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# Analysis of Progressively Type-II Inverted Generalized Gamma Censored Data and Its Engineering Application

1 Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi Arabia

2 Department of Statistics, St. Anthony’s College, Shillong, 793001, India

3 Faculty of Technology and Development, Zagazig University, Zagazig, 44519, Egypt

* Corresponding Author: Ahmed Elshahhat. Email:

(This article belongs to the Special Issue: Incomplete Data Test, Analysis and Fusion Under Complex Environments)

*Computer Modeling in Engineering & Sciences* **2024**, *141*(1), 459-489. https://doi.org/10.32604/cmes.2024.053255

**Received** 12 February 2024; **Accepted** 20 June 2024; **Issue published** 20 August 2024

## Abstract

A novel inverted generalized gamma (IGG) distribution, proposed for data modelling with an upside-down bathtub hazard rate, is considered. In many real-world practical situations, when a researcher wants to conduct a comparative study of the life testing of items based on cost and duration of testing, censoring strategies are frequently used. From this point of view, in the presence of censored data compiled from the most well-known progressively Type-II censoring technique, this study examines different parameters of the IGG distribution. From a classical point of view, the likelihood and product of spacing estimation methods are considered. Observed Fisher information and the delta method are used to obtain the approximate confidence intervals for any unknown parametric function of the suggested model. In the Bayesian paradigm, the same traditional inferential approaches are used to estimate all unknown subjects. Markov-Chain with Monte-Carlo steps are considered to approximate all Bayes’ findings. Extensive numerical comparisons are presented to examine the performance of the proposed methodologies using various criteria of accuracy. Further, using several optimality criteria, the optimum progressive censoring design is suggested. To highlight how the proposed estimators can be used in practice and to verify the flexibility of the proposed model, we analyze the failure times of twenty mechanical components of a diesel engine.## Keywords

Recently, we have come across several studies on inverse (or reciprocal) distributions of one or two parameters in the literature. Louzada et al. [1] recently emphasized that inverse distributions provide greater flexibility for fitting data and, in many cases, have been found to be better than many other standard distributions. The three parameter inverted generalized gamma (IGG) distribution was introduced by Hoq et al. [2] in the context of life testing experiments. As specific examples, the IGG distribution includes a number of lifespan distributions, including inverted gamma, inverted half-normal, inverted Weibull, inverted exponential, inverted Rayleigh, inverted Maxwell-Boltzmann, and inverted chi-square. The IGG distribution has been found to be a good alternative to other skewed distributions, including inverse gamma, inverse Weibull, and generalized inverse exponential distributions, and can be used to illustrate skewed data.

A random variable x is said to have IGG

and

where

and

where

The hazard rate function (4) has an unimodal shape, while the density function (2) is unimodal and right-skewed with heavy tails. These tails become thicker when the shape parameters are equal, and longer when the shape parameter

Besides Type-I, Type-II, and progressive Type-II censoring (PT2C) plans, several censoring mechanisms are available in the literature. An important advantage of PT2C is that it allows the experimenter to withdraw some live units before the experiment stops. The PT2C can be described as follows: Suppose we have

The maximum product spacing (MPS) was initially presented by Cheng et al. [4] and was utilized by Ranneby [5] as an alternative to the maximum likelihood (ML) estimation approach. In recent times, the MPS method has been employed in different contexts; see Zhu [6], Jeon et al. [7], Nassar et al. [8], Nassar et al. [9], and many others. According to Anatolyev et al. [10], the MPS approach adheres to the invariance property and outperforms likelihood estimators in small-sample scenarios for heavy and skewed distributions.

The limitations and challenges of this work are particularly related to the complexity of parameter estimation, computational intensity, small sample sizes, model power, hypothesis testing, and inference difficulties. Addressing these issues often requires advanced statistical techniques and careful consideration of the specific characteristics of the data and the censoring scheme.

Researchers pay little attention to analyzing new extended gamma lifetime models because of their complex results and expensive numerical evaluations. Aside from the work of Ramos et al. [11], as they studied Bayesian inferences based on non-informative priors using a complete sample, and no attempt has been made, to our knowledge, to estimate IGG parameters based on PT2C since the IGG distribution was introduced in the literature.

Given the usefulness and practicality of the IGG distribution, and with samples collected from the PT2C strategy, we state our main objectives in this paper as follows:

• Obtain the frequentist point estimators of the parameters (including

• Obtain the approximate confidence intervals (ACIs) for all parameters using the acquired ML estimators (MLEs) and MPS estimators (MPSEs).

