Different estimation techniques for constant-partially accelerated life tests of chen distribution using complete data

The issue of various estimation techniques in constant partially accelerated life tests with complete data is the main subject of this research. The Chen distribution is regarded as an item’s lifetime under use conditions. To estimate the distribution parameters and the acceleration factor, maximum likelihood estimation, least square estimation, weighted least square estimation, Cramér Von–Mises estimation, Anderson–Darling estimation, right-tail Anderson–Darling estimation, percentile estimation, and maximum product of spacing estimation are presented for classical estimation. For illustrative purposes, two real data sets are analyzed. The investigation of the two real data sets reveals that the suggested techniques are practical and can be used to solve some engineering-related issues. In order to compare the results of the several estimation techniques that have been offered based on mean square error and absolute average bias, a simulation study is presented at the end. When adopting the smallest values for mean square error and absolute average bias, this study demonstrates that maximum product of spacing estimation is the technique that is most effective among the alternatives in most cases.

estimation for the Kumaraswamy distribution using CPALT.Mahmoud et al. 15 proposed parameter estimation for the inverted generalized linear exponential distribution under CPALT using a progressive type-II censoring scheme.parameter estimation for the Nadarajah-Haghighi distribution based on the progressive type-II censoring scheme was investigated by 16 .For the modified Kies exponential distribution, Nassar and Alam 17 examined parameter estimation based on the CPALT utilizing a type-II censoring data.Based on progressive first failure type-II censored using CPALT, Eliwa and Ahmed 18 conducted a reliability analysis of the Lomax model.
On the other hand, the issue of various estimation techniques based on CPALT using complete data, which is the focus of this study, has not been adequately addressed.Additionally, a variety of natural phenomena, engineering problems, and clinical treatment produce a large amount of complete real data that are extremely important to our life.According to the aforementioned, the issue of various estimation methodologies based on CPALT and employing complete real data is of considerable relevance.
With an increasing or bathtub-shaped hazard rate function (HRF), Chen 19 suggested a two-parameter lifetime distribution.Due to the fact that the bathtub-shaped HRF serves as a useful conceptual model for electronic and machinery industries, it has received consideration from numerous researchers; see [20][21][22][23] .
The Chen distribution has some distinctive properties compared to other models with two parameters such as the fact that its HRF is bathtub-shaped and also the confidence intervals for the shape parameter and the joint confidence regions for the two parameters have closed form.Therefore, many researchers have studied its statistical inference based on ALT and PALT, see [24][25][26] .
The main objective of this research is to provide eight techniques of estimations for CPALT of Chen distribution based on the complete data, namely: maximum likelihood estimation (MLE), least square estimation (LSE), weighted least square estimation (WLSE), Cramér Von-Mises estimation (CVME), Anderson-Darling estimation (ADE), right-tail Anderson-Darling estimation (RADE), percentile estimation (PE), and maximum product of spacing estimation (MPSE).To illustrate the importance of the model in resolving various engineering issues, two complete real data sets are used.A simulation study is conducted to assess the performance of the suggested methods.Small, medium, and large sample sizes were used to compare the mean squared errors (MSE) and the absolute average bias (AAB) of the estimators' performances.
The sections of this study are arranged as follows."Basic assumptions and model description" presents the main concepts of CPALT.The MLE of Chen distribution using CPALT is studied in "Maximum likelihood estimation"."Least square and weighted least square estimations" discusses the estimation of Chen distribution using CPALT based on LSE and WLSE.The CVME is studied for Chen distribution under CPALT in "Cramér Von-Mises estimation".In "Anderson-Darling and right-tail Anderson-Darling estimations", the ADE and RTADE are presented to estimate the unknown parameters and accelerated factor of Chen distribution using CPALT.PE using CPALT for Chen distribution is studied in "Percentile estimation".In "Maximum product of spacing estimation", the MPSE is presented for Chen distribution under CPALT.Two uncensored real data sets are analyzed in "Numerical computations".In "Simulation study", the simulation study is covered.Conclusion remarks are reported in "Conclusion".

