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Modern reliability engineering accelerated life tests (ALT) and partially accelerated life tests (PALT) are widely used to obtain the timely information on the reliability of objects, products, elements, and materials as well as to save time and cost. The ALTs or PALTs are useful in determining the failed manners of the items at routine conditions by using the information of the data generated from the experiment. PALT is the most sensible method to be used for estimating both ordinary and ALTs. In this research, constant stress PALT design for the Fréchet distribution with type-I censoring has been investigated due to a wide applicability of the Fréchet distribution in engineering problems especially in hydrology. The distribution parameters and acceleration factor are obtained by using the maximum likelihood method. Fisher's information matrix is used to develop the asymptotic confidence interval estimates of the model parameters. A simulation study is conducted to illustrate the statistical properties of the parameters and the confidence intervals by using the R software. The results indicated that the constant stress PALT plan works well. Moreover, a numerical example is given to exemplify the performance of the proposed methods.

The continuous progress in manufacturing design makes the products and materials highly reliable [

Particularly, when test items are run at both normal and higher than normal stress conditions, the more suitable test to be applied is the partially accelerated life test (PALT) [

The commonly used methods in PALT are constant stress PALT (CSPALT) [

Additionally, censoring is very ordinary in life testing experiments. It frequently happens that the experiment is censored when the experimenter may not be able to examine the lifespan of all units put on the test because of time restrictions and other limitations on the data collection. It is usually used when a distribution of exact lifetimes is known only for a part of the test items, and the remainder of the lifetimes is known only to exceed certain values under a life test. The two most common types of censoring are type-I censoring (time censoring) and type-II censoring (failure censoring).

For a brief overview of PALT, literature is abundant on designing PALT. Bai and Chung [

In this paper, the CSPALT plan using type-I censoring with the assumption that the lifetimes of the test items at use condition follow Fréchet distribution which is considered. Maurice Fréchet was a French mathematician who had identified one possible limit distribution for the largest order statistic in 1927 [

The reliability function of Fréchet distribution is

The Fréchet distribution is applied to extreme events and linked to the modeling of several real-world phenomena, including human lifetimes, flood and seismic analyses, radioactive emissions, and maximum one-day rainfalls.

The rest of the study is arranged as follows. In Section

The following notations are used for model description:

In CSPALT, total test units “

The following assumptions are also made for CSPALT.

Under normal-use condition, lifespan of an object supports Fréchet distribution.

The lifespan of an object at accelerated condition is attained by ^{−1}

The lifespan

The lifespan

The lifetimes

The ML method is one of the most important and widely used methods in statistics. The main reasons are that the ML method is very vigorous and provides the estimates of parameters with good statistical properties such as consistency, asymptotic unbiased, asymptotic efficiency, and asymptotic normality.

Let

The likelihood function for

The total likelihood for

It is more convenient to work with the log-likelihood function. The log-likelihood function of (

Let

MLEs of

The above equations cannot be written in the closed form. So, an iterative procedure can be used to get MLEs. Here, we use the BFGS Quasi-Newton Optimization method which is available in the R software. Furthermore, for interval estimates of model parameters, we need the Fisher information matrix.

Here, it is hard to obtain an explicit solution to nonlinear equations, so an iterative method such as the Newton Raphson method [

The 100 (1 −

A simulation study is carried out to evaluate the performance of proposed estimators. It is performed by using the R software (3.2.3) for demonstrating the theoretical outcomes of the estimation problems. The performances of MLEs are evaluated through mean square error (MSE) and absolute bias of estimates. Also, 90% and 95% confidence limits are constructed for parameters and acceleration factors. For this purpose, several data sets are generated from two-parameter Fréchet distribution with sizes

The simulation steps are summarized below:

For estimation, the sample size is varied to see the effect of large and small samples are considered. The random samples are generated from Fréchet distribution by using transformation

The values for true parameters are taken as (

The “

The estimates of the model parameters and their corresponding summary statistics are obtained by the CSPALT model.

For the different combination of stresses, the MLEs for

Tables

Figures

Average estimates of CSPLAT with

Parameters | Estimates | MSE | Absolute bias | 95% CI | |
---|---|---|---|---|---|

50 | 1.5870 | 0.8109 | 0.8370 | −0.8226, 3.9966 | |

1.1830 | 0.1229 | 0.0169 | 0.49494, 1.8711 | ||

0.4159 | 0.3485 | 0.5840 | −1.2117, 2.0436 | ||

100 | 1.4755 | 0.5836 | 0.7352 | −0.5724, 3.5234 | |

1.2635 | 0.0715 | 0.0531 | 0.7797, 1.7473 | ||

0.4147 | 0.3465 | 0.5858 | −1.2112, 2.0408 | ||

150 | 1.4603 | 0.53274 | 0.7103 | −0.5569, 3.5019 | |

1.2914 | 0.04893 | 0.0776 | 0.8076, 1.7464 | ||

0.4136 | 0.34611 | 0.5864 | −1.2172, 2.0429 | ||

200 | 1.4386 | 0.49254 | 0.68863 | −0.5043, 3.3997 | |

1.2992 | 0.04160 | 0.09918 | 0.8576, 1.7275 | ||

0.4126 | 0.34663 | 0.58743 | −1.2125, 2.0402 | ||

300 | 1.4221 | 0.46364 | 0.67210 | −0.4592, 3.3088 | |

1.3089 | 0.03655 | 0.10886 | 0.8753, 1.7414 | ||

0.4128 | 0.34592 | 0.58724 | −1.2159, 2.0416 |

Average estimates of CSPLAT with

Parameters | Estimates | MSE | Absolute bias | 95% CI | |
---|---|---|---|---|---|

