In reliability engineering and lifetime analysis, many units of the product fail with different causes of failure, and some tests require stress higher than normal stress. Also, we need to design the life experiments which present methodology for formulating scientific and engineering problems using statistical models. So, in this paper, we adopted a partially constant stress accelerated life test model to present times to failure in a small period of time for Gompertz life products. Also, considering that, units are failing with the only two independent causes of failure and tested under type-I generalized hybrid censoring scheme the data built. Obtained data are analyzed with two methods of estimations, maximum likelihood and Bayes methods. These two methods are used to construct the point and interval estimators with the help of the MCMC method. The developed results are measured and compared under Monte Carlo studying. Also, a data set is analyzed for illustration purposes. Finally, some comments are presented to describe the numerical results.

The data under life-testing experiments may be complete or censoring data; the words of complete data set are used when the time to failure for all units under the test is obtained. But, the word censoring data is used when some but not all data about tested units is obtained. Type-I and type-II censoring schemes are from the oldest censoring schemes in life testing experiments. In the type-I censoring scheme, tested time is prefixed and the number of failure units is random. But, in type-II censoring scheme, the tested time is random and the number of failure units is prefixed. The two types of censoring have the lack of memory where the number of failure units may be very small or zero in type-I censoring scheme, but the total time of the test may be very large in type-II censoring scheme. And, the joint censoring scheme of type-I and type-II is called hybrid censoring scheme.

In the plan of type-I hybrid censoring scheme (type-I HCS),

For type-I GHCS,

For type-II GHCS,

In life testing experiments, the problem of obtaining sufficient information about the life of a product under recent technology is more difficult for a long-life product. Then, the related statistical inferences became more difficult. This problem can be solved with a good choice of the type of censoring scheme. Another solution to this problem is exposing the test unit to stress higher than normal stress conditions which is known as accelerated life tests (ALTs). Studies [

The main objective in this paper is adopting the type-I GHCS with a partially constant ALT model when test units fail with only two independent causes of failure and the failure time is distributed with Gompertz distribution (GD). The Gompertz lifetime population with random variable

The paper is organized as follows. Section

In this section, we present the list of abbreviations that are used in the paper as well as a complete description of the model mechanism and the corresponding likelihood function.

GD : Gompertz distribution

MLE : maximum likelihood estimation

ME : mean

PC : probability coverage

CDF : cumulative distribution function

HRF : hazard failure rate function

MH : Metropolis–Hastings algorithm

CI : credible intervals

MCMC : Markov chain Monte Carlo

MSE : mean squared error

ML : mean interval length

PDF : probability density function

SF : survival function

SEL : squared error loss

ACI : approximate confidence interval

CDF : cumulative distribution function

Suppose that

Considering that, tested units with CDF given by (

The Gompertz lifetime distribution with common shape parameters

Consider that the proportional hazard model (also named Cox model) is relevant to handle the effects of the environment or stress on the lifetime distribution. Thus, the survival function

Therefore,

The results of the point and asymptotic confidence intervals of model parameters are discussed in this section with MLE for two independent causes of failure.

Let

The point MLE of model parameters can be obtained after taking the partial derivatives of (

Then, the likelihood equations are reduced to one linear equation of

The Fisher information matrix is defined as the minus expectation of second derivatives from the log-likelihood function with respect to model parameters. In practice, the expectation problem of the second derivative is more difficult to practice; then, the approximate Fisher information presents a suitable approximation that is used to build interval estimation as follows. Let

Then, the approximate information matrix

Then,

In this section, we present the Bayes point and interval estimation for the unknown model parameters to formulate the available information in the form of statistical distribution. The available information is exposed in the prior information and information exposed in the data. So, we consider the independent gamma priors for Gompertz parameters {

Then, the joint prior density is given by

The joint posterior distribution can be formulated by

The Bayes estimate of model parameters depends on the posterior distribution and the choice of the loss function. Without loss of generality, considering squared error loss function (SEL), the Bayes estimators for any function

