This paper considers the Type I hybrid censoring and investigates the optimal value for the sample size which is assumed as a truncated binomial Random variable‎. ‎Rayleigh distribution is considered for the lifetime distribution‎. ‎Towards this end‎, ‎various factors can be considered and the most important is the sampling cost criterion‎. ‎Since the sample size is a Random variable‎, ‎the optimal parameter of the Random sample size is determined so that the total cost of the test does not exceed a pre-determined value‎. ‎Numerical calculations and a simulation study have been performed to evaluate the obtained results‎. ‎Finally‎, ‎the conclusion of the article is presented.

In designing an optimal life-testing experiment under a censoring setup‎, ‎the removal vector scheme is usually chosen by optimizing a suitable criterion function‎. ‎The criterion functions are usually constructed based on cost or variance functions‎, ‎and sometimes a combination of both‎. ‎This paper considers a multiple optimization problem in the context of Type-II progressive censoring with Random dependent removal lifetime experiment‎. ‎A simple simulation algorithm is presented for obtaining the optimal scheme in a multi-objective optimal problem under the Type-II progressive censoring with Random dependent removal model‎. ‎Several simulation studies are conducted to evaluate and compare the performance of the proposed strategy‎. ‎Finally‎, ‎some concluding remarks and future works are provided.

Introduction A general censoring scheme called progressive Type-II right censoring has been considered. The removal plan can be fixed or Random, chosen according to a discrete probability distribution. In many practical problems, not only does an experiment process determines inevitably to use Random removals, but also a fixed removal assumption may be cumbersome to analyze some results of statistical inference. The scenario of Random removals has been introduced by Yuen and Tse (1996) under the Weibull lifetime distribution and the discrete Uniform distribution for Random removals. Tse et al. (2000) discussed Binomial removals even though the parameter p enormously impressed the experiment time, and the Uniform and Binomial distributions were independent of the lifetime distribution. The limitations mentioned above motivate us to propose a new method for determining removals based on the failure times. Material and Methods Let the lifetimes of the n units placed on the life-test be distributed as two-parameter Weibull distribution. The proposed Random removals use the relationship between the Weibull and Exponential and are based on two approaches: the normalized spacings with Random and fixed coefficients according to progressively Type-II censored order statistics from the Exponential distribution. Wherein the time distance between consecutive failure times depends on the type of lifetime distribution and the number of units that will be removed after each failure are proportional to a root function of the difference between two last failure times divided by the time of the first failure. The joint probability mass functions of Random removals are also derived. The estimations of parameters are derived using different estimation procedures such as the maximum likelihood, maximum product spacing, and least-squares methods. The proposed Random removal schemes are compared to the discrete Uniform and the Binomial removal schemes via a Monte Carlo simulation study in terms of their biases, root mean squared errors of estimators, expected total test times and the Ratio of the Expected Experiment Time (REET) values. Finally, an innovative technique is introduced for deriving progressive type II censoring samples from a real data set. Results and Discussion From comparing the REET values, it is evident that a slight reduction in expected experiment time occurs when a large number of units are tested for lifetimes under Uniform and Binomial distributions with a considerable probability, p, especially for cases with decreasing failure rate ,> 1. Although the Binomial distribution with p < 0: 5 has relatively acceptable performance, two proposed approaches have smaller REET values, which decreases significantly as the sample size n increases. However, binomial removals perform better than uniform removals in terms of E(Xm: m: n). Still, the expected test time depends very much on the value of removal probability p. Conclusion It is shown that the expected total time under the Random coefficients has the most negligible value concerning other approaches and reduces the expected full time on the test.

A hybrid censoring scheme is a mixture of type I and type II censoring schemes. When n items are placed on a life test, the experiment terminates under type I or type II hybrid censoring scheme if either a prefixed censoring time T or the rth (1 £ r £ n is fixed) failure is first or later observed, respectively. In this paper, we investigate the decomposition of entropy in both hybrid censoring schemes. Entropy of type I hybrid censoring scheme is formulated and in order to determining the entropy of type II hybrid censoring scheme the available information are used. The results are then applied to some common life time distributions as illustrative examples.Finally, maximum entropy of the mentioned censoring schemes is discussed.

An important challenge in using progressive Type-II right censoring is to determine a removal scheme. It can be predetermined or Randomly chosen per discrete distributions. This paper considers the Random removal problem and proposes two scenarios for determining the removal vector without introducing any parameter to a model when progressively Type-II censored samples are available from the three-parameter Weibull distribution. The proposed scenarios are based on the normalized spacings with Random and fixed coefficients according to progressively Type-II censored order statistics from an exponential distribution. The joint probability mass functions of removal vectors are provided as well as expected experimental time under the proposed two methods. Moreover, the maximum likelihood estimators (MLEs) and corrected maximum likelihood estimators (corrected MLEs) of parameters are obtained. The new approaches are compared with the patterns of removal derived from the discrete uniform and binomial distributions using a Monte Carlo simulation study. This comparison is based on their estimated biases, estimated mean squared errors and expected total time on the experiment. Finally, a real data example is given to show the practical applications of the paper.

The two parameter exponential distribution is particularly important among statistical distributions due to its constant failure rate and has applications in the fields of medicine, biology, clinical trials, public health, engineering, economics, demographics, and life span data, and reliability. Due to the importance of life span data, two parameter exponential distribution with censored data has recently attracted the attention of many researchers, but so far the inference about the location parameter with Random censored data in the presence of outlier data has not been discussed. In this article, the location and scale parameters of two parameter exponential distribution under Random censoring with the presence of k outliers are estimated by Bayesian and classical methods. Due to the importance of the spatial parameter, when censoring the two parameter exponential distribution with the presence of outlier data, the spatial parameter is considered the same but the scale parameter is different. In the Bayesian estimation of parameters, the is checked using Gibbs sampling under the error squared loss function. We recommend used the Bayesian estimation.The generalized variance is given according to the dimensions of the parameters using the maximum likelihood method.