0977	0.0993	0.77828	0.97
	β	1.3021	0.1521	0.0412	0.1289	0.1102	0.5958	0.96
	λ	1.0520	0.1497	0.0224	0.1233	0.0991	0.5867	0.96
100	θ	1.8172	0.1775	0.0310	0.0819	0.0902	0.6954	0.97
	β	1.3164	0.1381	0.0388	0.1171	0.1025	0.5413	0.95
	λ	1.1213	0.1410	0.0205	0.1182	0.0904	0.5527	0.95
130	θ	1.8523	0.1702	0.0302	0.0788	0.0874	0.6451	0.96
	β	1.3234	0.1354	0.0365	0.1152	0.0963	0.5132	0.95
	λ	1.1205	0.1382	0.0197	0.1085	0.0811	0.5378	0.95

This implies an approximate variance-covariance matrix of $(\hat{θ}, \hat{β}, \hat{λ})$ is given by $\begin{matrix} \hat{V} = (\begin{matrix} {\hat{v}}_{11} & {\hat{v}}_{12} & {\hat{v}}_{13} \\ {\hat{v}}_{21} & {\hat{v}}_{22} & {\hat{v}}_{23} \\ {\hat{v}}_{31} & {\hat{v}}_{32} & {\hat{v}}_{33} \end{matrix}) = I^{- 1} (\hat{θ}, \hat{β}, \hat{λ}) \end{matrix}$ (20) Consequently, based on the large sample theory of the MLEs, the approximate 100 (1 - α)% ACIs of θ, β and λ can be constructed, respectively, as: $\begin{matrix} \hat{θ} \pm z_{\frac{α}{2}} \sqrt{{\hat{v}}_{11}}, \hat{β} \pm z_{\frac{α}{2}} \sqrt{{\hat{v}}_{22}}, \hat{λ} \pm z_{\frac{α}{2}} \sqrt{{\hat{v}}_{33}} \end{matrix}$ Where $z_{\frac{α}{2}}$ is the (1 - α)% quantile of the standard normal.

5 Simulation study

The purpose of this section is to evaluate the performance of the obtained point and interval estimates of the model parameters through a simulation study of the ALT presented in this paper with different combinations of true parameter values, sample sizes, stress levels and censored times. All the computations are performed using MATLAB 9.7 R2019b software. The simulation procedure is carried out as follows:

Generate a pseudo sample {u_i, i = 1, 2, … , n } from uniform distribution over the interval [0, 1]

Apply the inverse-cdf method to transform u_i to have expressions for x_ij using the following formula: $\begin{matrix} x_{ij} = {(\frac{2 θ^{2} ln (- (1 - u_{i}))}{λ^{2 j}})}^{\frac{1}{2 β}}, \\ i = 1, 2, \dots, n, j = 1, 2, \dots, m \end{matrix}$

The levels of stress are considered to be: m = 4, 5, and m = 6 with pre-identified censored times: T = 8, T = 10, and T = 12 respectively.

1000 random samples of sizes n = 30, 50, 80, 100, and 130 based on true parameter values (θ = 2, β = 1.5, λ = 1.2) are generated.

The point estimates are evaluated according to the average of their mean, their mean squared error (MSEs), their standard errors (SEs), their relative absolute biases (RABs) and their relative standard error (RSEs) statistical measures, while the derived 95% asymptotic confidence intervals are evaluated according to the average of their lengths and their coverage probabilities.