Integration in (

The problem estimation with the Bayesian approach with the help of the MCMC method is needed to build the posterior conditional distributions of model parameters as follows:

Therefore, the problem of building conditional distributions of (

The conditional distribution (

Step 1: begin with initial parameter values

Step 2: gamma distribution is used to generate

Step 3: normal proposal distribution is used to generate

Step 4: report the value of the parameters vector

Step 5: put

Step 6: repeat steps (2) to (5)

Step 7: determine the iteration number that is needed for the stationary state

and the corresponding variance is defined by

where

Step 8: the ordered value of the vector

In this section, we assess the developed results in classical MLE or Bayesian approach for different combinations of sample size

MEs and MSEs for

MLE | ||||||||
---|---|---|---|---|---|---|---|---|

Par. | MEs | MSEs | MEs | MSEs | MEs | MSEs | ||

5.0 | (70, 35, 35, 15, 30) | 0.1231 | 0.0825 | 0.1202 | 0.0811 | 0.1187 | 0.0642 | |

0.0624 | 0.0342 | 0.0611 | 0.0325 | 0.0600 | 0.0227 | |||

0.0872 | 0.0536 | 0.0851 | 0.0514 | 0.0817 | 0.0324 | |||

3.3124 | 1.2251 | 3.3100 | 1.2227 | 2.4250 | 1.0011 | |||

(70, 35, 35, 25, 30) | 0.1175 | 0.0724 | 0.1145 | 0.0692 | 0.1101 | 0.0592 | ||

0.0584 | 0.0300 | 0.0571 | 0.0297 | 0.0542 | 0.0201 | |||

0.0832 | 0.0482 | 0.0817 | 0.0491 | 0.0813 | 0.0287 | |||

2.8142 | 1.1243 | 2.7842 | 1.1145 | 2.254 | 0.9852 | |||

(70, 30, 40, 15, 30) | 0.1250 | 0.0831 | 0.1215 | 0.0831 | 0.1199 | 0.0651 | ||

0.0641 | 0.0357 | 0.0624 | 0.0342 | 0.0621 | 0.0240 | |||

0.0893 | 0.0548 | 0.0867 | 0.0528 | 0.0825 | 0.0329 | |||

3.3139 | 1.2263 | 3.3115 | 1.2233 | 2.4262 | 1.0026 | |||

(70, 30, 40, 25, 30) | 0.1182 | 0.0731 | 0.1132 | 0.0699 | 0.1122 | 0.0601 | ||

0.0591 | 0.0314 | 0.0582 | 0.0299 | 0.0556 | 0.0214 | |||

0.0850 | 0.0480 | 0.0840 | 0.0481 | 0.0825 | 0.0298 | |||

2.8155 | 1.1251 | 2.7856 | 1.1152 | 2.2539 | 0.9857 | |||

10 | (70, 35, 35, 15, 30) | 0.1241 | 0.0782 | 0.1199 | 0.0772 | 0.1101 | 0.0574 | |

