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Original Article

Open Access Peer-reviewed

Yasser M. Amer^{ }, Rania M. Shalabi

Received September 12, 2019; Revised October 24, 2019; Accepted November 08, 2019

Step Stress-Partially Accelerated Life Test SS-PALT under Type-II progressive censoring with Binomial or uniform removal assuming Inverse Lomax distribution has been presented. A comparison between both removals is shown. The Newton-Raphson method is applied to obtain maximum likelihood estimators MLE of the parameters and the optimal stress-change time which minimizes the generalized asymptotic variance. A simulation study is performed to illustrate the statistical properties of the parameters.

The engineers used the Partially accelerated life tests (PALTs) successfully to estimate the acceleration factor and thus extrapolating the accelerated data to normal conditions. In a PALT, items are tested at both normal and accelerated conditions. There are three Types of PALTs, these Types are step-stress, progressive-stress and constant-stress. Such testing conducted under stresses is called accelerated life test (ALT) or partially accelerated life test (PALT). In ALT, the units are put under stresses to get more failures in a short time. The main assumption in ALT is that the mathematical model relating the lifetime of the unit and the stress is known or can be assumed. In some cases, such model is neither known nor assumed. That is, ALT data cannot be extrapolated to normal use condition. So, in such cases, PALT is a more suitable test to be used to estimate the statistical model parameters.

In a SS-PALT, test unit starts at normal use condition for a specified time. If it does not fail at that time, it is putted under stress. Stress is repeatedly increased until the test unit fails or the test is terminated based on a certain censoring scheme. As indicated by ^{ 1}, the step-stress method can reduce the testing time and save a lot of manpower, material sources and money. Specifically, SS-PALT should be used for reliability analysis to save time and money especially when the test units are with high reliability and the mathematical model indicated above is unknown or cannot be assumed Partially accelerated life tests (PALT) have been studied by several authors under step-stress scheme. For more details, see ^{ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}. It is noted that no studies have been made on the step-stress PALT under progressive censoring. In this paper, we will combine progressive censoring and step-stress PALT to develop a step-stress PALT with Progressively Type-II Censored Data using the exponential distribution as a lifetime model.

When the experimenter does not observe the lifetimes of all test units the censored sampling arises in a life test. There are two censoring schemes, Type-I censoring and Type-II censoring. Both of these two censoring schemes do not allow for units to be removed from the test at the points other than the final termination point. This allowance may be desirable when a compromise between reduced the time of experimentation and the observation of at least some extreme lifetimes is sought. Progressively censored sampling allows to the experimenter to save time and cost. The most popular one is the progressive type-II censoring scheme and it can be briefly described as follows. Suppose n identical units are put on a life testing experiment. The integer k < n is prefixed, and r_{1},..., r_{k }are k prefixed non-negative integers such that of the surviving units are randomly selected and removed from the test. Similarly, at the time of the second failure, r_{2} units are chosen randomly from the remaining n−r_{1}−2 units and they are removed, and so on. This experiment terminates when the *m*^{th} failure occurs at time the remaining surviving units are all removed from the test.

Extensive work has been done on various aspects of different progressive censoring schemes. ^{ 17} and ^{ 18} considered the progressive Type-II censoring scheme with fixed But in some reliability experiments, the number of removals cannot be considered to be fixed. ^{ 19} and ^{ 20} considered progressive censoring with random (binomial) removals to estimate the unknown parameters of Weibull and Gompertz distribution using ordinary life testing. ^{ 21} used progressively Type-II censored data with binomial removals to estimate the parameters of exponential reliability model. ^{ 22} discuss step-stress partially-accelerated life test under progressive Type-II censoring with random removals.

The removals from the test are assumed to have binomial distributions. The lifetimes of the test units are considered to be exponential distributed. Recently, ^{ 23} provided Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Gompertz Distribution and the removals from the test are assumed to have binomial and uniform distributions. Also, ^{ 24} introduced the same on Frechet Distribution.

