  Triangular distributions are a well-known class of distributions that are
often used as an elementary example of a probability model. Maximum likelihood
estimation of the mode parameter of the triangular distribution over the unit
interval can be performed via an order statistics-based method. It had been
conjectured that such a method can be conducted using only a constant number of
likelihood function evaluations, on average, as the sample size becomes large.
We prove two theorems that validate this conjecture. Graphical and numerical
results are presented to supplement our proofs.
