  The progressive censoring scheme has received considerable amount of
attention in the last fifteen years. During the last few years joint
progressive censoring scheme has gained some popularity. Recently, the authors
Mondal and Kundu ("A new two sample Type-II progressive censoring scheme",
arXiv:1609.05805) introduced a balanced two sample Type-II progressive
censoring scheme and provided the exact inference when the two populations are
exponentially distributed. In this article we consider the case when the two
populations follow Weibull distributions with the common shape parameter and
different scale parameters. We obtain the maximum likelihood estimators of the
unknown parameters. It is observed that the maximum likelihood estimators
cannot be obtained in explicit forms, hence, we propose approximate maximum
likelihood estimators, which can be obtained in explicit forms. We construct
the asymptotic and bootstrap confidence intervals of the population parameters.
Further we derive an exact joint confidence region of the unknown parameters.
We propose an objective function based on the expected volume of this
confidence set and using that we obtain the optimum progressive censoring
scheme. Extensive simulations have been performed to see the performances of
the proposed method, and one real data set has been analyzed for illustrative
purposes.
