Note on “Parameters estimators of irregular right-angled triangular distribution”

Abstract

Simple estimators were given in (Kachiashvili & Topchishvili, 2016) for the lower and upper limits of an irregular right-angled triangular distribution together with convenient formulas for removing their bias. We argue here that the smallest observation is not a maximum likelihood estimator (MLE) of the lower limit and we present a procedure for computing an MLE of this parameter. We show that the MLE is strictly smaller than the smallest observation and we give some bounds that are useful in a numerical solution procedure. We also present simulation results to assess the bias and variance of the MLE.

Hybrid simplicial-randomized approximate stochastic dynamic programming for multireservoir optimization

“…Toward this end, we generate samples of grid points with the mode at the upper limit b, are given in [34], where it is shown that the MLE of b is also x (m) . However, by arguing as in [36], it is easily seen that x (m) is not an MLE of b for a right-angled triangular distribution with the mode at the lower limit. The true MLE is provided in Appendix C.…”

“…For the triangular distribution with mode at left TL(0, b), we see that the unbiased estimate and conĄdence interval limits based on the TR(0, b) order statistic x (m) are quite large compared to the other distributions. There might be an interest here in using an MLE estimate instead, which has small bias and smaller variance as pointed out in [36] thus allowing a smaller sample size for estimating the approximation error, and therefore fewer computations.…”