Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under Type-I and Type-II censoring

In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum likelihood estimators [^(q)]₁{hat{theta}_{1}} and [^(q)]₂{hat{theta}_{2}} of the parameters θ ₁ and θ ₂ have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively. Here, we prove that these estimators are stochastically monotone with respect to θ ₁ and θ ₂, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the parameters θ ₁ and θ ₂. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions under truncation, which is also of independent interest.