The comparative life testing for products from different production lines under joint censoring schemes has received some attention over the past few years. Mondal and Kundu recently used the balanced joint progressive type-II censoring scheme to discuss the comparative exponential and Weibull populations. This paper implements the balanced censoring scheme with a hybrid progressive type-I censoring scheme known as a balanced joint progressive hybrid type-I censoring scheme (BJPHCS). The life Lomax products’ model formulation from two different lines of production with BJPHCS is discussed. The model parameters are estimated under maximum likelihood estimation for point and the corresponding asymptotic confidence intervals. Under independent gamma priors, the Bayes estimators and associated credible intervals are obtained with the help of MCMC technique. The validity of the theoretical results developed in this paper for estimation problems is discussed through numerical example and Monte Carlo simulation study, which report the estimators’ quality. Finally, we give a brief comment describing the numerical results.

The quality of any life product has required putting some product units under a life testing experiment, and the life data may be complete or censoring. The test cost and time determine a suitable method that is used to obtain the data. Type-I and type-II censoring schemes are presented as the oldest censoring schemes. The experiment is run to prefixed time in type-I censoring scheme, and the number of failed units is random. However, the experiment is run to prefixed number of failures in the Type-II censoring scheme with a random experiment time. In practice, if we need to remove units at any time of the experiment for engineering, clinical studies, or other purposes, then a progressive censoring scheme is applied to improve productivity while keeping a high level; see, for more details, [

Suppose

In particular, when the product comes from different production lines, a joint censoring scheme appears as a suitable censoring scheme, and the obtaining data are used to determine the relative merits of life products in competing duration. Therefore, suppose two lines of production

For jointly progressive hybrid type-I censoring scheme, the total sample with size (

Balanced joint progressive type-II censoring is introduced early by [

This paper aims to use the progressive hybrid type-I censoring scheme under a balanced joint technique to present the BJPHCS. The BJPHCS is analytically easier to handle than another joint censoring scheme. Also, the properties of the estimators under BJPHCS can be stated more explicitly. Under this scheme, we construct the likelihood function for the model parameters. The lifetime information obtained from Lomax lifetime distribution under BJPHCS is used to present the two Lomax life distributions’ statistical inference. Also, the asymptotic confidence interval under the normality distribution of the estimates is constructed. The Bayes estimators of unknown model parameters with the corresponding credible intervals are developed. Different estimators are discussed and compared through the Monte Carlo simulation study and illustrated through a numerical example based on BJPHCS.

The paper is constructed with the following sections. The concepts and model of the formulation are built in Section

Suppose the product comes from different two lines

Schematic diagram of BJPHCS.

Hence, under observed sample

The reliability and hazard rate functions of Lomax lifetime distributions is also given by

The Lomax lifetime distribution has been introduced early in [

For the given joint sample

The joint likelihood function is defined by (

The likelihood equations can be obtained from logarithm (

Let

Also,

Equations (

The nonlinear equations (

For each of the parameters,

Fisher information matrix is defined as the minus expectation of the second derivative of log-likelihood function with respect to the model parameters’ vectors

Then, at confidence level

The Bayesian approach for parameters’ estimation depends on the prior information available about the parameters and the data presented. Gamma distribution characterized by different shapes depends on its parameters; this property has marked it to be a more suitable distribution in other cases. So, we consider the independent gamma prior for distribution of model parameters as follows:

Also, after obtaining the data in the form of balanced joint progressive hybrid type-I censoring data which formed by likelihood function, then, we construct all information about the parameters by posterior probability density defined by

Hence, the Bayes estimators have two integrables, which need approximation methods such as numerical integration or Lindley approximation. The more general case is applied in different Bayesian computation areas called as Markov chain Monto Carlo (MCMC) method.

The full conditional distributions of the parameters are given by

Then, the posterior distribution is reduced to two conditional gamma functions and two distributions more similar to normal distribution. Then, the posterior distribution (

Step 1: for given initial vectors

Step 2: generate two values

Step 3: with proposal normal distributions, generate two values

Step 4: construct the iteration vector

Step 5: change

Step 6: steps (2) to (5) are followed to get iteration procedure to

Step 7: to reach the stationary state, we need iteration number

as well as the corresponding posterior variance of

where

Step 8: the obtaining empirical distribution of

Discuss and illustrate the proposed methods in this paper, firstly, about the relation between choosing the prior information that related with true parameters’ values. The true parameters are chosen to satisfy that

The generated balanced joint progressive type-I data.

