sample sizes (Fig. 5).
80% in all cases.
Land (1972).
The minimum achieved coverage is greater than
These results are consistent with the findings of
Lognormal Estimator for Gamma Distributed Data
A major finding is that the bias of the lognormal estimator becomes
large for gamma distributed data as a becomes small or c becomes large.
In particular, a relative bias (100[Ave(E(x)) - EX]/EX) of about 25% is
present for a = 1 (Fig. 6).
The case a = 1 corresponds to the exponential distribution.
The relative bias of the lognormal estimator
becomes much worse for a < 1.
This result would be expected because the
shapes of the two distributions differ greatly for a < 1.
However, even
for c = 0.75 (o = 1.78), the relative bias of the lognormal estimator
for gamma distributed data is about 6%.
Also, the achieved coverage of
the confidence interval begins to decrease for c = 1.0 and n = 100
(Fig. 6).
Coverage becomes very poor for c > l.
Gamma Estimator for Lognormally Distributed Data
In contrast to the lognormal estimator for gamma distributed data, the
gamma estimator for lognormal data does quite well.
This estimator is
unbiased, and so the simulations showed. Also, the achieved coverage of
this estimator is good for small sample sizes (n = 5) (Fig. 7), whereas
the coverage of the lognormal estimator is usually significantly less
than the predicted 95% for n = 5.
However, the average confidence
interval width is usually greater for the arithmetic mean.
The achieved
coverage of the arithmetic mean drops as c increases, but never below
75%, even for n = 5 and c = 2.
Also for c = 0.75, the average confidence
interval length becomes about the same for the two estimators.
Robustness
The same general conclusions discussed in the preceding four sections
also hold when the lognormal and gamma estimators are applied to a
compound lognormal or gamma distribution with the expected value selected
from a uniform distribution,
The arithmetic mean still provides an
unbiased estimate of EX in all cases, while the lognormal estimator
provides an unbiased estimate when the variate is uniform-lognormal
distributed, but not for the case when the variate is uniform-gamma
distributed.
Confidence interval coverage for both estimators is always greater than
70% when there is negligible bias.
Generally, the arithmetic mean had
better coverage for the smaller sample sizes, while the lognormal esti-
mator had better coverage for n = 100.
Also, the lognormal estimator
tended to have better coverage when c > 1.0.
Of course, the average
confidence interval width was also greater for the lognormal estimator
when the coverage was larger than the gamma estimator.
For the case
EX = 1.0 and uniform-lognormal data, Fig. 8 provides the reader with
some feel for the relationship between sample size c and coverage for
the two estimators considered.
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