Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Abstract

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions$\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form$\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta$ where$\mathop = \limits^\mathcal{D}$ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i−1)] degrees of freedom and$\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$

The paper also derives the distribution of the statisticV=F ₁ sin σ−F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.