Table 2
Simulation results using GP with ALT for PRD when m = 5, T = 10

Sample size(n) Parameter mean SE MSE RAB RSE length coverage

30 θ 1.8662 0.1523 0.0232 0.0669 0.0762 0.5970 0.95

β 1.2814 0.1712 0.0531 0.1521 0.1408 0.6703 0.97

λ 1.0974 0.1161 0.0135 0.0855 0.0774 0.4551 0.95

50 θ 1.8811 0.1306 0.0171 0.05911 0.0653 0.5214 0.95

β 1.2865 0.1693 0.0501 0.1490 0.1382 0.6636 0.96

λ 1.1075 0.0993 0.0952 0.0771 0.0662 0.3892 0.94

80 θ 1.8866 0.1137 0.0129 0.0557 0.0569 0.4457 0.95

β 0.1299 0.1605 0.0482 0.1402 0.1301 0.6291 0.96

λ 1.1129 0.0886 0.0079 0.0726 0.0591 0.3475 0.94

100 θ 1.9021 0.1022 0.0106 0.0510 0.0503 0.4006 0.95

β 1.3001 0.1586 0.0437 0.1372 0.1278 0.6217 0.96

λ 1.1323 0.0807 0.0064 0.0678 0.0509 0.3163 0.93

130 θ 1.9203 0.0673 0.0102 0.0487 0.0482 0.3895 0.94

β 1.3551 0.1498 0.0417 0.1338 0.1223 0.6021 0.96

λ 1.1300 0.0765 0.0051 0.0597 0.0477 0.2845 0.93

Sample size(n)	Parameter	mean	SE	MSE	RAB	RSE	length	coverage
30	θ	1.8662	0.1523	0.0232	0.0669	0.0762	0.5970	0.95
	β	1.2814	0.1712	0.0531	0.1521	0.1408	0.6703	0.97
	λ	1.0974	0.1161	0.0135	0.0855	0.0774	0.4551	0.95
50	θ	1.8811	0.1306	0.0171	0.05911	0.0653	0.5214	0.95
	β	1.2865	0.1693	0.0501	0.1490	0.1382	0.6636	0.96
	λ	1.1075	0.0993	0.0952	0.0771	0.0662	0.3892	0.94
80	θ	1.8866	0.1137	0.0129	0.0557	0.0569	0.4457	0.95
	β	0.1299	0.1605	0.0482	0.1402	0.1301	0.6291	0.96
	λ	1.1129	0.0886	0.0079	0.0726	0.0591	0.3475	0.94
100	θ	1.9021	0.1022	0.0106	0.0510	0.0503	0.4006	0.95
	β	1.3001	0.1586	0.0437	0.1372	0.1278	0.6217	0.96
	λ	1.1323	0.0807	0.0064	0.0678	0.0509	0.3163	0.93
130	θ	1.9203	0.0673	0.0102	0.0487	0.0482	0.3895	0.94
	β	1.3551	0.1498	0.0417	0.1338	0.1223	0.6021	0.96
	λ	1.1300	0.0765	0.0051	0.0597	0.0477	0.2845	0.93

Table 3

Simulation results using GP with ALT for PRD when m = 6, T = 12

Sample size(n)	Parameter	mean	SE	MSE	RAB	RSE	length	coverage
30	θ	1.9177	0.0952	0.0093	0.0411	0.0476	0.3731	0.94
	β	1.3221	0.1679	0.0482	0.1407	0.1382	0.6581	0.96
	λ	1.1362	0.0716	0.0051	0.0532	0.0477	0.2806	0.92
50	θ	1.9227	0.0794	0.0063	0.0386	0.0397	0.3113	0.93
	β	1.3312	0.1509	0.0389	0.1322	0.1206	0.5915	0.95
	λ	1.1351	0.0618	0.0039	0.0510	0.0418	0.2422	0.92
80	θ	1.9310	0.0708	0.0060	0.0384	0.0329	0.2775	0.92
	β	1.3374	0.1481	0.0347	0.1289	0.1172	0.5805	0.95
	λ	1.1421	0.0582	0.0032	0.0482	0.0392	0.2281	0.92
100	θ	1.9326	0.0684	0.0057	0.0363	0.0317	0.2681	0.92
	β	1.3522	0.1406	0.0315	0.1100	0.1018	0.5511	0.95
	λ	1.1876	0.0615	0.0043	0.0318	0.0301	0.2410	0.92
130	θ	1.9442	0.0661	0.0053	0.0314	0.0295	0.2553	0.91
	β	1.3841	0.1220	0.0287	0.0993	0.1006	0.5321	0.95
	λ	1.9002	0.0587	0.0037	0.0287	0.0274	0.2366	0.91

It is observed from Tables 1–3 that point estimates are performed well in terms of the specified statistical measures and their efficiencies increase with an increase in the sample size and the indicated termination times of the life test that are ordinary increase with the increase of stress levels. This is expected because we will obtain more failures, which leads to greater consistency among the estimators. We also notice that the average lengths of the asymptotic intervals are relatively small and their coverage probabilities exceed the nominal 95% significance level in most results, but as expected, when the average lengths of these intervals become relatively small, there is a relative decrease in their coverage probabilities up to 91%. This confirmed the validity of the maximum likelihood estimation method for estimating the model parameters.

6 Optimum test plan

In performing the ALT using the GP, in addition to estimating the life of the produced units, one of the recurrent problems in designing the life test is to determine a reasonable value for the acceleration factor λ. This will assist in improving the accuracy of evaluating the reliability of the tested units at the normal stress level. In this section, we approach the finding of an optimal value for the parameter λ using the A-optimality criterion (the trace optimality criterion).