In this paper, we will use SS-PAL under progressive Type-II censoring with random removals. The removals from the test are assumed to have binomial and uniform distributions. The lifetimes of the test units are considered to be Inverse Lomax distributed. Also, we will determine the optimal stress change time which minimizes the generalized asymptotic variance of the MLE of parameters. Section 2 presents the Inverse Lomax distribution and the assumptions of the partially accelerated model. Estimation of model parameters is given in Section 3; in Section 4 and 5 simulation study results and conclusion are given.

The Lomax or Pareto II (the shifted Pareto) distribution was proposed in ^{ 25}. This distribution has found wide applications especially in analysis of the business failure life time data, income and wealth inequality, size of cities, actuarial science, medical and biological sciences, engineering, lifetime and reliability modeling. In lifetime, the Lomax model belongs to the family of decreasing failure rate in ^{ 26}.

The Inverse Lomax distribution (ILD) belongs to inverted family of distributions and found to be very flexible to analyze the situation where the non-monotonicity of the failure rate has been realized in ^{ 27}. If a random variable Y has Lomax distribution, then has an Inverse Lomax distribution (ILD) ^{ 28}. (ILD) has an application in stochastic modeling of decreasing failure rate life components. Like other distributions belonging to the family of generalized Beta distribution, the (ILD) also has application in economics and actuarial sciences ^{ 29}. (ILD) was implemented on geophysical databases ^{ 30} on the sizes of land fires in the California state. ^{ 31}, carried out research work regarding the statistical inference and Prediction on (ILD) through Bayesian inferences. ^{ 32} considered the (ILD) to possess the Lorenz ordering relationship between ordered statistics.

The probability density function (PDF) and the cumulative distribution function (CDF) for (ILD) respectively are as follows:

(1) |

(2) |

The following assumptions are used throughout the paper:

1. *n* identical and independent units are put on the life test.

2. The lifetime of each unit has an exponential distribution.

3. The test is terminated at the *m*^{th} failure, where *m* is prefixed

4. Each of the *n* units is first run under normal use condition. If it does not fail or remove from the test by a pre-specified time . it is put under accelerated condition (stress).

5. At the *i*^{th} failure a random number of the surviving units, are randomly selected and removed from the test. Finally, at the *m*^{th} failure the remaining surviving units are all removed from the test and the test is terminated.

6. The lifetime, say *X*, of a unit under SS-PALT can be rewritten as

In this paper, we will have assumed that, probability density function (PDF) of *X* is given by

(3) |

In addition, the survival functions (SF) under normal and accelerate use conditions respectively is given by

(4) |

And

(5)

The number of units removed from the test at each failure time follows a binomial distribution and any individual unit being removed is independent of others but with the same probability that is,* **and for** ** * an*d** *

Let , denote the observation obtained form a progressively Type-II censored sample with random removals in a SS-PALT. Here . Given the pre-determined number of removals the conditional likelihood function of the observations takes the following form

(6) |

From (1), (2) and (3) is inserted in (6) and simplify, we get

(7) |

Where

And

The number of units removed at each failure time follows a binomial distribution such that

and for

(8) |

At the parameter is known, the likelihood of the sample of size n is given as

(9) |

Where,

That is,

(10) |

The log-likelihood function can be written as follows:

(11) |

First partial derivatives are derived to obtain the estimate of the parameters and acceleration factor . The log likelihood which is associated with (10) is given

Where,

The following likelihood equations are obtained by equating the partial derivatives of with respect to and to zero:

(13) |

(14) |

(15) |

(16) |

From Equation (16), is estimated as follow

(17) |

There is no closed-from solution to this system of equations (13-15), so we will solve for and iteratively, using the Newton-Raphson method, a tangent method for root finding. In our case we will estimate iteratively

(18) |

where g is the vector of normal equations for which we want with and and G is the matrix of second derivatives

(19) |

The Newton-Raphson algorithm converges, as our estimates of and change by less than a tolerated amount with each successive iteration, to and .

The bias and the root of mean squared error (RMSE) of an estimator of the parameter , easily obtained by

(20) |

The number of units removed from the test at each failure time follows a uniform distribution and any individual unit being removed is independent of others but with the same probability p. that is, and for i=1,2,..3. and .

The number of units removed at each failure time follows a uniform distribution such that

(21) |

And for

where, the joint probability distribution of and is given by

(22) |

where **.**

It is clear that does not depend on the parameters and , hence the maximum likelihood estimators can be derived directly by maximizing the equations (9) and then solving the equations (13-16).

