ABSTRACT
We must now solve equation (5.5.6) and the last equation of (5.5.5) for estimates 8 and-y. Accordingly, we choose a first approximation -y 1 < x1:N· We substitute this value into (5.5.6), solve for 81, and calculate 61 from the second equation of (5.5.5). We then test these approximations by substitu}ion into the
~ast equation of (5.5.5). If this result is zero, then "f 1 = 1. 8 1 = 8, and 61 = 6, and no further calculations are necessary. Otherwise, we select a second approximation and proceed through a second cycle of calculations. We continue until we find two approximations 'Y; and 'Yi such that the absolute value h, - 'Y;\ is sufficiently small and such that
Final estimates are obtai~ed by linear interpolation between"{; and 'Yj• and between 8; and 8j. We calculate e for the second equation of (5.5.5) as
and
(5.5.7)
(5.5.8)
(5.5.9)
(5.5.10)
Maximum likelihood estimators of parameters of the W eibull distribution are subject to regularity restrictions that limit their usefulness. They are valid only if 8 > 1, and for the three-parameter distribution the asymptot_ic variancecovariance matrix is not valid unless o > 2 (a3 < 0.63). In an effort to circumvent these disadvantages of the MLE, Cohen Whitten, and Ding (1984) proposed modified moment estimators which employ the first-order statistic as an estimator of the threshold parameter in the three-parameter Weibull when samples are complete. When samples are censored, it seems more appropriate to employ a similar modification of the maximum likelihood estimators, and that is the topic for consideration in this section. The estimating equations are a In L/a8 = 0, a In uae = 0, and E(XI:N) = xi:N• and in their expanded form, these equations become
To solve these equations simultaneously for the MMLE, we proceed as for the MLE and with the first approximation, 'Y~> substituted into equation (5 .5.6), we solve for o1• We then "test" by substituting these approximations into the third equation of (5.6.1). If this equation is satisfied, then no further calculations are required. Otherwise, we continue with additional approximations to 'Y until, as in the case of the MLE, we find two approximations "/; and "/j in a sufficiently narrow interval such that"/; :§ 'Y :§ "/j· Final estimates are then obtained by linear interpolation, as in the case of the MLE.