• As an alternative to the ML function in the Bayesian paradigm, the MPS function is also taken into consideration. Next, both ML and MPS functions are explored in a Bayesian framework for all unknown parameters under squared error loss (SEL) with independent gamma priors.

• Implement Markov chain Monte Carlo (MCMC) methodology with Metropolis–Hastings (M–H) sampler to calculate the offered Bayes point estimates as well as Bayes confidence intervals (BCIs).

• Various optimality metrics are assessed to provide the optimal PT2C plan.

• Extensive Monte Carlo simulations are performed to evaluate the performance of the proposed techniques.

• An engineering data set representing diesel engine failure times is analyzed to show the applicability of the proposed methodologies in a real-life scenario.

The rest of the article is organized as follows: In Sections 2 and 3, we discuss ML- and MPS-based estimation. In Section 4, we discuss Bayesian inferences. Simulation investigations are highlighted in Section 5. In Section 6, the optimal PT2C is discussed. One real data set is analyzed in Section 7. Finally, various observations are made in Section 8.

This section considers the ML estimation to provide the MLEs along with their ACIs of the IGG parameters

Suppose

where

Using (5), without constant terms, the log-likelihood function

where

The MLEs

and

where

and

To prove the convergence of the MLEs

In this subsection, the

It is established that

where

To find the

where

Hence, once we obtain the variances

respectively, where

3 Product of Spacings Estimation

This section examines the MPS estimation approach to acquire the MPSEs and ACIs of

Substituting (2) and (1) into the product of spacing (PS) function of PT2C, we get the joint PS function of

where

By maximizing the following logarithmic MPS function (e.g.,

where

Solving the following three normal equations will yield the MPSEs of

and

where

Since the proposed MPSEs from (13)–(15) lack closed form solutions, an iterative method such as Newton-Raphson can be easily used to derive the MPSEs numerically. It is important to note that the MPSEs and MLEs of

Now, to highlight the convergence of the MPSEs

To get the two bounds

It follows that

Replacing

respectively, where

In this section, under the PT2C scheme, the Bayes framework for unknown IGG parameters based on LF and PF methods is discussed. Prior information plays an influential role in obtaining Bayes’ estimate. We now consider the independent gamma priors (say,

where

Suppose

where the Bayes estimator

4.1 Posterior Density via LF-Based

Using (5) and (18), the joint posterior PDF via the LF (say,

Using an arbitrary function

All triple integrals in (21) do not have a closed form. Therefore, we will implement the M-H algorithm. From (20), we note that the full conditionals of

4.2 Posterior Density via PS-Based

Using (11) and (18), the joint posterior PDF via PS-based (say,

Certainly, from (22), the Bayes estimator via PS-based (say

The performance of the proposed estimation methodologies is highlighted in this part. According to the algorithm proposed by Balakrishnan et al. [13] for various levels of

Once the simulated PT2C samples are acquired, the classical (point and 95% interval) estimations are offered. Following two independent information requirements called prior-mean and prior-variance, we can easily assign certain values to the proposed prior parameters; see Kundu [14]. Therefore, two different sets for the hyperparameter values of

To conduct the M-H sampling procedure, 12,000 MCMC samples are gathered from the posterior functions (20) and (22). For each simulated Markov chain, the first 2000 variates are removed to ignore the effect of the selection of initial guesses. In all proposed numerical simulations, we assume that all IGG model parameters are unknown. We also assume that the practitioner records the failure time of all test units

The performances of the point theoretical results of

where

All calculations are performed in

• In general, the proposed point (or interval) estimates of

• As

• Comparing the proposed point approaches, it is clear that:

– The MLE (or MCMC-LF) of

– The performance of

• Comparing the proposed interval approaches, it is clear that:

– The ACI-LF estimates of

– The AILs of ACI-PS estimates of

• The CPs of the BCI estimates are (in most cases) greater than the specified nominal level, while the ACI estimates are lower (or closer) to the specified nominal level.

• Due to the availability of gamma priors, Bayes’ MCMC-LF (or MCMC-PS) estimates of all unknown parameters outperform ML (or MPS) estimates. The same comment is also drawn in the context of comparing the BCI-LF (or BCI-PS) with the ACI-LF (or ACI-PS).

• Obviously, the variance of Prior-II is less than that of Prior-I. Therefore, all Bayes estimates of all unknown parameters developed from Prior-II using LF (or PS) data outperform those obtained based on other methods.