The asymptotic Fisher information matrix can be formulated as follows;

(23) |

In relation to the asymptotic variance-covariance matrix of the ML estimators of the parameters, it can be approximated by numerically inverting the above Fisher's information matrix .

In this section we explore the choice of in a SS-PALT with Type-II progressive censoring. We propose one selection criterion which enable one to choose the optimal value of . The proposed criterion is based on the determinant of the Fisher's information matrix. Maximizing that determinant is equivalent to minimizing the generalized asymptotic variance (GAV) of the MLE of the model parameters. The GAV is the reciprocal of the determinant of the Fisher's information matrix F, for example, see ^{ 33}. That is,

(24) |

So, the optimal value of is chosen such that the determinant of the Fisher's information matrix is maximized and then the GAV is minimized. This is called the D-optimality criterion.

A simulation study is performed to obtain MLEs of and . Also, to study the properties of these estimates through the root of the mean squared errors (RMSEs),) and the confidence intervals for different sample sizes. Moreover, we will determine the optimal stress change time which minimizes the generalized asymptotic variance of the MLE of parameters. To perform the simulation study, we followed the same steps has been introduced in ^{ 24};

a) First specify the value of n and m.

b) The value of the parameters are chosen to be α=2, θ=0.5, β=2.3,τ=2, p=0.4*.*

c) Generate a random sample with size n and censoring size m with random removals, from the random variable X given by (3).

d) Generate a group value and also where and .

e) For different sample sizes n= 20, 50, 80 and 100, compute the ML estimates.

The bias and the root of mean squared error (RMSE) are obtained associated with the MLE of the parameters, optimal value of and also the Optimal GAV of the MLEs of the model parameters are obtained numerically for each sample size.

This paper presented the SS-PALT under Type-II progressive censoring with Binomial or uniform removals assuming (ILD). Comparison between both removals is shown. The Newton-Raphson method is applied to obtain MLE estimators of the parameters and the optimal stress-change time which minimizes the GAV.

The numerical study for obtaining the optimum plan for binomial removal is tabulated in Table 1 for different sample size and Table 2 describes uniform removal for possible values of the parameters. We derive the MLE of the parameters. Also, we compute the RMSE associated with the MLE. The above results it is easy to find that for the fixed values of the parameters, the error and optimal time decrease with increasing sample size n. Performance of testing plans and model assumptions are usually evaluated by the properties of the maximum likelihood estimates of model parameters. From the numerical results, we can conclude that both the average value of and the average value of GAV for Type-II progressive censoring are getting close to those of complete sample with the bigger m and close faster for bigger n. Hence from the numerical result, we can conclude that estimates of binomial or uniform are stable with relatively small RMSE with increasing sample size. Therefore, the test design obtained here is a robust design and work well for binomial or uniform removal.

[1] | Rao, R., 1992, Equivalence of the tampered random variables and tampered failure rate models in ALT for a class of life distribution having the setting the clock back to zero property, Communication in Statistics – Theory and Methods, Vol. 21, No. 3, 647-664. | ||

In article | View Article | ||

[2] | Abdel-Ghaly, El-Khodary, A. E. H. and Ismail, A. A., 2003, Estimation and Optimal Design in Step Partially Accelerated Life Tests for the Pareto Distribution using Type-II Censoring, the Proceedings of the 15th annual conference on Statistics and Computer Modeling in Human and Social Sciences, Faculty of Economics and Political Science, Cairo University, Egypt, 16-29. | ||

In article | |||

[3] | Abdel-Ghaly, El-Khodary, A. E. H. Ismail, and A. A., 2007, Estimation and Optimum Constant Stress Partially Accelerated Life Test Plans for a Compound Pareto Distribution with Type-I Censoring, InterStat, Electronic Journal, Nov., # 2. | ||

In article | |||

[4] | Abdel-Ghaly, A., El-Khodary, E. H. and Ismail, A. A., 2008, Maximum Likelihood Estimation and Optimal Design in Step Partially Accelerated Life Tests for the Pareto Distribution with Type-I Censoring, InterStat, Electronic Journal, Jan., # 2. | ||

