Far East Journal of Theoretical Statistics
Volume 4, Issue 2, Pages 391 - 422
(December 2000)
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MAXIMUM
LIKELIHOOD
AND THE WEIBULL DISTRIBUTION
K. O. Bowman (U. S. A.) and L. R. Shenton (U. S. A.)
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Abstract: The Weibull distribution has three parameters, location a, scale b
and shape c. Maximum likelihood estimators are and
solutions may not always exist; for example the location estimate
must be less than the smallest member of the sample. We consider three
estimation problems: (1) Estimation of one parameter when the other two are
assumed to be known. (2) Estimating the scale and shape parameters when the
location parameter is known. (3) Estimating the three parameters simultaneously.
Results being based on the covariance matrix and its cofactors, we give
explicit expressions for the asymptotic bias, 2nd order variances, skewness to
order and
asymptotic kurtosis to order 1/N, N being the sample size. Except
for the simultaneous estimation of a, b, c, the expressions for these
asymptotic moments and moment ratios are simple in form involving gamma and
Riemann Zeta functions. They provide a new basic supplement to our knowledge of
maximum likelihood estimator moments.
A surprising discovery is the part played by the location parameter whenever
it has to be estimated. For the three parameter estimation case it is already
known that asymptotic covariance only exist if c > 2. It turns
out that the asymptotic skewness only exist if c > 3 and the asymptotic
kurtosis only exist if c > 4. This applies to the asymptotic
distribution of and
The source of this characteristic is the singularity appearing in the
expectation of logarithmic derivatives. When less than 3 parameters are to be
estimated the problem arises whenever intrudes.
For the 3 parameter case, a new expression is developed for the asymptotic
variance of Lastly, wherever possible
simulation studies are invoked for verification purposes. |
Keywords and phrases: covariance matrix, gamma functions,
Hessian matrix, Newton’s algorithm, poly-gamma
function, singularities, Taylor series. |
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