Basic assumptions.
• Under use conditions, the lifetimes of test units follow the Chen distribution and are independent and identi- cally distributed.The probability density function (PDF) and the cumulative distribution function (CDF) of Chen distribution are given by and respectively.The survival function (SF) and HRF are given by: and respectively.Equation (4) can be used to demonstrate how the hazard rate function for the Chen distribution can have two different shapes: an increasing shape for ξ ≥ 1 and > 0 and a bathtub shape for ξ < 1 and . For additional properties, see 27 .
• Under the acceleration condition, the lifetimes of test units follow the Chen distribution and are independent and identically distributed.The HRF of test unit can be given by h 2 (t) = ϕ h 1 (t) , where ϕ > 1 is the accelera- tion factor.Then the HRF, SF, CDF, and PDF can be written as Least square and weighted least square estimations.LSE and WLSE are presented for estimating the beta distribution's parameters in the paper by 28 .With the use of CPALT, these techniques will be used to estimate the unknown parameters and the accelerated factor of Chen distribution under complete data.For this purpose, take lifetimes T (ji) , i = 1, . . ., m j , j = 1, 2 to be two complete ordered samples from Chen distribution under CPALT.Then the LSE of the unknown parameters and the accelerated factor can be obtained by minimizing the following function w.r.t. the unknown parameters , ξ , and the accelerated factor ϕ .Upon using ( 2) and ( 7), the equation ( 14) can be written as Additionally, the following non-linear equations can be solved to yield the LSEs of the unknown parameters and the accelerated factor.and where and One can obtain WLSEs for Chen distribution using CPALT under complete data by minimizing the following function ; , ξ , ϕ) are given by ( 15), ( 16), ( 17), ( 18) and ( 19) respectively.
Percentile estimation.Kao 32 was the one who first proposed the PE, which has been used to estimate the unknown parameters of Weibull distribution.In order to apply this technique in this subsection to obtain the PE of Chen distribution using CPALT, the following equation needs to be minimized where w.r.t. the unknown parameters , ξ , and the accelerated factor ϕ or by solving the following non-linear equations: and Maximum product of spacing estimation.The method of MPSE developed by 33,34 is applied in this subsection to estimate the Chen distribution using CPALT under complete data.To obtain the MPSE of Chen distribution under CPALT, the following function needs to maximize w.r.t. the unknown parameters , ξ , and the accelerated factor ϕ , where D ji is the uniform spacings of a random sample from the Chen distribution under CPALT and defined by Upon using Eqs.( 2) and ( 7) into Eq.( 20), one can show that Vol:.( 1234567890 www.nature.com/scientificreports/So, the MPSE of Chen distribution using CPALT can be obtained also by solving the following non-linear equations ∂M ∂ = 0 , ∂M ∂ξ = 0 , and ∂M ∂ϕ = 0.

Numerical computations
To illustrate the computation of methods presented in the previous section, two real life data sets are presented.
Data set 1: ordered times to failure.The data presented in 35 expressing the required failure times for ten steel samples under the influence of four stress levels are used in this subsection.Accordingly, only two levels of stress, 0.87 and 0.99 ( 10 6 psi), are used as the use condition and the accelerated condition after being modified to meet the problem being examined, see Table 1.First, the MLE is used under complete data to check the validity of the Chen D to fit the data set for use and accelerated conditions.The Kolmogorov-Smirnov (K-S) distance and the corresponding p value are obtained for use and accelerated conditions.The results are summarized in Table 2. From Table 2, the Chen distribution provides a good fit to the data under use and accelerated conditions.Figure 1 also displays the empirical CDF and the fitted CDF of the Chen distribution using MLE in the use and accelerated conditions.
The estimation methods, which are given in "Maximum likelihood estimation" to "Maximum product of spacing estimation", are used to obtain the estimates of the unknown parameters and the accelerated factor under   www.nature.com/scientificreports/CPALT using the ordered times to failure data.The estimates based on real data sets under different methods of estimation are tabulated in Table 3.
Data set 2: oil breakdown times of insulating fluid.The data set from 36 that details the insulating fluid's oil breakdown times under high test voltages are considered in this subsection after being modified to meet the problem being examined.Accordingly, only two levels of stress, 30 and 32 kV, are used as the use condition and the accelerated condition, see Table 4.
First, the MLE is used under complete data to check the validity of the Chen distribution to fit the data set for use and accelerated conditions.The Kolmogorov-Smirnov (K-S) distance and the corresponding p value are obtained for use and accelerated conditions.The results are summarized in Table 5.From Table 5, the Chen distribution provides a good fit to the data under use and accelerated conditions.The empirical CDF and the fitted CDF of the Chen distribution using MLE under use and accelerated conditions are also shown in Fig. 2.
The estimation methods, which are given in "Maximum likelihood estimation" to "Maximum product of spacing estimation", are used to obtain the estimates of the unknown parameters and the accelerated factor under CPALT using the oil breakdown times of insulating fluid data.The estimates based on real data sets under different methods of estimation are tabulated in Table 6.