• Comparison of the proposed Scheme-1 (first stage) and Scheme-3 (last stage), it is clear that the point estimates of

• Regarding the interval estimates of

• As a tip, to obtain accurate estimates of any unknown life parameter when the proposed censored data are present, the experimenter should record an appropriate effective sample size, taking into account the total cost of the test.

• To sum up, the Bayes M-H procedure is recommended to study the unknown parameters of life of the IGG model in the presence of PT2C data.

The previous sections dealt with the derivation of point and interval estimations, using both frequentist and Bayesian MCMC estimations, for the parameters of life of the IGG distribution when samples are gathered from the PT2C strategy. Following Ng et al. [17], when the design of removal items

Thus, to select the optimal progressive censoring (OPC) plan, Table 11 reports common criteria for this purpose. Regarding the A-and D-optimality criteria, our objective is to reduce the trace and determinant values of estimated variances and covariances developed from the LF and PS methods. Further, the goal of F-optimality is to maximize the observed values of the Fisher matrices in relation to the MLE (or MPSE) (say,

This section analyzes a useful real data set consisting of the failure times (in weeks) of 20 mechanical components of a diesel engine to demonstrate the performance of the proposed methodologies; see Murthy et al. [20]. For computational convenience, we multiply each original data unit by one hundred, and the new transformed data set is: 6.70, 6.80, 7.60, 8.10, 8.40, 8.50, 8.50, 8.60, 8.90, 9.80, 9.80, 11.4, 11.4, 11.5, 12.1, 12.5, 13.1, 14.9, 16.0, and 48.5.

First, the IGG distribution is fitted to the complete diesel engine data with eleven inverted lifetime distributions (for

(1) Generalized inverted Gompertz (GIG(

(2) Alpha-power inverse-Weibull (APIW(

(3) Generalized inverse-Weibull (GIW(

(4) Exponentiated inverted-Weibull (EIW(

(5) Generalized inverted-exponential (GIE(

(6) Generalized inverted half-logistic (GIHL(

(7) Inverted exponentiated Rayleigh (IER(

(8) Inverted Nadarajah–Haghighi (INH(

(9) Inverted Gompertz (IGo(

(10) Inverse-Weibull (IW(

(11) Inverse gamma (IG(

To evaluate the feasibility of the IGG model in comparison with other popular models, several information measures are implemented, namely: (i) Kolmogorov-Smirnov (KS) statistic (with its p-value); (ii) Anderson-Darling (AD); (iii) Cramer von Mises (CvM); (iv) estimated negative log-likelihood (ENL); (v) Akaike information (AI); (vi) consistent Akaike information (CAIC); and (vii) Hannan-Quinn information (HQI).

The MLEs (with their standard-errors (SEs)) of

To capture the behavior of the estimated PDFs and CDFs of the IGG and its competitive distributions using the diesel engine data, we have provided two plots: (a) representing the histogram and the fitted densities and (b) representing the fitted/empirical reliability lines; see Fig. 4. It demonstrates that the IGG distribution reflects the overall structure of the histograms and confirms the numerical results presented here.

Using the complete diesel engine data, for different choices of

Tables 14 and 15 indicate that the computed estimates of

Now, using Table 13, the selection of the OPC plan is explored. So, from (10) and (16), the criteria A, D, and F are evaluated; see Table 16. It shows that the scheme

In this paper, various estimates of the inverted generalized gamma parameters

Acknowledgement: The authors would desire to express their thanks to the editor and the three anonymous referees for useful suggestions and valuable comments. The authors would also like to express their full thanks to the Deanship of Scientific Research and Libraries, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding after Publication, Grant No. (RPFAP-34-1445) for supporting this project.

Funding Statement: This research project was funded by the Deanship of Scientific Research and Libraries, Princess Nourah bint Abdulrahman University, through the Program of Research Project Funding after Publication, Grant No. (RPFAP-34-1445).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Refah Alotaibi, Sanku Dey, Ahmed Elshahhat; data collection: Refah Alotaibi, Ahmed Elshahhat; analysis and interpretation of results: Ahmed Elshahhat; draft manuscript preparation: Refah Alotaibi, Sanku Dey. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available within the paper.

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

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Appendix A: Fisher-LF Elements

Differentiating (6) with respect to the unknown parameters

and

where

and

Appendix B: Fisher-PS Elements

Differentiating (12) with respect to the unknown parameters

and

where

## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*141*

*(1)*, 459-489. https://doi.org/10.32604/cmes.2024.053255

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 141, no. 1, pp. 459-489, 2024. https://doi.org/10.32604/cmes.2024.053255

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