Bayesian Prediction for Chris-Jerry Model Using Unified Progressive Hybrid Censored Sample
Subject Areas : Operation ResearchHanieh Panahi 1 * , Sahar Hosseinikhah Choshaly 2
1 - Department of Mathematics and Statistics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2 - Department of Management, Islamic Azad University, Lahijan Branch, Lahijan, Iran
Keywords: Bayesian point Prediction, HPD predictive, Chris-Jerry distribution, Unified Progressive Hybrid Censoring.,
Abstract :
From a real analysis standpoint, modeling hinges on the choice of the most suitable analysis method, whether it be a frequentist or Bayesian approach, to obtain an updated model. Modeling lifetime data with heavy tail has been a problem among many management researchers. Also, predicting unobserved data based on available data is one of the most important challenges in various sciences such as management and engineering. Bayesian predictive point in Chris-Jerry distribution is considered, in this paper. Te observed data is censored using a unified progressive hybrid censoring scheme. For evidence of the effectiveness of the given methodology, an application is explored.
[1] Gorny, J. and Cramer, E. (2018). Modularization of hybrid censoring schemes and its application to unified progressive hybrid censoring, Metrika. 81(2), 173-210.
[2] Nagy, M. and Alrasheedi, A.F. (2022). Classical and Bayesian Inference Using Type-II Unified Progressive Hybrid Censored Samples for Pareto Model, Appl Bionics Biomech. 2073067.
[3] Dutta, S. and Kayyal, S. (2024). Estimation and prediction for Burr type III distribution based on unified progressive hybrid censoring scheme, Journal of Applied Statistics, 1-33.
[4] Nik, A.S., Asgharzadeh, A. and Raqab, M. (2021). Estimation and prediction for a new Pareto-type distribution under progressive type-II censoring, Mathematics and Computers in Simulation, 190,508-530.
[5] Asadi, S. Panahi, H. Parviz, P. (2024). Estimation for Inverse Burr Distribution under Generalized Progressive Hybrid Censored data with an application to Wastewater Engineering Data. REVSTAT-Statistical Journal.
[6] Elshahhat, A., and Ashour, S. (2016). Bayesian and Non-Bayesian Estimation for Weibull Parameters Based on Generalized Type-II Progressive Hybrid Censoring Scheme. Pakistan Journal of Statistics and Operation Research, 12(2), 213-226.
[7] Onyekwere, CK. and Obulezi, OJ. (2022). Chris-Jerry Distribution and Its Applications, Asian Journal of Probability and Statistics, 20(1): 16-30.
[8] Chidera F. Innocent, Omoruyi A. Frederick, Edidiong M. Udofia, kechukwu J. Obulezi, Chinyere P. Igbokwe (2023). Estimation of the Parameters of the Power Size Biased Chris-Jerry Distribution, International Journal of Innovative Science and Research Technology, 5: 423-436.
[9] Okechukwu J. Obulezi, Ifean yi C. Anabike, Etaga Harrison O. (2023). Marshall-Olkin Chris-Jerry Distribution and its Applications, International Journal of Innovative Science and Research Technology, 8, 20-31.
[10] Chinedu, E.Q.; Chukwudum, Q.C.; Alsadat, N.; Obulezi, O.J.; Almetwally, E.M.; Tolba, A.H. New Lifetime Distribution with Applications to Single Acceptance Sampling Plan and Scenarios of Increasing Hazard Rates. Symmetry 2023, 15, 1881.