0.0615 | 0.0302 | 0.0574 | 0.0289 | 0.0564 | 0.0200 | |||

0.0860 | 0.0489 | 0.0841 | 0.0470 | 0.0821 | 0.0241 | |||

3.3002 | 0.9945 | 3.0001 | 0.9880 | 2.2135 | 0.8852 | |||

(70, 35, 35, 25, 30) | 0.1177 | 0.0623 | 0.1199 | 0.0711 | 0.1082 | 0.0452 | ||

0.0562 | 0.0278 | 0.0541 | 0.0255 | 0.0522 | 0.0185 | |||

0.0811 | 0.0350 | 0.0810 | 0.0442 | 0.0817 | 0.0211 | |||

2.7142 | 0.9799 | 2.6452 | 0.9777 | 2.2101 | 0.8800 | |||

(70, 30, 40, 15, 30) | 0.1271 | 0.0782 | 0.1199 | 0.0792 | 0.1117 | 0.0574 | ||

0.0632 | 0.0321 | 0.0588 | 0.0301 | 0.0575 | 0.0215 | |||

0.0871 | 0.0499 | 0.0857 | 0.0490 | 0.0829 | 0.0254 | |||

3.3022 | 0.9962 | 3.0025 | 0.9898 | 2.2128 | 0.8870 | |||

(70, 30, 40, 25, 30) | 0.1181 | 0.0636 | 0.1214 | 0.0718 | 0.1099 | 0.0460 | ||

0.0577 | 0.0291 | 0.0562 | 0.0263 | 0.0538 | 0.0197 | |||

0.0815 | 0.0370 | 0.0819 | 0.0460 | 0.0822 | 0.0215 | |||

2.7155 | 0.9812 | 2.6471 | 0.9772 | 2.2122 | 0.8817 |

MLs and PCs for 95% interval estimation

MLE | ||||||||
---|---|---|---|---|---|---|---|---|

Par. | ML | PC | ML | PC | ML | PC | ||

5.0 | (70, 35, 35, 15, 30) | 0.421 | 0.90 | 0.399 | 0.90 | 0.284 | 0.91 | |

0.124 | 0.89 | 0.134 | 0.91 | 0.114 | 0.92 | |||

0.215 | 0.88 | 0.200 | 0.90 | 0.141 | 0.91 | |||

4.125 | 0.90 | 4.101 | 0.91 | 3.521 | 0.96 | |||

(70, 35, 35, 25, 30) | 0.321 | 0.90 | 0.311 | 0.92 | 0.184 | 0.92 | ||

0.101 | 0.91 | 0.094 | 0.91 | 0.089 | 0.92 | |||

0.187 | 0.90 | 0.160 | 0.93 | 0.118 | 0.92 | |||

3.421 | 0.91 | 3.405 | 0.91 | 3.011 | 0.93 | |||

(70, 30, 40, 15, 30) | 0.435 | 0.89 | 0.412 | 0.90 | 0.295 | 0.91 | ||

0.135 | 0.89 | 0.155 | 0.89 | 0.127 | 0.92 | |||

0.220 | 0.89 | 0.214 | 0.91 | 0.161 | 0.93 | |||

4.131 | 0.89 | 4.117 | 0.91 | 3.528 | 0.90 | |||

(70, 30, 40, 25, 30) | 0.341 | 0.90 | 0.318 | 0.92 | 0.197 | 0.93 | ||

0.118 | 0.90 | 0.112 | 0.90 | 0.096 | 0.91 | |||

0.199 | 0.90 | 0.172 | 0.92 | 0.124 | 0.92 | |||

3.434 | 0.90 | 3.414 | 0.91 | 3.018 | 0.91 | |||

10 | (70, 35, 35, 15, 30) | 0.314 | 0.90 | 0.305 | 0.91 | 0.200 | 0.93 | |

0.095 | 0.91 | 0.088 | 0.91 | 0.078 | 0.92 | |||

0.147 | 0.90 | 0.130 | 0.92 | 0.095 | 0.91 | |||

4.098 | 0.91 | 4.001 | 0.91 | 3.324 | 0.95 | |||

(70, 35, 35, 25, 30) | 0.275 | 0.93 | 0.266 | 0.92 | 0.170 | 0.94 | ||

0.082 | 0.91 | 0.076 | 0.93 | 0.051 | 0.92 | |||

0.095 | 0.93 | 0.081 | 0.92 | 0.068 | 0.93 | |||

4.072 | 0.91 | 3.865 | 0.92 | 3.214 | 0.94 | |||

(70, 30, 40, 15, 30) | 0.331 | 0.91 | 0.322 | 0.91 | 0.211 | 0.91 | ||

0.112 | 0.90 | 0.093 | 0.92 | 0.085 | 0.94 | |||

0.162 | 0.90 | 0.148 | 0.92 | 0.099 | 0.94 | |||

4.107 | 0.92 | 4.014 | 0.90 | 3.341 | 0.94 | |||

(70, 30, 40, 25, 30) | 0.282 | 0.90 | 0.271 | 0.90 | 0.173 | 0.94 | ||

0.087 | 0.91 | 0.085 | 0.93 | 0.064 | 0.93 | |||

0.107 | 0.92 | 0.092 | 0.91 | 0.072 | 0.95 | |||

4.078 | 0.92 | 3.870 | 0.94 | 3.223 | 0.92 |

In this section, we choose the set of data generated from GD with respect to type-I GHCS and accelerated under partially constant stress ALTs. The random sample is generated from two GDs over the following algorithms.

Step 1: let the total sample take the size

Step 2: suppose that the parameter vectors are randomly chosen to be

