Estimating and Planning Accelerated Life Test Using Constant Stress for Generalized Logistic Distribution under Type-I Censoring

Abstract

The optimal designs and statistical inference of accelerated life tests under type-I are studied for constant stress-accelerated life tests (CSALTs). It is assumed that the lifetime at design stress has generalized logistic distribution. The scale parameter of the lifetime distribution at constant stress levels is assumed to be an inverse power law function of the stress level. The maximum likelihood (ML) estimators of the model parameters, Fisher information matrix, the asymptomatic variance-covariance matrix, the confidence bounds, the predictive value of the scale parameter, and the reliability function under the usual conditions are obtained under type-I censoring. Moreover, the optimal design of the accelerated life tests is studied according to the D-optimality criterion to specify the optimal censoring time. Finally, the numerical studies are introduced to illustrate the proposed procedures.

1. Introduction

Accelerated life testing (ALT) is becoming important and widely used in many fields (such as in manufacturing industries to assess or demonstrate component and subsystem reliability) because of rapidly changing technologies, higher customer expectations for better reliability, and the need for rapid device development. Briefly, ALT is a method for estimating the reliability of devices at normal use conditions from failure data obtained at severe conditions. The failure data is analyzed in terms of a suitable physical statistical model to obtain desired information about a device or its performance under normal use conditions.

Commonly, all available test data obtained from ALT is used in statistical analysis. However, the obtained data may be incomplete or it may include uncertainty about the failure time. There are three types of possible censoring schemes, right censoring, left censoring, and interval censoring. Moreover, the most common schemes are time-censoring and failure-censoring. Time-censored data is also known as type-I censored. It occurs when the life test is terminated at a specified time, before all units have failed. Data is failure censored (or type-II censored) if the test is terminated after a specified number of failures.

ALT can take the form of usage rate acceleration (for devices that do not operate continuously under normal conditions such as home appliances) or overstress acceleration (for devices with very high or continuous usage such as communication satellites, computers, and monitors). In real life, different types of stress loading may be considered when performing an accelerated test. The common types are constant stress, step stress, and progressive stress.

The most common stress loading is constant stress. In constant stress-accelerated life test (CSALT), the stress is kept at a constant level of stress throughout the life of the test; that is, each unit is run at a constant high stress level until the occurrence of failure or the observation is censored. Practically, most devices such as lamps, semiconductors, and microelectronics are run at a constant stress. Many authors have studied statistical inference of CSALT; for example, see Lawless [1], McCool [2], Bai and Chung [3], Bugaighis [4], Watkins [5], Abdel-Ghaly et al. [6], and El-Dessouky [7].

Before launching a new product, the manufacturer is always faced with decisions regarding the optimum method to estimate the reliability of the product or the service. Moreover, a test plan needs to be developed to obtain appropriate and sufficient information in order to accurately estimate the reliability performance at operating conditions, significantly reduce test times and costs, and achieve other objectives. The appropriate criteria for choosing a test plan depend on the purpose of the experiment (Meeker et al. [8]).

Optimum CSALT plans were studied for different lifetime distributions based on different censoring scheme; for example, Nelson and Kielpinski [9] studied optimum ALT plans for normal and lognormal life distributions. Nelson [10] reviewed statistically optimal and compromise plans for the single stress ALT planning problem. Yang [11] proposed an optimal design of 4-level constant-stress ALT plans considering different censoring times. Ding et al. [12] dealt with Weibull distribution.

The term Generalized Logistic (GL) distribution is used as the name for several different families of probability distributions (see Johnson et al. [13]). The main feature of the GL distribution is that new parameters were introduced to control both location and scale. It allows for a greater degree of flexibility and it is expected to be useful in many more practical situations (Nadarajah and Kotz [14]), for example, in extreme value event evaluation, in hydrological risk analysis, and in a quanta response data, and to model the data with a unimodal density (for more details, see Mathai and Provost [15], Tolikas [16], Alkasasbeh and Raqab [17], Tolikas and Gettinby [18], Shabri et al. [19], and Tolikas [20]).

Alkasasbeh and Raqab [17] and Shin et al. [21] were interested in estimating the parameters of the GL distribution by ML and other methods while Rai et al. [22] employed the genetic algorithm to derive the unit hydrograph by using nine distributions including GL distribution. Kim et al. [23] derived the probability plot correlation coefficient test statistic for the GL distribution. Shabri et al. [19] introduced the method of trimmed L-moments as an alternative way in estimating the flood for higher return period and derived the trimmed L-moments for the GL distribution. However, no studies have been made on estimating or planning (optimal design) CSALT using the GL distribution.