0.03075 | 0.21431 | 0.23253 | 0.32229 | 0.39942 | 0.51726 | 0.80195 | 0.81130 |
---|---|---|---|---|---|---|---|

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

0.83083 | 0.86332 | 1.29308 | 1.30155 | 1.46959 | 1.48689 | 1.66629 | 2.03257 |

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

2.16232 | 2.73166 | 2.75379 | 2.91897 | 4.33301 | 7.74818 | — | — |

1 | 1 | 0 | 0 | 1 | 1 | — | — |

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

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

0.7215 | 0.8097 | (0, 1.9320) | 1.9320 | (0.3969, 1.4132) | 1.01626 | |

2.8368 | 2.8453 | (0, 3.9947) | 3.9947 | (1.4185, 4.8630) | 3.4445 | |

0.4443 | 1.0680 | (0, 9.5860) | 9.5860 | (0.5572, 1.7875) | 1.23033 | |

1.2865 | 3.2739 | (0, 3.7570) | 3.7570 | (1.8051, 5.2599) | 3.45486 |

The simulation number and the corresponding histogram of

The simulation number and the corresponding histogram of

The simulation number and the corresponding histogram of

The simulation number and the corresponding histogram of

This section has examined the best choice of sample size, censoring scheme, and ideal test time. This discussion is reported for the classical ML and Bayes estimators of Lomax lifetime distribution under a balanced joint progressive hybrid type-I censoring scheme. Hence, the developed theoretical results are compared and assessed with constructed Monte Carlo simulation studies. The point estimators are measured with mean estimate values (ME) and square rote of simulated variance (

The commonly used problem in life products with different production lines is measuring the relative merits of the products competing for the duration. The main aim is to use censoring that can be applied to obtain the information in determining time. Recently, a joint censoring scheme and a specially balanced joint censoring scheme can be applied for this problem. BJPHC is proposed for obtaining information about lifetime Lomax products. In this paper, we consider the MLEs of unknown model parameters and the corresponding approximate confidence intervals. Also, we consider Baye’s estimation of the unknown parameters under consideration of the gamma priors on the unknown parameters. The Bayes estimates are computed with squared error loss functions. The Bayes estimators’ explicit forms cannot be obtained, so the MCMC approach is impalement to get the parameter estimation. The results are compared by using Monte Carlo simulation between the performances of the MLEs and approximate Bayes estimates under the assumptions of noninformative and informative priors. The Bayes estimates under the MCMC technique have been required to generate from the posterior distribution. The simulated MCMC sample plot and the corresponding histograms match quite well with the theoretical posterior density functions. The results can be extended to different types of loss functions. The proposed model and the results can also be easily extended for several members of the exponential family. Finally, we should mention that the products in Tables

The life experiments under balanced jointly progressive type-I censoring scheme present a good source for obtaining the information about Lomax lifetime products

The results improve for larger values of experiment total time

The Bayes estimation with informative prior distribution performs better than ML and noninformative prior method

The results of MLE are closed to the Bayes estimates for noninformative prior

For increase, the effective sample size

Censoring scheme with middle censoring performs better than other censoring schemes

The method of MCMC for approximation Bayesian estimate serves well, especially in dimensional cases

MEs and

( | Pa | ML | |||||||
---|---|---|---|---|---|---|---|---|---|

AVs | MSEs | AVs | MSEs | AVs | MSEs | ||||

1.5 | (35,30) | (25,(5, 0^{(24)})) | 3.832 | 1.254 | 3.710 | 1.114 | 3.654 | 0.984 | |

1.421 | 0.426 | 1.332 | 0.420 | 1.340 | 0.325 | ||||

2.315 | 0.754 | 2.374 | 0.701 | 2.214 | 0.654 | ||||

1.985 | 0.513 | 1.854 | 0.500 | 1.852 | 0.452 | ||||

(25,(0^{(10)}, 5, 0^{(10)})) | 3.784 | 1.124 | 3.725 | 1.012 | 3.611 | 0.897 | |||

1.389 | 0.400 | 1.325 | 0.387 | 1.351 | 0.301 | ||||

2.301 | 0.715 | 2.312 | 0.698 | 2.223 | 0.578 | ||||

1.852 | 0.480 | 1.811 | 0.471 | 1.777 | 0.399 | ||||

(45,45) | (30,(15, 0^{(29)})) | 3.452 | 0.985 | 3.411 | 0.979 | 3.412 | 0.745 | ||

1.301 | 0.345 | 1.311 | 0.341 | 1.310 | 0.298 | ||||

2.298 | 0.621 | 2.313 | 0.602 | 2.205 | 0.495 | ||||

1.811 | 0.407 | 1.815 | 0.399 | 1.712 | 0.300 | ||||

(1.5, 30,(0^{(14)}, 15, 0^{(15)})) | 3.415 | 0.956 | 3.415 | 0.949 | 3.364 | 0.711 | |||

1.289 | 0.323 | 1.300 | 0.310 | 1.277 | 0.250 | ||||

2.250 | 0.598 | 2.313 | 0.578 | 2.211 | 0.428 | ||||

1.783 | 0.385 | 1.811 | 0.370 | 1.719 | 0.287 | ||||

3.0 | (30,35) | (25,(5, 0^{(24)})) | 3.500 | 0.928 | 3.490 | 0.915 | 3.331 | 0.697 | |

1.233 | 0.301 | 1.288 | 0.299 | 1.215 | 0.210 | ||||

2.218 | 0.511 | 2.310 | 0.503 | 2.200 | 0.401 | ||||

1.760 | 0.322 | 1.781 | 0.315 | 1.702 | 0.262 | ||||

(25,(0^{(10)},5, 0^{(10)})) | 3.512 | 0.901 | 3.495 | 0.899 | 3.312 | 0.656 | |||

