Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 41 / 5 / b

Put $A={\sum }_{i=1}^3{S}_{i1}$, $B={\sum }_{i=1}^3{S}_{i2}$ and $d={\lambda }_1-{\lambda }_2$. Independence of the pairs makes their likelihood contributions multiply. The three fully observed pairs contribute $({\lambda }_1{\lambda }_2{)}^3{e}^{-{\lambda }_{1}A-{\lambda }_{2}B}$; each remaining pair contributes its hypoexponential distribution density. Thus the observed-data log-likelihood isAn equivalent expression, convenient for calculations, isThe apparent singularity as $d\downarrow 0$ cancels between the last sum and $-5\log d$. The resulting equal-rate limit is finite for positive observed times.