= Monotonicity bound for a mixed-censoring likelihood
{title2=$L(S)\leq(1-u)(u-v)^2uv$}
Suppose three exact event times $t_1<t_2<t_3$ coexist with a right-censored time $b$ and an interval $(a,c]$, ordered as $t_1<a<b<t_2<c<t_3$. Set $u=S(a)$ and $v=S(c)$. Monotonicity bounds the three event masses by $1-u$, $u-v$ and $v$, while $S(b)\leq u$. Thus the mixed-censoring likelihood is bounded by the displayed expression. Equality is attained by putting all mass at $t_1,t_2,t_3$. The reduced log-likelihood is strictly concave for $0<v<u<1$, and its stationary equations yield $u=4/5$, $v=4/15$. The fitted masses are $1/5$, $8/15$, $4/15$.
Back to article page