This paper is organized as follows. In Section 2, the underlying distribution and the test method are described. Section 3 introduces the ML estimators of the model parameters with their properties under type-I censoring. Optimum censoring time test plan is developed in Section 4. Finally, the simulation studies needed for illustrating the theoretical results are presented in Section 5.

2. The Model

2.1. The Generalized Logistic Distribution

The probability density function (pdf) of a three-parameter-generalized logistic distribution (Molenberghs and Verbeke [24]) is given by 𝑓(𝑥)=𝛼𝛾𝑒^𝛼𝑥1+𝛾𝜃𝑒^𝛼𝑥^−(𝜃+1),−∞<𝑥<∞,𝛼,𝛾,𝜃>0.(2.1) The reliability function takes the form 𝑅(𝑥)=1+𝛾𝜃𝑒^𝛼𝑥^−𝜃,−∞<𝑥<∞,𝛼,𝛾,𝜃>0,(2.2) and the corresponding failure rate is given by ℎ(𝑥)=𝛼𝛾𝑒^𝛼𝑥1+𝛾𝜃𝑒^𝛼𝑥⁻¹,−∞<𝑥<∞,𝛼,𝛾,𝜃>0.(2.3)

2.2. Assumptions

We assume the following assumptions for the CSALT procedure.(i)A total of 𝑁 units are divided into 𝑛₁,𝑛₂,…,𝑛_𝑘 units where ∑^𝑘𝑗=1𝑛_𝑗=𝑁.(ii)There are 𝑘 levels of high stress 𝑉_𝑗, 𝑗=1,…,𝑘 in the experiment, and 𝑉_𝑢 is the stress under usual conditions, where 𝑉_𝑢<𝑉₁<⋯<𝑉_𝑘.(iii)Each 𝑛_𝑗,𝑗=1,…,𝑘 unit in the experiment is run at a prespecified constant stress 𝑉_𝑗,𝑗=1,…,𝑘.(iv)It is assumed that the stress affected only on the scale parameter of the underlying distribution.(v)The failure times 𝑥_𝑖𝑗,𝑖=1,…,𝑛_𝑗 and 𝑗=1,…,𝑘, at stress levels 𝑉_𝑗,𝑗=1,…,𝑘, are the 3-parameter-generalized logistic distribution with probability density function: 𝑓𝑥_𝑖𝑗,𝛼_𝑗,𝛾,𝜃=𝛼_𝑗𝛾𝑒^{𝛼𝑗𝑥𝑖𝑗}1+𝛾𝜃𝑒^{𝛼𝑗𝑥𝑖𝑗}^−(𝜃+1)−∞<𝑥_𝑖𝑗,<∞,𝛼_𝑗,𝛾,𝜃>0,𝑖=1,…,𝑛_𝑗,𝑗=1,…,𝑘.(2.4)(vi)The scale parameter 𝛼_𝑗, 𝑗=1,…,𝑘, of the underlying lifetime distribution (2.4) is assumed to have an inverse power law function on stress levels, that is, 𝛼_𝑗=𝐶𝑆^𝑃𝑗,𝐶,𝑃>0,where𝑆_𝑗=𝑉^∗𝑉_𝑗𝑉_∗,=^𝑘_𝑗=1𝑉^𝑏𝑗_𝑗,𝑏_𝑗=𝑛^𝑗∑^𝑘𝑗=1𝑛_𝑗,(2.5) where 𝐶 is the constant of proportionality, and 𝑃 is the power of the applied stress.

3. Maximum Likelihood (ML) Estimation

In addition to the common assumptions in Section 2.2, we assume that the experiment is terminated at a prespecified censoring time 𝑇_𝑗. Thus, the corresponding likelihood function will be as in the following form: 𝐿=^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝐶𝑆^𝑃𝑗𝛾𝑒^{𝐶𝑆𝑃𝑗𝑥𝑖𝑗}1+𝛾𝜃𝑒^{𝐶𝑆𝑃𝑗𝑥𝑖𝑗}^−(𝜃+1)^𝛿𝑖𝑗1+𝜃𝛾𝑒^{𝐶𝑆𝑃𝑗𝑇𝑗}^−𝜃^{1−𝛿𝑖𝑗},(3.1) where 𝛿_𝑖𝑗 is an indicator variable, such that 𝛿_𝑖𝑗=1if𝑥^𝑖𝑗0if𝑥_𝑖𝑗≤𝑇_𝑗,>𝑇_𝑗.(3.2)

