Maximum likelihood estimation in the 3-parameter Weibull distribution. A look through the generalized extreme-value distribution

The main contribution of this paper is to provide reasonable confidence intervals for maximum likelihood estimates of percentile points in dielectric breakdown voltage probability distributions. The Weibull distributions which include threshold values are often used in breakdown voltage distributions. In some breakdown voltage samples, there exist cases in which the maximum likelihood estimates of all the three Weibull parameters diverge; these cases, in limiting forms, correspond to Gumbel distributions. However, computing confidence intervals of the percentile point estimates using the observed information matrix might be impossible when the distribution models are assumed to be the Weibull type. For such cases, adoption of the generalized extreme-value distribution as an extension of the Weibull distribution allows us to assess this troublesome problem. A Monte Carlo simulation using the newly developed generalized extreme-value distribution parameter estimation code reveals the property of the percentile point estimates for extreme-value type distributions, when the Weibull shape parameter is large. Biases and root mean squared errors of percentile point estimates are investigated both for the maximum likelihood estimates and for some closed form estimates; maximum likelihood estimates provide reasonable confidence intervals. For convenience, a demonstrative example with an estimation algorithm is provided