1.230 | 0.298 | 1.282 | 0.281 | 1.210 | 0.199 | ||||

2.214 | 0.497 | 2.301 | 0.490 | 2.213 | 0.395 | ||||

1.752 | 0.303 | 1.777 | 0.307 | 1.713 | 0.249 | ||||

(45,45) | (30,(15, 0^{(29)})) | 3.515 | 0.889 | 3.475 | 0.852 | 3.300 | 0.601 | ||

1.217 | 0.265 | 1.225 | 0.256 | 1.212 | 0.187 | ||||

2.210 | 0.460 | 2.298 | 0.455 | 2.215 | 0.361 | ||||

1.744 | 0.297 | 1.778 | 0.296 | 1.710 | 0.215 | ||||

(30,(0^{(14)},15, 0^{(15)})) | 3.421 | 0.860 | 3.470 | 0.844 | 3.278 | 0.593 | |||

1.210 | 0.241 | 1.211 | 0.239 | 1.210 | 0.161 | ||||

2.212 | 0.448 | 2.210 | 0.437 | 2.200 | 0.339 | ||||

1.737 | 0.281 | 1.755 | 0.277 | 1.712 | 0.203 |

MILs and PCs of Lomax distributions with

( | Pa | ML | |||||||
---|---|---|---|---|---|---|---|---|---|

MILs | PCs | MILs | PCs | MILs | PCs | ||||

1.5 | (35,30) | (25,(5, 0)) | 7.235 | (0.77) | 7.027 | (0.88) | 6.548 | (0.89) | |

3.368 | (0.88) | 3.287 | (0.88) | 3.001 | (0.90) | ||||

5.214 | (0.89) | 5.147 | (0.89) | 4.512 | (0.89) | ||||

4.512 | (0.89) | 4.325 | (0.90) | 3.541 | (0.90) | ||||

(25,(0^{(10)},5, 0^{(10)})) | 7.122 | (0.89) | 6.920 | (0.90) | 6.442 | (0.90) | |||

3.265 | (0.89) | 3.182 | (0.89) | 2.905 | (0.90) | ||||

5.118 | (0.91) | 5.100 | (0.91) | 4.470 | (0.89) | ||||

4.410 | (0.89) | 4.360 | (0.90) | 3.480 | (0.91) | ||||

(45,45) | (30,(15, 0^{(29)})) | 7.001 | (0.90) | 6.780 | (0.90) | 6.311 | (0.92) | ||

3.170 | (0.89) | 3.165 | (0.92) | 2.798 | (0.92) | ||||

5.024 | (0.90) | 5.003 | (0.91) | 4.304 | (0.90) | ||||

4.312 | (0.90) | 4.299 | (0.91) | 3.320 | (0.91) | ||||

(1.5, 30,(0^{(14)},15, 0^{(15)})) | 6.902 | (0.91) | 6.700 | (0.91) | 6.140 | (0.93) | |||

3.045 | (0.90) | 3.022 | (0.92) | 2.666 | (0.92) | ||||

4.920 | (0.89) | 4.901 | (0.93) | 4.123 | (0.94) | ||||

4.211 | (0.91) | 4.200 | (0.90) | 3.245 | (0.92) | ||||

3.0 | (30,35) | (25,(5, 0^{(24)})) | 6.450 | (0.90) | 6.320 | (0.90) | 5.824 | (0.90) | |

2.842 | (0.90) | 2.850 | (0.92) | 2.601 | (0.92) | ||||

4.754 | (0.90) | 4.701 | (0.91) | 4.013 | (0.93) | ||||

4.032 | (0.90) | 4.045 | (0.91) | 3.122 | (0.91) | ||||

(25,(0^{(10)},5, 0^{(10)})) | 6.361 | (0.91) | 6.311 | (0.92) | 5.715 | (0.94) | |||

2.715 | (0.91) | 2.711 | (0.92) | 2.514 | (0.92) | ||||

4.620 | (0.92) | 4.601 | (0.92) | 3.992 | (0.93) | ||||

3.911 | (0.90) | 3.920 | (0.93) | 3.009 | (0.93) | ||||

(45,45) | (30,(15, 0^{(29)})) | 6.155 | (0.90) | 6.140 | (0.93) | 5.560 | (0.92) | ||

2.511 | (0.94) | 2.531 | (0.92) | 2.347 | (0.92) | ||||

4.413 | (0.91) | 4.422 | (0.93) | 3.870 | (0.94) | ||||

3.741 | (0.90) | 3.700 | (0.91) | 2.891 | (0.91) | ||||

(30,(0^{(14)},15, 0^{(15)})) | 6.035 | (0.92) | 6.011 | (0.93) | 5.453 | (0.95) | |||

2.415 | (0.91) | 2.427 | (0.95) | 2.215 | (0.92) | ||||

4.340 | (0.94) | 4.311 | (0.93) | 3.745 | (0.93) | ||||

3.654 | (0.92) | 3.643 | (0.92) | 2.779 | (0.96) |

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

The authors have no conflicts of interest regarding the publication of the paper.

This study was funded by the Deanship of Scientific Research, Taif University, KSA (research project no. 1-440-6179).