The log-likelihood function is ln𝐿=ln𝐶^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗+𝑃^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗ln𝑆_𝑗+ln𝛾^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗+𝐶^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝑆^𝑃𝑗𝑥_𝑖𝑗−(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗ln1+𝛾𝜃𝑒^{𝐶𝑆𝑃𝑗𝑥𝑖𝑗}−𝜃^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗ln1+𝜃𝛾𝑒^{𝐶𝑆𝑃𝑗𝑇𝑗}.(3.3)

3.1. ML Estimation of the Parameters

The first derivatives of the log-likelihood function (3.3) with respect to the unknown parameters 𝐶, 𝑃, 𝛾, and 𝜃 are 𝜕ln𝐿𝜕𝐶=𝐶^𝑘−1_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗+^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝑆^𝑃𝑗𝑥_𝑖𝑗−(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉_𝑖𝑗−𝜃^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍_𝑗𝜕ln𝐿,𝜕𝑃=^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗ln𝑆_𝑗+𝐶^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜎_𝑖𝑗−(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗ln𝑆_𝑗𝜉_𝑖𝑗−𝜃^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗ln𝑆_𝑗𝑍_𝑗𝜕ln𝐿,𝜕𝛾=𝛾1^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗−(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜈_𝑖𝑗−𝜃^𝑘𝜕ln𝐿_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇_𝑗,𝜕𝜃=^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜃+1𝜃𝜈^𝑖𝑗−𝜋_𝑖𝑗+^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇_𝑗−Λ_𝑗,(3.4) where 𝜉_𝑖𝑗=𝑠^𝑃𝑗𝑥_𝑖𝑗𝜈_𝑖𝑗, 𝜈_𝑖𝑗=(1+(𝜃/𝛾)𝑒^{−𝐶𝑆𝑃𝑗𝑥𝑖𝑗})⁻¹, 𝑍_𝑗=𝑆^𝑃𝑗𝑇_𝑗𝜇_𝑗, 𝜇_𝑗=(1+(𝜃/𝛾)𝑒^{−𝐶𝑆𝑃𝑗𝑇𝑗})⁻¹, 𝜋_𝑖𝑗=−ln(1−𝜈_𝑖𝑗), Λ_𝑗=−ln(1−𝜇_𝑗), and 𝜎_𝑖𝑗=𝑆^𝑃𝑗𝑥_𝑖𝑗ln𝑆_𝑗.