In article | |||

[5] | Abdel-Ghani, M. M., 2004, The estimation problem of the Log-Logistic parameters in step partially accelerated life tests using Type-I censored data, The National Review of Social Sciences, 41(2), 1-19. | ||

In article | |||

[6] | Aly, H. M. and Ismail, A. A., 2008, Optimum Simple Time-Step Stress Plans for Partially Accelerated Life Testing with Censoring, Far East Journal of Theoretical Statistics, 24(2), 175 - 200. | ||

In article | |||

[7] | Ismail, A., 2004, The Test Design and Parameter Estimation of Pareto Lifetime Distribution under Partially Accelerated Life Tests, Ph.D. Thesis, Department of Statistics, Faculty of Economics & Political Science, Cairo University, Egypt. | ||

In article | |||

[8] | Ismail, A., 2006, On the Optimal Design of Step-Stress Partially Accelerated Life Tests for the Gompertz Distribution with Type-I Censoring, InterStat, Electronic Journal, June, # 1. | ||

In article | |||

[9] | Ismail, A., 2014, Inference for a step-stress partially accelerated life test model with an adaptive Type-II progressively hybrid censored data from Weibull distribution, Journal of Computational and Applied Mathematics ,260, 533–542. | ||

In article | View Article | ||

[10] | Lone, S. A., Rahman, A. and Islam, A., 2016, Estimation in StepStress Partially Accelerated Life Tests for the MukherjeeIsla Distribution Using Time Constraint, International Journal of Modern Mathematical Sciences,14(3): 227-238. | ||

In article | |||

[11] | Lone, S. A., Rahman, A. and Islam, A., 2017, Step Stress Partially Accelerated Life Testing Plan for Competing Risk Using Adaptive Type-I Progressive Hybrid Censoring, Pakistan Journal of Statistics, vol. 33(4), 237-248. | ||

In article | |||

[12] | Lone, S. A., Rahman, A. and Islam, A., 2018, Step-Stress Partially Accelerated Life Testing Plan for Rayleigh Distribution Using Adaptive Type-II ProgressiveHybrid Censoring. | ||

In article | View Article | ||

[13] | Mohie El-Din, M. M., Amein, M. M., El-Attar H. E. and Hafez, E. H., 2016, Estimation in Step-Stress Accelerated Life Testing for Lindely Distribution with Progressive First-Failure Censoring, Journal of Statistics Applications & Probability, Vol 5, No. 3, 393-398. | ||

In article | View Article | ||

[14] | Nasser, S. G. and Elharouna, N. M., 2019, Inference for exponentiated Weibull distribution under constant stress partially accelerated life tests with multiple censored, Communications for Statistical Applications and Methods, vol. 26, No. 2, 131–148. | ||

In article | View Article | ||

[15] | Soliman, A. A., Ahmed, E. A. Abou-Elheggag, N. A. and Ahmed, S. M., 2017, Step-Stress Partially Accelerated Life Tests Model in Estimation of Inverse Weibull Parameters under Progressive Type-II Censoring, Applied Mathematics & Information Sciences, vol.11, No. 5, 1369-1381. | ||

In article | View Article | ||

[16] | Wang, F. K., Cheng Y. F. and Lu, W. L., 2012, Partially Accelerated Life Tests for the Weibull Distribution Under Multiply Censored Data, Communications in Statistics - Simulation and Computation, vol.41, NO (9), 1667-1678. | ||

In article | View Article | ||

[17] | Cohen, C. 1963, Progressively Censored Samples in the Life Testing, Technometerics, 5, 327-339. | ||

In article | View Article | ||

[18] | Cohen, C. and Niggard, N. J., 1977, Progressively Censored Sampling in the Three Parameter Gamma Distribution, Technometerics, 19, 333-340. | ||

In article | View Article | ||

[19] | Tse S. K., Yang, C. and Yuen, H. K., 2000, Statistical Analysis of Weibull Distributed Lifetime Data under Type-II Progressive Censoring with Binomial Removals. Journal of Applied Statistics, vol. 27, no. 8, pp. 1033-1043. | ||

