Bias-Corrected Maximum Likelihood Estimators of the Parameters of the Unit-Weibull Distribution

Abstract

It is well known that the maximum likelihood estimates (MLEs) have appealing statistical properties. Under fairly mild conditions their asymptotic distribution is normal, and no other estimator has a smaller asymptotic variance.
However, in finite samples the maximum likelihood estimates are often biased estimates and the bias disappears as the sample size grows.
Mazucheli, Menezes, and Ghitany (2018b) introduced a two-parameter unit-Weibull distribution which is useful for modeling data on the unit interval, however its MLEs are biased in finite samples.
In this paper, we adopt three approaches for bias reduction of the MLEs of the parameters of unit-Weibull distribution.
The first approach is the analytical methodology suggested by Cox and Snell (1968), the second is based on parametric bootstrap resampling method, and the third is the preventive approach introduced by Firth (1993).
The results from Monte Carlo simulations revealed that the biases of the estimates should not be ignored and the bias reduction approaches are equally efficient. However, the first approach is easier to implement.
Finally, applications to two real data sets are presented for illustrative purposes.