Re: Weibull distribution: reference for maximum likelihood inference
- From: Jack Tomsky <jtomsky@xxxxxxxxxxxxx>
- Date: Fri, 18 Apr 2008 00:07:22 EDT
Hi everybody,
I would be interested in reading a reference on the
maximum likelihood
estimation of the parameters of the Weibull
distribution. I would also
gladly read about unbiased estimation, interval
estimation, and tests
for the Weibull. The estimation of the parameters of
the Weibull
distribution is not very well covered in the
textbooks with which I am
familiar (Degroot and Schervish, Casella and Berger,
Hogg and Craig.)
Sometimes, there will be one or two problems about
the Weibull, which
is nice, but not enough.
I greatly appreciate your help!
Lydia
I have a paper, "Computer iterative estimation of reliability -- extracting information from interval data", Computers and Industrial Engineering, Vol. 11, pp. 581-585, 1986.
It deals with the more general case of interval data where we observe a_i <= t_i <= b_i. Some a_i may be zero and some b_i may be infinity. The paper deals with large-sample confidence intervals for functions of a and p, the scale and shape parameters of the Weibull.
Particular functions of a and p of interest include the following measures of central tendency.
MEAN = a*Gamma((p+1)/p)
MEDIAN = a*(ln 2)^(1/p)
MODE = max(a*((p-1)/p)^(1/p), 0)
and, of course, a and p separately, as well as the cdf
F(t) = 1-exp(-(t/a)^p).
The large-sample procedure to obtain a confidence interval for h(a,p), which mathematically is based on inverting the likelihood ratio test, is to numerically obtain the two-dimensional region
S(a,p): ln L(a,p) >= ln L(ahat, phat) + ln(1- Q)
where Q = 1-exp(-(z((1+P)/2))^2/2)
and P is the confidence level.
Then numerically search along the boundary of S(a,p) and determine the largest and smallest values of h(a,p). These form the end-points of the confidence interval for h(a,p).
If your data does not involve interval data, simply use the density function evaluated at t_i in your likelihood function.
Jack
.
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- Weibull distribution: reference for maximum likelihood inference
- From: lydiajones94
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