In article | View Article | ||

[20] | Wu, S.J., Chen, Y.J. and Chang, C.T., 2007, Statistical Inference Based on Progressively Censored Samples with Random Removals from the Burr Type XII Distribution. Journal of Statistical Computation and Simulation, vol. 77, no. 1, pp. 19-27. | ||

In article | View Article | ||

[21] | Sarhan, and Abuammoh, A., 2008, Statistical Inference using progressively Type-II censored data with random scheme, International Mathematical Forum, 3, no. 35,1713 - 1725. | ||

In article | |||

[22] | Ismail, A., 2009, Optimal Design of Step-Stress Life Test with Progressively Type-II Censored Exponential Data, International Mathematical Forum, 4, no 40, 1963 - 1976 | ||

In article | |||

[23] | Abdel Hady, D. H. 2019, Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Gompertz Distribution, American Journal of Applied Mathematics and Statistics, 2019, Vol. 7, No. 1, 37-42 | ||

In article | |||

[24] | Shahab, S., Anwar, S. and Islam, A., 2015, Optimal Design of Step Stress Partially Accelerated Life Test Under Progressive Type-Ii Censored Data with Random Removal for Frechet Distribution, RT&A, vol.10, # 02 (37) | ||

In article | |||

[25] | Lomax, K. S., 1954, Business failures: Another example of the analysis of failure data, Journal of the American Statistical Association 49, 847–852. | ||

In article | View Article | ||

[26] | Chahkandi, M. and Ganjali, M., 2009, On some lifetime distributions with decreasing failure rate, Computational Statistics and Data Analysis 53, 4433–4440. | ||

In article | View Article | ||

[27] | Singh, S. K., Singh, U., Kumar, D., 2012, Bayes estimators of the reliability function and parameters of inverted exponential distribution using informative and non-informative priors. Journal of Statistical computation and simulation, vol.83, no. 12, pp. 2258–2269. | ||

In article | View Article | ||

[28] | Singh Yadav, A., Singh U. and Singh, S.K., 2016. On Hybrid Censored Inverse Lomax Distribution: Application to The Survival Data. Statistica, Anno LXXVI, n. 2, 2016, pp. 185-203. | ||

In article | View Article | ||

[29] | Kleiber C. and Kotz S., 2003. Statistical size distributions in economics and actuarial sciences. John Wiley & Sons, Inc., Hoboken, New Jersey. | ||

In article | View Article | ||

[30] | McKenzie, D., Miller, C. Falk, DA., 2011, The Landscape ecology of fire. Springer, New York. | ||

In article | View Article | ||

[31] | Rahman, J., Aslam, M., Ali, S., 2013, Estimation and prediction of inverse Lomax model via Bayesian approach. Caspian Journal of Applied Sciences Research, 2(3), pp. 43–56. | ||

In article | |||

[32] | Kleiber, C., 2004. Lorenz ordering of order statistics from log-logistic and related distributions. Journal of Statistical Planning and Inference. 120, 13-19. | ||

In article | View Article | ||

[33] | Bai, D. S., Chung, S. W. and Chun, Y. R., 1993, Optimal design of partially accelerated life tests for the Lognormal distribution under Type-I censoring. Reliability Eng. & Sys. Safety, 40, 85-92. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2019 Yasser M. Amer and Rania M. Shalabi

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Yasser M. Amer, Rania M. Shalabi. Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Inverse Lomax Distribution. *American Journal of Applied Mathematics and Statistics*. Vol. 7, No. 5, 2019, pp 171-177. http://pubs.sciepub.com/ajams/7/5/3

Amer, Yasser M., and Rania M. Shalabi. "Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Inverse Lomax Distribution." *American Journal of Applied Mathematics and Statistics* 7.5 (2019): 171-177.

Amer, Y. M. , & Shalabi, R. M. (2019). Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Inverse Lomax Distribution. *American Journal of Applied Mathematics and Statistics*, *7*(5), 171-177.

Amer, Yasser M., and Rania M. Shalabi. "Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Inverse Lomax Distribution." *American Journal of Applied Mathematics and Statistics* 7, no. 5 (2019): 171-177.