Simulation study
The principal reason for this section is to compare the estimators of the parameters by utilizing MSE and AAB.
For varying values of m 1 and m 2 (number of two samples for use and accelerated conditions), a large num- ber N = 10,000 of complete samples are generated from Chen distribution under use and accelerated condi- tions.Take the true values of , ξ , and ϕ as ( , ξ , ϕ) = (1.5, 1.5, 2) , ( , ξ , ϕ) = (1.5, 2, 1.5) , ( , ξ , ϕ) = (1.5, 2, 2) , and ( , ξ , ϕ) = (2, 3, 3) .To carry out the numerical study, the following steps are required: 1 Generate two independent random samples of sizes m 1 and m 2 from Uniform (0,1) distribution using Ran- domReal[] in mathematica (U j1 , U j2 , . . ., U jm j ), j = 1, 2 .With different choice of m 1 , m 2 , and different values of the parameters and accelerated factor, the two complete samples are generated from the inverse CDFs F 1 (t) and F 2 (t) for use and accelerated conditions respectively as follow: , where i = 1, 2, 3, . . ., m j 2 Across using the results obtained in "Maximum likelihood estimation" to "Maximum product of spacing estimation", the different estimates of the unknown parameters and accelerated factor are calculated.using the package FindRoot[] in Mathematica or using the Newton-Raphson algorithm.3 Repeat Steps 1 − 2 , N = 10,000 times.4 Calculate the AEs, MSEs and AABs of the unknown parameters and accelerated factor from where, ˆ is the parameter estimation for .
The results obtained from the numerical comparison study between different methods based on MSEs and AABs for all estimates are presented in Tables 7, 8, 9, 10, 11 and 12.These tables amply demonstrate that: • It is clear from Tables 7, 8, 9, 10, 11 and 12 that with an increase in m 1 , m 2 , the MSEs and AABs decrease for all estimates as expected.• It is clear also that MLE improves for the better in terms of small values of MSE and AAB and becomes one of the best estimates for large sample in relation to the parameter .• We also find that the MPSE outperforms alternative techniques in most cases for parameter .
• For the parameter ξ , we find that MLE is the best estimate based on the lowest values of MSE and AAB.
• As for the parameter ϕ , we find that MPSE is the best estimate, followed by MLE according to MSE and AAB.
• Taking MSE and AAB into consideration, the MPSE technique outperforms alternative techniques in most cases.• In view of the results of the simulation study, we recommend the use of MPSE, MLE, PE and ADE to estimate CPALT under the complete data when taking MSE and AAB into consideration.

Conclusion
In this paper, the problem of various techniques of estimations under complete sample in CPALT has been studied.Eight methodologies of classical estimation, namely, MLEs, LSEs, WLSEs, CVMEs, ADEs, RTADEs, PEs, and MPSEs, have been considered to estimate the unknown parameters and the accelerated factor of Chen distribution under CPALT.The proposed methodologies were demonstrated using two real data sets, demonstrating their applicability as they can be applied to address several engineering-related problems.Additionally, in order to compare these methodologies with various sample sizes and various sets of the unknown parameters, and the accelerated factor, a comprehensive simulation analysis has been carried out.The AEs, MSEs, and AAB under complete data using CPALT have been calculated.According to the MSEs and AABs values computed from the simulation study, the MPSE is the most effective methodology among the alternatives in most cases for all parameters.Based on the results of the simulation study, it can be demonstrated that the MPSE, MLE, PE, and ADE methods can be recommended for estimating the parameters and accelerated factor for the CPALT of Chen distribution when complete data is available.With the help of the suggested methodology and the results of this investigation, some future studies can be presented, such as: https://doi.org/10.1038/s41598-023-42055-8 https://doi.org/10.1038/s41598-023-42055-8

9886 Figure 1 .
Figure 1.(a) The fitted CDF of Chen distribution under use condition for ordered times to failure data.(b) The fitted CDF of Chen distribution under accelerated condition for ordered times to failure data.

Figure 2 .
Figure 2. (a) The fitted CDF of Chen distribution under use condition for oil breakdown times of insulating fluid.(b) The fitted CDF of Chen distribution under accelerated condition for oil breakdown times of insulating fluid data.

Table 1 .
The ordered times to failure under two stress levels.

Table 2 .
The ML estimates of parameters, the K-S values and the associated p values under use and accelerated conditions.

Table 4 .
Oil breakdown times of insulating fluid.

Table 5 .
The ML estimates of parameters, the K-S values and the associated p values under use and accelerated temperatures.

Table 6 .
The different methods of estimations for ( , µ, θ, γ ) using real data.Actually, 6 value sets of the parameters and accelerated factor have been used in the simulation study.There- fore, a number of additional value sets can be used to extend the simulation study and examine their influence on the MSEs and AABs values.

Table 7 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations (1.5,1.5,2).

Table 8 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations (1.5,2,1.5).

Table 9 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations (1.5,2,2).

Table 10 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations(2,4,3).

Table 11 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations(2,3,4).

Table 12 .
The average estimators, the mean square error, and the absolute average bias of different methods of estimations(2,3,3).