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.

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.