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[1] | Rao, R., 1992, Equivalence of the tampered random variables and tampered failure rate models in ALT for a class of life distribution having the setting the clock back to zero property, Communication in Statistics – Theory and Methods, Vol. 21, No. 3, 647-664. | ||

In article | View Article | ||

[2] | Abdel-Ghaly, El-Khodary, A. E. H. and Ismail, A. A., 2003, Estimation and Optimal Design in Step Partially Accelerated Life Tests for the Pareto Distribution using Type-II Censoring, the Proceedings of the 15th annual conference on Statistics and Computer Modeling in Human and Social Sciences, Faculty of Economics and Political Science, Cairo University, Egypt, 16-29. | ||

In article | |||

[3] | Abdel-Ghaly, El-Khodary, A. E. H. Ismail, and A. A., 2007, Estimation and Optimum Constant Stress Partially Accelerated Life Test Plans for a Compound Pareto Distribution with Type-I Censoring, InterStat, Electronic Journal, Nov., # 2. | ||

In article | |||

[4] | Abdel-Ghaly, A., El-Khodary, E. H. and Ismail, A. A., 2008, Maximum Likelihood Estimation and Optimal Design in Step Partially Accelerated Life Tests for the Pareto Distribution with Type-I Censoring, InterStat, Electronic Journal, Jan., # 2. | ||

In article | |||

[5] | Abdel-Ghani, M. M., 2004, The estimation problem of the Log-Logistic parameters in step partially accelerated life tests using Type-I censored data, The National Review of Social Sciences, 41(2), 1-19. | ||

In article | |||

[6] | Aly, H. M. and Ismail, A. A., 2008, Optimum Simple Time-Step Stress Plans for Partially Accelerated Life Testing with Censoring, Far East Journal of Theoretical Statistics, 24(2), 175 - 200. | ||

In article | |||

[7] | Ismail, A., 2004, The Test Design and Parameter Estimation of Pareto Lifetime Distribution under Partially Accelerated Life Tests, Ph.D. Thesis, Department of Statistics, Faculty of Economics & Political Science, Cairo University, Egypt. | ||

In article | |||

[8] | Ismail, A., 2006, On the Optimal Design of Step-Stress Partially Accelerated Life Tests for the Gompertz Distribution with Type-I Censoring, InterStat, Electronic Journal, June, # 1. | ||

In article | |||

[9] | Ismail, A., 2014, Inference for a step-stress partially accelerated life test model with an adaptive Type-II progressively hybrid censored data from Weibull distribution, Journal of Computational and Applied Mathematics ,260, 533–542. | ||

In article | View Article | ||

[10] | Lone, S. A., Rahman, A. and Islam, A., 2016, Estimation in StepStress Partially Accelerated Life Tests for the MukherjeeIsla Distribution Using Time Constraint, International Journal of Modern Mathematical Sciences,14(3): 227-238. | ||

In article | |||

[11] | Lone, S. A., Rahman, A. and Islam, A., 2017, Step Stress Partially Accelerated Life Testing Plan for Competing Risk Using Adaptive Type-I Progressive Hybrid Censoring, Pakistan Journal of Statistics, vol. 33(4), 237-248. | ||

In article | |||

[12] | Lone, S. A., Rahman, A. and Islam, A., 2018, Step-Stress Partially Accelerated Life Testing Plan for Rayleigh Distribution Using Adaptive Type-II ProgressiveHybrid Censoring. | ||

In article | View Article | ||

[13] | Mohie El-Din, M. M., Amein, M. M., El-Attar H. E. and Hafez, E. H., 2016, Estimation in Step-Stress Accelerated Life Testing for Lindely Distribution with Progressive First-Failure Censoring, Journal of Statistics Applications & Probability, Vol 5, No. 3, 393-398. | ||

In article | View Article | ||

[14] | Nasser, S. G. and Elharouna, N. M., 2019, Inference for exponentiated Weibull distribution under constant stress partially accelerated life tests with multiple censored, Communications for Statistical Applications and Methods, vol. 26, No. 2, 131–148. | ||

In article | View Article | ||

[15] | Soliman, A. A., Ahmed, E. A. Abou-Elheggag, N. A. and Ahmed, S. M., 2017, Step-Stress Partially Accelerated Life Tests Model in Estimation of Inverse Weibull Parameters under Progressive Type-II Censoring, Applied Mathematics & Information Sciences, vol.11, No. 5, 1369-1381. | ||