Since the first derivative equations (3.4) are nonlinear equations, their solutions will be obtained numerically by using the Math-Cade program as will be seen in Section 5.1. The second partial derivatives of the log-likelihood function (3.3) with respect to the parameters 𝐶, 𝑃, 𝛾, and 𝜃 are as follows: 𝜕²ln𝐿𝜕𝐶²=−1𝛾𝛾∑^𝑘𝑗=1∑^𝑛𝑗_𝑖=1𝛿_𝑖𝑗𝐶²+𝜃(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉^2𝑖𝑗𝑒^{−𝐶𝑆𝑃𝑗𝑥𝑖𝑗}+𝜃^2𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍^2𝑗𝑒^{−𝐶𝑆𝑃𝑗𝑇𝑗}𝜕₂,ln𝐿𝜕𝑃²=−𝐶^k_𝑛𝑗=1^𝑗_𝑖=1+𝜃_𝑘𝛿_𝑖𝑗ln𝑆_𝑗𝜎_𝑖𝑗𝐶(𝜃+1)𝜉_𝑖𝑗1−𝜈_𝑖𝑗−1+(𝜃+1)𝜉_𝑖𝑗ln𝑆_𝑗_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍_𝑗𝜕₂ln𝑆_𝑗𝐶Ω_𝑗1−𝜇_𝑗+ln𝑆_𝑗,ln𝐿𝜕𝛾²=−1𝛾²^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗−(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜈^2𝑖𝑗−𝜃^𝑘𝜕₂_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇^2𝑗,ln𝐿𝜕𝜃²=−1𝜃𝜃1^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜈_𝑖𝑗(1−𝜃)+(1+𝜃)1−𝜈_𝑖𝑗−^𝑘𝜕₂_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇^2𝑗,ln𝐿𝜕𝐶𝜕𝑃=−^𝑘_𝑛𝑗=1^𝑗_𝑖=1+𝜃_𝑘𝛿_𝑖𝑗(𝜃+1)𝜉_𝑖𝑗𝐶𝜎_𝑖𝑗1−𝜈_𝑖𝑗+ln𝑆_𝑗−𝜎_𝑖𝑗𝜕₂_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍_𝑗𝐶Ω_𝑗1−𝜇_𝑗+ln𝑆_𝑗,ln𝐿𝜕𝐶𝜕𝛾=−𝛾1(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉_𝑖𝑗1−𝜈_𝑖𝑗+𝜃^𝑘𝜕₂_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍_𝑗1−𝜇_𝑗,ln𝐿𝜕𝐶𝜕𝜃=−^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉_𝑖𝑗1−(𝜃+𝜃1)1−𝜈^𝑖𝑗+^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇_𝑗𝑍_𝑗𝜕₂,ln𝐿𝜕𝑃𝜕𝛾=−𝛾𝐶(𝜃+1)^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉_𝑖𝑗1−𝜈_𝑖𝑗ln𝑆_𝑗+𝜃^𝑘𝜕₂_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝑍_𝑗1−𝜇_𝑗ln𝑆_𝑗,ln𝐿𝜕𝑝𝜕𝜃=−𝐶^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜉_𝑖𝑗ln𝑆_𝑗1−1+𝜃⁻¹1−𝜈_𝑖𝑗+^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇_𝑗𝑍_𝑗ln𝑆_𝑗𝜕₂,ln𝐿𝜕𝛾𝜕𝜃=−𝛾1^𝑘_𝑛𝑗=1^𝑗_𝑖=1𝛿_𝑖𝑗𝜈_𝑖𝑗1−1+𝜃⁻¹1−𝜈_𝑖𝑗+^𝑘_𝑛𝑗=1^𝑗_𝑖=11−𝛿_𝑖𝑗𝜇^2𝑗,(3.5) where Ω_𝑗=𝑆^𝑃𝑗𝑇_𝑗ln𝑆_𝑗.

Therefore, the elements of the Fisher information matrix for the MLE can be obtained as the expectations of the negative of the second partial derivatives, that is,𝐹=⎛⎜⎜⎜⎝𝑓¹¹𝑓₁₂𝑓₁₃𝑓₁₄𝑓₂₂𝑓₂₃𝑓₂₄𝑓₃₃𝑓₃₄𝑓₄₄⎞⎟⎟⎟⎠=−𝐸⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜕²ln𝐿𝜕𝑐²𝜕²ln𝐿𝜕𝑐𝜕𝑝𝜕²ln𝐿𝜕𝑐𝜕𝛾𝜕²ln𝐿𝜕𝜕𝑐𝜕𝜃₂ln𝐿𝜕𝑝²𝜕²ln𝐿𝜕𝑝𝜕𝛾𝜕²ln𝐿𝜕𝜕𝑝𝜕𝜃₂ln𝐿𝜕𝛾²𝜕²ln𝐿𝜕𝜕𝛾𝜕𝜃₂ln𝐿𝜕𝜃²⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(3.6) The asymptotic variance-covariance matrix for the MLE is defined as the inverse of the Fisher information matrix (3.6), that is, Σ=𝐹⁻¹.(3.7)

3.2. Prediction of the Scale Parameter and the Reliability Function

To predict the value of the scale parameter 𝛼_𝑢 under the usual condition stress 𝑉_𝑢, the invariance property of MLE is used (for more details see Meeker et al. [8]), that is, 𝛼=𝐶𝑆_𝑢𝑃,where𝑆_𝑢=𝑉^∗𝑉_𝑢𝑉_∗,=^𝑘_𝑗=1𝑉^𝑏𝑗_𝑗,𝑏_𝑗=𝑛^𝑗∑^𝑘𝑗=1𝑛_𝑗.(3.8)

The MLE of the reliability function at the lifetime 𝑥₀ under the usual condition stress 𝑉_𝑢, is given by 𝑅_𝑢𝑥₀=1+𝛾𝜃𝑒𝛼^𝑢𝑥0^−𝜃.(3.9)

4. Optimum Test Plan

Before starting an accelerated life test (which is sometimes an expensive and difficult endeavor), it is advisable to have a plan that helps in accurately estimating reliability at operating conditions while minimizing test time and cost.