50 | 2.1466 | 1.4601 | 1.12465 | −1.1840, 5.5040 | |

0.7112 | 0.46457 | 0.38876 | −0.9333, 2.2885 | ||

0.5080 | 0.24811 | 0.49196 | −0.8467, 1.8718 | ||

100 | 1.9824 | 1.04411 | 0.98242 | −0.8141, 4.7996 | |

0.7513 | 0.28392 | 0.34868 | −0.4538, 1.9765 | ||

0.5161 | 0.23707 | 0.48393 | −0.8336, 1.8634 | ||

150 | 1.9443 | 0.94151 | 0.94430 | −0.7090, 4.5968 | |

0.8099 | 0.23370 | 0.29013 | −0.2890, 1.9258 | ||

0.5156 | 0.23648 | 0.48444 | −0.8350, 1.8635 | ||

200 | 1.9318 | 0.90238 | 0.93182 | −0.6499, 4.4828 | |

0.8066 | 0.21864 | 0.29344 | −0.2114, 1.9076 | ||

0.5160 | 0.23574 | 0.48405 | −0.8345, 1.8628 | ||

300 | 1.9043 | 0.84035 | 0.90430 | −0.6121, 4.4129 | |

0.8315 | 0.18293 | 0.26852 | −0.1070, 1.8217 | ||

0.5166 | 0.23464 | 0.48344 | −0.8333, 1.8619 | ||

500 | 1.8991 | 0.82207 | 0.89914 | −0.5948, 4.3840 | |

0.8488 | 0.16980 | 0.25116 | −0.0358, 1.7549 | ||

0.5145 | 0.23628 | 0.48553 | −0.8374, 1.8635 |

Average estimates of CSPLAT with

Parameters | Estimates | MSE | Absolute bias | 95% CI | |
---|---|---|---|---|---|

50 | 1.7092 | 0.9542 | 0.9092 | −0.9064, 4.3248 | |

1.2032 | 0.2057 | 0.0032 | 0.3143, 2.0921 | ||

0.3449 | 0.2111 | 0.4551 | −0.9225, 1.6123 | ||

100 | 1.5806 | 0.6579 | 0.7806 | −0.6258, 3.7871 | |

1.2256 | 0.1198 | 0.0257 | 0.5454, 1.9059 | ||

0.3519 | 0.2028 | 0.4480 | −0.8932, 1.5971 | ||

150 | 1.5592 | 0.6060 | 0.7592 | −0.5721, 3.6905 | |

1.2563 | 0.1070 | 0.0563 | 0.6059, 1.9067 | ||

0.3465 | 0.2070 | 0.4535 | −0.9124, 1.6055 | ||

200 | 1.5407 | 0.5723 | 0.7407 | −0.5344, 3.6158 | |

1.2771 | 0.0829 | 0.0772 | 0.6928, 1.8614 | ||

0.3498 | 0.2058 | 0.4502 | −0.8998, 1.5994 | ||

300 | 1.5204 | 0.5324 | 0.7203 | −0.4893, 3.5301 | |

1.2906 | 0.0682 | 0.0905 | 0.7487, 1.8324 | ||

0.3481 | 0.2049 | 0.4519 | −0.9058, 1.6019 | ||

500 | 1.5081 | 0.5092 | 0.7081 | −0.4623, 3.4784 | |

1.2925 | 0.0534 | 0.0926 | 0.8045, 1.7806 | ||

0.3467 | 0.2040 | 0.4502 | −0.9103, 1.6038 |

Interval estimates of the parameters.

Parameters | 90% CI | 95% CI |
---|---|---|

−0.6142, 3.7657 | −0.2623, 3.4137 | |

0.26319, 2.5764 | 0.4491, 2.3905 | |

−0.5521, 0.9953 | −0.4277, 0.8710 |

Graph of MSE versus sample size when

Graph of MSE versus sample size when

Graph of MSE versus sample size when

For illustration, we assume that the values of Fréchet constant stress model parameters are known and use the simulated data based on

The 90% and 95% interval estimates for the model parameters and acceleration factor are provided in Table

The asymptotic variance-covariance matrix of

The 95% asymptotic confidence interval for

From the results, we observe that the point estimates are stable. We also examine that the results support theoretical findings of CSPALT for Fréchet distribution.

From Tables

Some pertinent suggestions are also given for future research which is tantamount to provide a pathway for future researchers in the field of PALT. This work can be extended for type-II censoring of CSPALT using Fréchet distribution.

The data used in this article are freely available upon request from the authors and citing this paper in your manuscripts.

The authors declare that they have no conflicts of interest to report regarding the present study.