In article | View Article | ||

[16] | Wang, F. K., Cheng Y. F. and Lu, W. L., 2012, Partially Accelerated Life Tests for the Weibull Distribution Under Multiply Censored Data, Communications in Statistics - Simulation and Computation, vol.41, NO (9), 1667-1678. | ||

In article | View Article | ||

[17] | Cohen, C. 1963, Progressively Censored Samples in the Life Testing, Technometerics, 5, 327-339. | ||

In article | View Article | ||

[18] | Cohen, C. and Niggard, N. J., 1977, Progressively Censored Sampling in the Three Parameter Gamma Distribution, Technometerics, 19, 333-340. | ||

In article | View Article | ||

[19] | Tse S. K., Yang, C. and Yuen, H. K., 2000, Statistical Analysis of Weibull Distributed Lifetime Data under Type-II Progressive Censoring with Binomial Removals. Journal of Applied Statistics, vol. 27, no. 8, pp. 1033-1043. | ||

In article | View Article | ||

[20] | Wu, S.J., Chen, Y.J. and Chang, C.T., 2007, Statistical Inference Based on Progressively Censored Samples with Random Removals from the Burr Type XII Distribution. Journal of Statistical Computation and Simulation, vol. 77, no. 1, pp. 19-27. | ||

In article | View Article | ||

[21] | Sarhan, and Abuammoh, A., 2008, Statistical Inference using progressively Type-II censored data with random scheme, International Mathematical Forum, 3, no. 35,1713 - 1725. | ||

In article | |||

[22] | Ismail, A., 2009, Optimal Design of Step-Stress Life Test with Progressively Type-II Censored Exponential Data, International Mathematical Forum, 4, no 40, 1963 - 1976 | ||

In article | |||

[23] | Abdel Hady, D. H. 2019, Optimal Design of Step Stress Partially Accelerated Life Test under Progressive Type-II Censored Data with Random Removal for Gompertz Distribution, American Journal of Applied Mathematics and Statistics, 2019, Vol. 7, No. 1, 37-42 | ||

In article | |||

[24] | Shahab, S., Anwar, S. and Islam, A., 2015, Optimal Design of Step Stress Partially Accelerated Life Test Under Progressive Type-Ii Censored Data with Random Removal for Frechet Distribution, RT&A, vol.10, # 02 (37) | ||

In article | |||

[25] | Lomax, K. S., 1954, Business failures: Another example of the analysis of failure data, Journal of the American Statistical Association 49, 847–852. | ||

In article | View Article | ||

[26] | Chahkandi, M. and Ganjali, M., 2009, On some lifetime distributions with decreasing failure rate, Computational Statistics and Data Analysis 53, 4433–4440. | ||

In article | View Article | ||

[27] | Singh, S. K., Singh, U., Kumar, D., 2012, Bayes estimators of the reliability function and parameters of inverted exponential distribution using informative and non-informative priors. Journal of Statistical computation and simulation, vol.83, no. 12, pp. 2258–2269. | ||

In article | View Article | ||

[28] | Singh Yadav, A., Singh U. and Singh, S.K., 2016. On Hybrid Censored Inverse Lomax Distribution: Application to The Survival Data. Statistica, Anno LXXVI, n. 2, 2016, pp. 185-203. | ||

In article | View Article | ||

[29] | Kleiber C. and Kotz S., 2003. Statistical size distributions in economics and actuarial sciences. John Wiley & Sons, Inc., Hoboken, New Jersey. | ||

In article | View Article | ||

[30] | McKenzie, D., Miller, C. Falk, DA., 2011, The Landscape ecology of fire. Springer, New York. | ||

In article | View Article | ||

[31] | Rahman, J., Aslam, M., Ali, S., 2013, Estimation and prediction of inverse Lomax model via Bayesian approach. Caspian Journal of Applied Sciences Research, 2(3), pp. 43–56. | ||

In article | |||

[32] | Kleiber, C., 2004. Lorenz ordering of order statistics from log-logistic and related distributions. Journal of Statistical Planning and Inference. 120, 13-19. | ||

In article | View Article | ||

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