Applying the A optimality criterion, the optimum value of λ can be obtained by maximizing the sum of the diagonal elements in the Fisher information matrix (see [6]). Substituting for the parameters (θ, β) by their MLEs $(\hat{θ}, \hat{β})$ , the optimum value (λ^∗) of λ is obtained by solving the following nonlinear equation with respect to λ. $f (λ) = \frac{\partial (\sum_{i = 1}^{3} I_{ii})}{\partial λ} |_{(\hat{θ}, \hat{β})} = 0$ (21) Where: $\begin{matrix} \frac{\partial I_{11}}{\partial λ} = \\ \frac{6}{θ^{4}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {jx}_{ij}^{2 β} λ^{2 j - 1}) + \\ j (n - r_{j}) T^{2 β} λ^{2 j - 1} ⌉_{(\hat{θ}, \hat{β})} \end{matrix}$ $\begin{matrix} \frac{\partial I_{22}}{\partial λ} = \\ - \frac{4}{θ^{2}} ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} j (ln (x_{ij}))^{2} x_{ij}^{2 β} λ^{2 j - 1}) + \\ j (n - r_{j}) {{(ln (T))}^{2} T}^{2 β} λ^{2 j - 1} ⌉_{(\hat{θ}, \hat{β})} \end{matrix}$ $\begin{matrix} \frac{\partial I_{33}}{\partial λ} = - \frac{1}{θ^{2}} \times \\ ⌈ \sum_{j = 1}^{m} (\sum_{i = 1}^{r_{j}} {j (2 j - 1) (2 j - 2) x}_{ij}^{2 β} λ^{2 j - 3}) + \\ j (2 j - 1) (2 j - 2) (n - r_{j}) T^{2 β} λ^{2 j - 3} ⌉ \end{matrix}$ Consequentially, the optimum reliability function at the stress level j, j = 1, 2, …, m is given by $R^{*} (t) = exp (- \frac{t^{2 \hat{β}} {λ^{*}}^{2 j}}{2 {\hat{θ}}^{2}}), t > 0$ (22) Based on true parameter values (θ = 2, β = 1.5) as indicated in the simulation study, Table 4 lists the values of λ^∗ and Figures 1–4 present the theoretical, the MLE and the optimum reliability function with different combinations of (n, m).

Table 4

The obtained values of λ^∗ with different combinations of (n, m) based on the simulation study

Levels of stress
Sample size	1	2	3	4	5	6
n	λ ^∗
30	1.1449	1.1476	1.1502	1.1572	1.1591	1.1606
50	1.1473	1.1502	1.1575	1.1608	1.1627	1.1638
80	1.5052	1.5882	1.1601	1.1623	1.1654	1.1691
100	1.5227	1.6031	1.1688	1.1710	1.1752	1.1906

Fig. 1

The theoretical, the MLE and the optimum reliability function with (n = 50, m = 4).

Fig. 2

The theoretical, the MLE and the optimum reliability function with (n = 80, m = 4).

Fig. 3

The theoretical, the MLE and the optimum reliability function with (n = 50, m = 5).

Fig. 4

The theoretical, the MLE and the optimum reliability function with (n = 80, m = 6).

As it clearly appears from the results in Table 4, with the increase in sample size and the number of stress levels, we observe that the optimum value λ^∗ of λ converges to the true value λ = 1.2 which is proposed in the simulation study.

Moreover, Figure 4 shows that the optimum reliability function gives a better estimation for the theoretical reliability function than the reliability function estimated by MLE. This also confirmed that the optimum selection of the GP ratio requires a strict modification in the estimation procedures.

7 Conclusion

Using the GP with the constant ALT under Type-I censored data, this article deals with the likelihood estimation approach for the parameters of the resulting model when the life of the tested units is assumed to be the PRD. The Newton-Raphson method is employed to obtain the ML point estimates. On the other hand, the interval estimates are obtained based on the asymptotic normality of the MLEs. The characteristics of the point estimates are evaluated in terms of MSE, SE, RAB and RSE accuracy measures, while the interval estimates are evaluated according to their average lengths and their coverage probabilities through a simulation study.

Simulation results indicated that the RABS and the MSEs of the ML point estimates decrease as the sample sizes are decreases. This confirms that the ML point estimates provide asymptotically normally distributed and consistent estimators. On the other hand, for a fixed sample size n, with an increase in the termination time T, the MSE, SE, RAB and RSE of the ML point estimates are decreased. This will also increase the efficiency in estimating the model parameters.

In addition, the derived Fisher information matrix is investigated by the A-optimality criterion to obtain optimum values for the acceleration ratio of the GP, λ that gave a significantly better estimation of the reliability function at a given termination time than the ML point estimate.

Concluding remarks from this work indicated the PRD to be an appropriate life model for the inspected units of ALT using time-censored data.

Future researchers may extend this work using Bootstrap and Bayesian estimation methods based on other progressively censored schemes. Other distributions (continuous or discrete) can also be considered to include more applications in reliability and survival analysis.

Footnotes

Acknowledgment

The authors would like to express their deepest gratitude to the anonymous referees for their insightful recommendations and remarks.

Conflicts of Interest

The authors declare no conflict of interest.

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