Step 3: the prior information is almost selected to satisfy

Step 4: generate two samples of size

Step 5: step 4 is repeated for distribution (

Step 6: compute

The data under used and accelerated conditions with its cause of failure is obtained from Tables

The generated data under used conditions.

0.1844 | 0.1900 | 0.2027 | 0.5829 | 0.7032 | 0.9103 | 1.2755 | 1.4515 | 1.5733 | 1.6700 |

0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

1.747 | 1.7529 | 1.7958 | 1.9991 | 2.0656 | 2.3661 | 2.3773 | 2.458 | 2.5426 | 2.9961 |

1 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 |

3.3552 | 3.3924 | 3.4199 | 3.4412 | 3.4652 | 3.8778 | 4.1908 | 4.2606 | 4.811 | 4.9416 |

1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |

The generated data under accelerated conditions.

0.0746 | 0.1233 | 0.1298 | 0.1596 | 0.1950 | 0.3845 | 0.4421 | 0.5080 | 0.6562 | 0.8073 |

1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 0 |

0.9045 | 0.9650 | 1.1242 | 1.3739 | 1.4121 | 1.5416 | 1.5892 | 1.6191 | 1.6475 | 1.6705 |

0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |

1.7388 | 1.7909 | 1.8932 | 2.1856 | 2.2012 | 2.2866 | 2.2870 | 2.6332 | 2.6470 | 2.7982 |

0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

Point and 95% confidence and credible intervals (ACIs and CIs) of MLE Bayes estimates.

Pa.s | 95% ACIs | Length | 95% CIs | Length | ||
---|---|---|---|---|---|---|

0.7431 | 0.7271 | (0.2894, 1.1967) | 0.9073 | (0.3131, 1.1825) | 0.8694 | |

0.2808 | 0.2689 | (0.0454, 0.5162) | 0.4708 | (0.0644, 0.5542) | 0.3898 | |

0.3229 | 0.3076 | (0.0571, 0.5888) | 0.5317 | (0.1327, 0.5749) | 0.4422 | |

1.4976 | 1.7884 | (0.5732, 2.4220) | 1.8488 | (0.8346, 3.4357) | 2.6011 |

Simulation number of

The histogram of

Simulation number of

The histogram of

Simulation number of

The histogram of

Simulation number of

The histogram of

Modern technology products have a long period of time, and information about the life product is more difficult. Then, to overcome this problem, the experimenter determined the censoring scheme that serves this problem. In this paper, we choose type-I GHCS which keeps the minimum number needed in statistical inference in a small period of time. Also, this type of censoring is applied with the concept of partially constant ALTs for units that fail under two independent causes of failure and units that have Gompertz lifetime distribution. Moreover, we have seen that the proposed model can be easily extended for different populations. Also. we can mention that we can use a more general class of prior information such as priors with log-concave density functions. The simulation study is conducted, and the results are reported in Tables

The proposed model is more acceptable

The results for the large value of

The results are getting better for increasing values of increasing (

The results are getting better for closed values of (

The results under MLEs and noninformative priors are closed

Bayesian estimation under informative prior is better than MLEs

The results for the selected set of parameters are more acceptable

No data were used to support the findings of this study.

The authors declare that they have no conflicts of interest.