In this section, we determine the best choice of the values of the censoring times 𝑇_𝑗, 𝑗=1,…,𝑘, according to the D-Optimality criterion. The criterion is based on the minimization of the determinant of Fisher information matrix of the MLE of the model parameters (Gouno [25]). Therefore, the optimal value of 𝑇_𝑗 at each stress level 𝑉_𝑗, 𝑗=1,…,𝑘, can be obtained by solving the following equation: 𝜕||𝐹||𝜕𝑇_𝑗=0,𝑗=1,…,𝑘.(4.1) The determinant of 𝐹 and the derivation of (4.1) are placed in the appendix. To get the optimum values of the stress change time 𝑇₁ and 𝑇₂ that minimized |𝐹| of the MLE under the stress level 𝑉_𝑗, 𝑗=1,2, the numerical solution is obtained as will be shown in Section 5.2.

5. Simulation Studies

This section presents the numerical solutions to obtain the MLE of the unknown parameters 𝐶, 𝑃, 𝛾, and 𝜃, their mean squared errors (MSEs), relative absolute biases (RAB), Lower Bound (LB), Upper Bound (UB), the estimated of scale parameter 𝛼, and reliability function 𝑅(𝑥₀) under normal use conditions 𝑉_𝑢. Also, it presents the numerical solutions to determine the best choice values of the stress change time 𝑇^∗1 and 𝑇^∗2.

5.1. MLE under Type-I Censoring

In this section, the numerical solution is performed according to the following steps.(i)For given values of 𝐶, 𝑃, and stress level 𝑉_𝑗, 𝑗=1,2,3, the values of 𝛼_𝑗, 𝑗=1,2,3 are calculated according to (2.5).(ii)Generate a random sample of size 𝑁 from the 3-parameter-generalized logistic distribution and obtain the random variables 𝑥_𝑖𝑗, {𝑖=1,…,𝑛_𝑗,𝑗=1,…,𝑘} for given values of 𝑛_𝑗, 𝑇_𝑗, 𝑗=1,…,𝑘, and different initial guesses of the true parameters 𝛼, 𝛾, 𝜃, say 𝛼₀, 𝛾₀, 𝜃₀.(iii)Based on the values of 𝑛_𝑗, 𝑇_𝑗, 𝑉_𝑗, 𝑥_𝑖𝑗, {𝑖=1,…,𝑛_𝑗,𝑗=1,…,𝑘}, and 𝑉_𝑢, the MLE and their MSE, RAB, LB, and UB, in addition to 𝛼_𝑢 and 𝑅_𝑢(𝑥₀), are obtained.(iv)The steps are repeated more than 150 times until getting the MLE as shown in Table 1.

Table 1 

The MLE, RAB, and MSE.

The numerical results which are placed in Tables 1 to 4 are based on 𝑛₁=29, 𝑛₂=10, 𝑛₃=2, 𝑇₁=4, 𝑇₂=3, 𝑇₃=2, 𝑉₁=0.75, 𝑉₂=1.5, 𝑉₃=2.25, and 𝑉_𝑢=0.5.

From the results of Tables 1, 2, 3 and 4, we observe that the MSE of the scale parameter 𝛼_𝑗, 𝑗=1,2,3, decreases as the stress value 𝑉_𝑗, 𝑗=1,2,3, increases. In general, the MSE decrease as the values of 𝐶₀ and 𝛾₀ decreases at the same values of 𝑃₀ and 𝜃₀, while in the variance-covariance matrix we observe that the covariance between 𝐶 and 𝑃 is the smallest one and converges to zero. The reliability decrease when the mission time 𝑥₀ increases. It is reduced when the values of 𝐶₀ decrease and the values of 𝑃₀, 𝛾₀, and 𝜃₀ increase at the same mission time.

Table 2 

The confidence intervals.

Table 3 

The asymptotic variance-covariance matrix.

Table 4 

Estimates 𝛼 and 𝑅(𝑥₀) under normal conditions.