Islamic Azad University Rasht Branch ISSN: 2588-5723 E-ISSN:2008-5427
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Optimization Iranian Journal of Optimization Volume 15, Issue 4, 2023, 251-255 Research Paper |
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Online version is available on: www.ijo.rasht.iau.ir
Bayesian Prediction for Chris-Jerry Model Using Unified Progressive Hybrid Censored Sample
Hanieh Panahi1* and Sahar Hosseinikhah choshaly 2
1 Department of Mathematics and Statistics, Lahijan Branch, Islamic Azad University, Lahijan, Iran
2 Department of Management, Lahijan Branch, Islamic Azad University, Lahijan, Iran
Revise Date: 25 December 2024 Abstract
Keywords: Bayesian point Prediction HPD predictive Chris-Jerry distribution Unified Progressive Hybrid Censoring
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INTRODUCTION
Prediction problems arise naturally in important fields of statistical research and have been very useful in many areas of application such as management, quality control, life experiments and so on. In experiments where failure information is available only on a part of the data, the sample are said to be censored data. Nowadays, the quality and lifetime of any industrial product have become higher because of modern science and technology. Thus, it has become more challenging to increase product qualities with reliability testing in a short period with low expenditure. In this context, censoring is sensible and important to study the failure data in a reliability experiment. Time (Type-I) and (failure) censoring are the most popular censorship techniques. To reduce the cost and duration associated with experiment, progressively Type-II censoring has been developed, which allows the researcher to eliminate items at periods other than the end time. Also, Gorny & Cramer (2018) introduced unified progressive hybrid censoring scheme, where the experimental time may exceed the predetermined time T and the effective sample sizes m, k are prefixed. Let 
and 
be the two time points. This experiment will be stopped at
. In recent years, this censoring and its generalization have been introduced and studied quite extensively (Nagy & Alrasheedi, 2022; Dutta & Kayyal, 2024; Nik et al., 2021; Asadi et al., 2024; Elshahhat & Ashour, 2016).
The Chris-Jerry distribution is a heavy-tailed distribution for analyzing real data (Onyekwere & Obulezi, 2022). It has been introduced as a mixture of two distributions called the exponential (with parameter
) and the gamma (with scale and shape parameters as 3 and
, respectively) when the mixing proportion is taken as
. The positive random variable X, is said to be the Chris-Jerry random variable with parameter 
denoted by
, if the cumulative distribution function (cdf) is
and the probability density function (pdf) is given as
For more details, see Onyekwere and Obulezi (2022), Innocent et al. (2023), Obulezi et al. (2023), and Chinedu et al. (2023). For various parameter values, the probability density function (pdf) plot can be shown below.
Fig. 1. Pdf plot of Chris-Jerry distribution
Let 
denote a unified progressive hybrid censored sample from a 
distribution, with censoring scheme
. Then, the joint density function of unified progressive hybrid censored order statistics is given by:
The remaining portion of the paper is structured as follows. Bayesian predictors and predictive intervals are proposed in Section 2. Section 3 evaluates the estimators’ performance via simulations. Finally, in Section 4, we conclude the study with a brief discussion.
BAYESIAN APPROACH
The Bayes estimation (BE) of the Chris-Jerry distribution under the unified progressive hybrid will be covered in this section. It is assumed that the random variable follows gamma prior distribution with hyper-parameters 
. The prior of can be given as:
The corresponding posterior density given the observed data 
is given by:
Thus, under the squared error loss function, the BE of any function on model parameters say 
, can be obtained as the posterior mean as:
Bayesian Prediction
This section deals with the point and interval prediction for 
-th order statistic out of 
removed units based on the unified progressive hybrid sample with progressive censoring scheme. The conditional density function of
, can be formulated as:
Here,
. Considering the cumulative distribution function and density function of the Chris-Jerry model, the conditional density function of 
as follows:
So, the Bayesian predictive density of can be written as:
To avoid the complexity of the integration given in (9), the Monte Carlo Markov chain samples 
, for 
have been used. So, the -th ordered statistic from the censored data is given by:
SIMULATION STUDIES
This Section is devoted to the comparative study of the proposed estimates under different unified progressive hybrid censored schemes. The simulation is carried out using R software. We considered different values for n, m, k and 
to generate 104 unified progressive hybrid censored schemes from the Chris-Jerry model. For given 
, different censoring schemes are adopted as:
Ø CSI : 
Ø CSII : 
Ø CSIII: 
Table 1: Bayesian point predictors and HPD predictive for.
