H. Hirose, MAXIMUM-LIKELIHOOD-ESTIMATION IN THE 3-PARAMETER WEIBULL DISTRIBUTION- A LOOK THROUGH THE GENERALIZED EXTREME-VALUE DISTRIBUTION, IEEE transactions on dielectrics and electrical insulation, 3(1), 1996, pp. 43-55
The main contribution of this paper is to provide reasonable confidenc
e intervals for maximum likelihood estimates of percentile points in d
ielectric breakdown voltage probability distributions. The Weibull dis
tributions 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 Weibu
ll parameters diverge; these cases, in limiting forms, correspond to G
umbel distributions. However, computing confidence intervals of the pe
rcentile point estimates using the observed information matrix might b
e impossible when the distribution models are assumed to be the Weibul
l type. For such cases, adoption of the generalized extreme-value dist
ribution as an extension of the Weibull distribution allows us to asse
ss 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 extrem
e-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 c
losed form estimates; maximum likelihood estimates provide reasonable
confidence intervals. For convenience, a demonstrative example with an
estimation algorithm is provided.