5.2. Optimum Test Plan of Censoring Times

In order to obtain the optimum values of the censoring times 𝑇₁ and 𝑇₂, a numerical solution is performed by generating a random sample of size 𝑁 from the 3-parameter-generalized logistic distribution. The random variables 𝑥_𝑖𝑗{𝑖=1,…,𝑛_𝑗,𝑗=1,2} for given values of 𝑛_𝑗, 𝑇_𝑗, 𝑗=1,2, and different values of (𝐶₀, 𝑃₀, 𝛾₀, 𝜃₀) are obtained. Therefore, the optimum values of the stress change time 𝑇₁, 𝑇₂, say 𝑇^∗1, 𝑇^∗2, and the Generalized Asymptotic Variance (GAV) are computed. The numerical results are placed in Tables 5, 6, 7, 8, and 9, which are also based on 𝑛₁=29, 𝑛₂=10, 𝑇₁=4, 𝑇₂=3, 𝑉₁=0.75, 𝑉₂=1.5, and 𝑉_𝑢=0.5.

Table 5 

The optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at (𝐶₀=1, 𝑃₀=1, 𝛾₀=1.5, 𝜃₀=1).

Table 6 

The optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at (𝐶₀=1.1, 𝑃₀=1, 𝛾₀=1.3, 𝜃₀=1).

Table 7 

The optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at (𝐶₀=1.25, 𝑃₀=1.2, 𝛾₀=1.2, 𝜃₀=1).

Table 8 

The optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at (𝐶₀=1.2, 𝑃₀=1.2, 𝛾₀=1.5, 𝜃₀=1.2).

Table 9 

The optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at (𝐶₀=1, 𝑃₀=1.1, 𝛾₀=1.4, 𝜃₀=1).

As seen, the results show the optimum values of 𝑇^∗1, 𝑇^∗2, and GAV at different values of (𝐶₀, 𝑃₀, 𝛾₀, 𝜃₀) and different values of 𝑁. In general, we observe that GAV decreases as the sample size increases.

6. Conclusion

The GL distribution has been extensively used in many different areas and it is very useful in a wide variety of applications, especially in the analysis of survival data. In addition, it is used as the name for several different families of probability distribution. This paper presented the Maximum Likelihood method of the parameter estimation with type-I censoring. The data failure times at each stress level are assumed to follow the 3-parameter-generalized logistic distribution with scale parameter that is an inverse power law function. The ML estimation, Fisher’s information matrix, the asymptomatic variance-covariance matrix, the prediction of the value of the scale parameter, and the reliability function under the usual conditions stress were obtained for various combinations of the model parameters. In additional, the corresponding optimum value of the stress change time is obtained numerically by the D-optimality criterion. Since standard Logistic, four-parameter-extended GL, four-parameter-extended GL type-I, two-parameter GL, type-I GL, Generalized Log-logistic, standard Log-logistic, Logistic Exponential, Exponentiated Exponential (for 𝑥>0), Generalized Burr, Burr III, and Burr XII distributions are special cases from the GL distribution, then their results of the MLE and optimum test plan become particular cases of the results obtained here.

Appendix

The determinant of 𝐹 is ||𝐹||=+𝑓𝑓₃₃𝑓₄₄₂₃−𝑓²³⁴𝑓₁₁𝑓₂₂−𝑓²¹²−𝑓₂₃𝑓₄₄−𝑓₂₄𝑓₃₄𝑓₁₁𝑓₂₃−𝑓₁₂𝑓₁₃+𝑓₁₃𝑓₃₄−𝑓₂₄𝑓₃₃𝑓₁₁𝑓₂₄−𝑓₁₂𝑓₁₄−𝑓₁₃𝑓₄₄−𝑓₁₄𝑓₃₄𝑓₁₃𝑓₂₂−𝑓₁₂𝑓₂₃𝑓₃₄−𝑓₃₃𝑓₁₄𝑓₁₄𝑓₂₂−𝑓₁₂𝑓₂₄−𝑓₁₃𝑓₂₄−𝑓₂₃𝑓₁₄𝑓₁₄𝑓₂₃−𝑓₁₃𝑓₂₄.(A.1)

