A New Approach to Maximum Likelihood Estimation of the Three‐parameter Gamma and Weibull Distributions

Summary A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.