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| k | Censoring Schemes |
| Point Prediction | Interval Prediction | |
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(60,30) | 10 | CS I | 1 | 1.67345 | 1.234 | 1.875 |
(60,30) | 10 |
| 2 | 1.74324 | 1.289 | 1.952 |
(60,30) | 10 | CS II | 1 | 1.34832 | 1.098 | 1.599 |
(60,30) | 10 |
| 2 | 1.37869 | 1.112 | 1.543 |
(60,30) | 10 | CS III | 1 | 1.70546 | 1.249 | 1.903 |
(60,30) | 10 |
| 2 | 1.72384 | 1.265 | 1.937 |
(60,30) | 15 | CS I | 1 | 1.79345 | 1.293 | 1.917 |
(60,30) | 15 |
| 2 | 1.83552 | 1.365 | 2.013 |
(60,30) | 15 | CS II | 1 | 1.39894 | 1.212 | 1.587 |
(60,30) | 15 |
| 2 | 1.43760 | 1.268 | 1.621 |
(60,30) | 15 | CS III | 1 | 1.82543 | 1.378 | 2.014 |
(60,30) | 15 |
| 2 | 1.86089 | 1.500 | 2.112 |
CONCLUSION
Modeling heavy-tailed data has been a problem among many researchers. One of the common features among standard heavy-tailed distributions is the parsimonious validity of the models in terms of the number of parameters. The new one-parameter Chris-Jerry distribution furnishes a better fit compared to other heavy tailed models, called Lindley, Exponential and Pareto distributions. The Bayes point and interval prediction for observable unified progressive hybrid censored data from the Chris-Jerry model are derived. The results show that the Bayes point predictors lie between the lower and the upper bounds in each case. The lengths of the prediction intervals are decreasing by increasing the effective sample size (k). It has also been observed that different censoring schemes do not have much effect on the simulation results.
REFERENCES
Asadi, S., Panahi, H., & Parviz, P. (2024). Estimation for inverse Burr distribution under generalized progressive hybrid censored data with an application to wastewater engineering data. REVSTAT-Statistical Journal.
Chinedu, E. Q., Chukwudum, Q. C., Alsadat, N., Obulezi, O. J., Almetwally, E. M., & Tolba, A. H. (2023). New lifetime distribution with applications to single acceptance sampling plan and scenarios of increasing hazard rates. Symmetry, 15, Article ID 1881.
Dutta, S., & Kayyal, S. (2024). Estimation and prediction for Burr type III distribution based on unified progressive hybrid censoring scheme. Journal of Applied Statistics, 1–33.
Elshahhat, A., & Ashour, S. (2016). Bayesian and non-Bayesian estimation for Weibull parameters based on generalized Type-II progressive hybrid censoring scheme. Pakistan Journal of Statistics and Operation Research, 12(2), 213–226.
Gorny, J., & Cramer, E. (2018). Modularization of hybrid censoring schemes and its application to unified progressive hybrid censoring. Metrika, 81(2), 173–210.
Innocent, C. F., Frederick, O. A., Udofia, E. M., Obulezi, I. J., & Igbokwe, C. P. (2023). Estimation of the parameters of the power size biased Chris-Jerry distribution. International Journal of Innovative Science and Research Technology, 5, 423–436.
Nik, A. S., Asgharzadeh, A., & Raqab, M. (2021). Estimation and prediction for a new Pareto-type distribution under progressive Type-II censoring. Mathematics and Computers in Simulation, 190, 508–530.
Nagy, M., & Alrasheedi, A. F. (2022). Classical and Bayesian inference using Type-II unified progressive hybrid censored samples for Pareto model. Applied Bionics and Biomechanics, Article ID 2073067.
Obulezi, O. J., Anabike, I. C., & Harrison, E. O. (2023). Marshall-Olkin Chris-Jerry distribution and its applications. International Journal of Innovative Science and Research Technology, 8, 20–31.
Onyekwere, C. K., & Obulezi, O. J. (2022). Chris-Jerry distribution and its applications. Asian Journal of Probability and Statistics, 20(1), 16–30.