The Weibull Distribution: A Handbook - Index of

17.1 Classical estimation approaches 559
RIDER (1961) gave the following solution of the system (17.3a–c):
�bj = d1 d2 − d3 ± � d2 3 − 6d1 d2 d3 − 4d2 1 d22 + 4d31 d3 + 4d3 3
d2 1 − d2
where for simplification he has set
The estimator of the mixing proportion is
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dj = m ′ j Γ
�
1 + j
�
; j = 1,2,3.
c
, (17.4a)
�p = d1 −�b2 �b1 −� . (17.4b)
b2
We now turn to the general case where all five parameters p,b1,b2,c1,c2 are unknown. In
order to avoid the solution of a system of five dependent non–linear equations
m ′ r = p b r �
1 Γ 1 + r
�
+ (1 − p)b
c1
r �
2 Γ 1 + r
�
; r = 1,... ,5.
c2
FALLS (1970) suggested to estimate p by the graphical approach of KAO (1959):
1. Plot the sample CDF for the mixed data on WEIBULL–probability–paper (see
Fig. 9/4) and visually fit a curve among these points (= WEIBULL plot).
2. Starting at each end of the WEIBULL plot, draw two tangent lines and denote them
by � p F1(x) and � (1 − p)F2(x), which are estimates of p F1(x) and (1 − p)F2(x),
respectively.
3. At the intersection of both tangent lines drop a vertical line on the percent scale which
gives the estimate �p of p.
The solution of m ′ r = µ ′ r (r = 1,... ,4) with p substituted by �p will not be unique. When
confronted with more than one set of acceptable estimates �b1, �b2,�c2,�c2, FALLS (1970) suggested
to adopt PEARSON’s procedure and chose the set which produces the closest agreement
between m ′ 5 and the theoretical moment µ′ 5 when evaluated by the estimates.
17.1.2 Estimation by maximum likelihood
We will first present the solution of the special case of a two–fold mixture (Sect. 17.1.2.1)
and then turn to the general case with more than two subpopulations (Sect. 17.1.2.2).
17.1.2.1 The case of two subpopulations
When we have postmortem data that are type–I censored, T being the censoring time,
there are r1 failures in subpopulation 1 and r2 in subpopulation 2, found in a sample of n