The derivative of |𝐹| for obtaining the optimum value of the censoring time 𝑇_𝑗, 𝑗=1,…,𝑘 is given as follows: 𝜕||𝐹||𝜕𝑇_𝑗=+𝑓𝑓₃₃𝑓_4433−𝑓²³⁴𝑓^11𝑓₂₂+𝑓₁₁𝑓^22−2𝑓₁₂𝑓^12−𝑓_23𝑓₄₄+𝑓₃₃𝑓^44−2𝑓^34𝑓₃₄𝑓₁₁𝑓₂₂−𝑓²¹²−𝑓₂₃𝑓₄₄+𝑓₂₃𝑓^44−𝑓^24𝑓₃₄−𝑓₂₄𝑓^34𝑓₁₁𝑓₂₃−𝑓₁₂𝑓₁₃+𝑓₂₃𝑓₄₄−𝑓₂₄𝑓₃₄𝑓^11𝑓₂₃+𝑓₁₁𝑓^23−𝑓^12𝑓₁₃−𝑓₁₂𝑓^13+𝑓_23𝑓₃₄−𝑓₂₄𝑓₃₃𝑓^11𝑓₂₄+𝑓₁₁𝑓^24−𝑓^12𝑓₁₄−𝑓₁₂𝑓^14−𝑓₁₃𝑓₃₄+𝑓₂₃𝑓^34−𝑓^24𝑓₃₃−𝑓₂₄𝑓^33𝑓₁₁𝑓₂₄−𝑓₁₂𝑓₁₄−𝑓_13𝑓₄₄−𝑓₁₄𝑓₃₄𝑓^13𝑓₂₂+𝑓₁₃𝑓^22−𝑓^12𝑓₂₃−𝑓₁₂𝑓^23+𝑓₁₃𝑓₄₄+𝑓₁₃𝑓^44−𝑓^14𝑓₃₄−𝑓₁₄𝑓^34𝑓₁₃𝑓₂₂−𝑓₁₂𝑓₂₃+𝑓_13𝑓₃₄−𝑓₃₃𝑓₁₄𝑓^14𝑓₂₂+𝑓₁₄𝑓^22−𝑓^12𝑓₂₄−𝑓₁₂𝑓^24−𝑓₁₃𝑓₃₄+𝑓₁₃𝑓^34−𝑓^33𝑓₁₄−𝑓₃₃𝑓^14𝑓₁₄𝑓₂₂−𝑓₁₂𝑓₂₄−𝑓_13𝑓₂₄−𝑓₂₃𝑓₁₄𝑓^14𝑓₂₃+𝑓₁₄𝑓^23−𝑓^13𝑓₂₄−𝑓₁₃𝑓^24𝑓₂₄+𝑓₁₃𝑓^24−𝑓^23𝑓₁₄−𝑓₂₃𝑓^14𝑓₁₄𝑓₂₃−𝑓₁₃𝑓₂₄,(A.2) where𝑓^11=𝜃²𝛾^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆_𝑗𝑃𝑓𝑍_𝑗1−𝜇_𝑗𝐶𝑆_𝑗𝑃𝑇_𝑗1−2𝜇_𝑗+2_22,=𝐶𝜃^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗ln𝑆_𝑗²𝑓_33𝜇_𝑗𝐶𝑆^𝑗𝑃𝑇_𝑗1−𝜇_𝑗1−𝐶𝑍_𝑗+1+𝐶𝑆_𝑗𝑃𝑇_𝑗1−𝜇_𝑗²,=−2𝐶𝜃𝛾^2𝑛𝑗_𝑖=1𝑓_441−𝛿_𝑖𝑗𝑆_𝑗𝑃1−𝜇_𝑗𝜇^2𝑗,=−2𝐶𝜃^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆_𝑗𝑃1−𝜇_𝑗𝜇^2𝑗𝑓_12,=𝜃^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝑓_13ln𝑆_𝑗𝜇_𝑗𝐶𝑆^𝑗𝑃𝑇_𝑗1−𝜇_𝑗1−𝐶𝑍_𝑗+1+𝐶𝑆_𝑗𝑃𝑇_𝑗1−𝜇_𝑗²,=𝜃𝛾^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝑓𝜇_𝑗1−𝜇_𝑗1+𝐶𝑆_𝑗𝑃𝑇_𝑗1−2𝜇_𝑗_14,=^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝑓𝜇_𝑗21+2𝐶𝑆_𝑗𝑃𝑇_𝑗1−𝜇_𝑗_23,=𝐶𝜃𝛾^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝑓ln𝑆_𝑗𝜇_𝑗1−𝜇_𝑗1+𝐶𝑆_𝑗𝑃𝑇_𝑗1−2𝜇_𝑗_24,=𝐶^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝑓ln𝑆_𝑗𝜇_𝑗21+2𝐶𝑆_𝑗𝑃𝑇_𝑗1−𝜇_𝑗_34,=2𝐶𝛾^𝑛𝑗_𝑖=11−𝛿_𝑖𝑗𝑆^𝑃𝑗𝜇^2𝑗1−𝜇_𝑗.(A.3)