The main points from the Weibull distribution experiments given in Table 7.2 are as follows:

(i) When the three-parameter Weibull distribution is correct, the three discrimina­

tion criteria and the estimated three-parameter Weibull distribution are identical in

terms of BIAS and RRMSE for both the maximum and 98% quantities for all values of

the shape parameter (namely, 0.5, 1, 2 and 3). Underfitting the correct Weibull distri­

bution with its two-parameter variant yields considerably larger BIAS and RRMSE for

all values of the shape parameter, although the degree of error is reduced as the value

of the shape parameter is increased. Thus, estimating the correct three-parameter

distribution is not preferable in terms of BIAS and RRMSE as compared with using

the three discrimination criteria, although underfitting the three-parameter Weibull

distribution by setting the location parameter

to zero yields much larger BIAS and

RRMSE values in all cases considered. This result is broadly similar in qualitative

terms to those obtained for the gamma distribution although, in the latter case, esti­

mating the correct distribution is preferred in terms of BIAS for a large value of the

shape parameter.

(ii) As compared with the case of the gamma distribution, overfitting the correct

two-parameter Weibull distribution yields some surprising results in terms of BIAS,

especially for larger values of the shape parameter. When the shape parameter is 0.5,

the estimated two-parameter Weibull distribution has much lower BIAS values than

those obtained using the discrimination criteria and the estimated three-parameter

Weibull distribution for both the maximum and 98% quantities. As the shape param­

eter is increased from 0.5 to 1, the correctly estimated two-parameter distribution is

still preferred for the maximum quantity but is inferior to SIC, which favours the lower

dimensioned model, for the 98% quantity. When the shape parameter is increased to

2 or 3, all three discrimination criteria are preferred to the estimated two-parameter

Weibull distribution in terms of BIAS. Indeed, when the shape parameter is set to

3, even the overfitted three-parameter distribution has lower BIAS than the correctly

fitted two-parameter variant for both the maximum and 98% quantities. In terms of

RRMSE, the correctly estimated two-parameter distribution is preferred in all cases,

although its superiority is diminished as the shape parameter is increased. While there

is not a noticeable difference between the overfitted th ree-p aram eter distribution and th e th ree discrim ination criteria, SIC is always second best to th e estim ated two- pa­ ram eter d istrib u tio n , and th e estim ated th ree-p aram eter distribution is generally the worst. Q ualitatively, th e results are reasonably sim ilar to those for th e gam m a distri­ bution in th a t overfitting does not generally increase th e BIAS and RRM SE values for the m axim um and 98% quantities for th e Weibull distribution. However, it is interest­ ing to note th a t estim atin g th e correct tw o-param eter W eibull distribution does not always yield th e sm allest BIAS value relative to th e th ree discrim ination criteria, or even to th e overfitted th ree-p aram eter distribution.

Finally, Table 7.3 contains th e results from experim ents for the lognorm al d istri­ bution. Since th e lognorm al d istribution has th e opposite behaviour to the gam m a and W eibull d istrib u tio n s as th e shape p aram eter is increased, it is useful for com par­ ative purposes to exam ine th e results as th e shape p aram eter is decreased rath er than increased. T he principal points to note from th e tab le are as follows:

(i) W hen th e th ree-p aram eter lognorm al distribution is correct (7 = 1) and the shape p a ra m ete r is 1.2 or 0.9, th e correctly fitted th ree-p aram eter d istribution and th e th ree discrim ination criteria have identical BIAS and RRM SE values for both the m axim um and 98% q u an tities, whereas the underfitted tw o-param eter lognorm al dis­ trib u tio n has su b stan tially higher values for BIAS and RRM SE. A lthough th e under­ fitted tw o-param eter d istrib u tio n still has the largest BIAS and RRM SE values when th e shape p a ra m ete r is reduced to 0.5 or 0.4, the oth er four m ethods do not rem ain identical. W hen th e shape p aram eter is 0.5, the LR m ethod has by far th e smallest BIAS for th e m axim um q u a n tity while SIC has th e largest; for th e 98% quantity, AIC and th e correctly fitted th ree-p aram eter distribution have th e sm allest BIAS, and SIC again has th e largest. These rankings are not m aintained when th e shape param eter is reduced to 0.4. T h e sm allest BIAS values for th e m axim um and 98% quantities are AIC and th e estim ated th ree-p aram eter lognorm al distribution, respectively, w ith SIC th e worst of th e four m ethods in each case. On th e basis of RRM SE, however, the th ree-p aram eter lognorm al distribution has the sm allest value, followed closely by

AIC, with SIC the worst of the four methods, for both the maximum and 98% quan­

tities when the shape parameter is 0.5 or 0.4. The poor performance of SIC, which

favours the more parsimonious two-parameter lognormal variant of the correct model,

is consistent with the findings for the gamma and Weibull distributions, as is the re­

sult that underfitting the model will generally lead to much larger values of BIAS and

RRMSE for both the maximum and 98% quantities. Moreover, estimating the correct

distribution is the preferred strategy, at least in terms of RRMSE, and sometimes also

for BIAS, relative to using the three discrimination criteria to determine which of the

three- and two- parameter distributions should be used.

(ii) Similar observed patterns to the above do not hold when the correct model is the

two-parameter lognormal distribution (7 = 0). For example, when the shape parameter

is 1.2, the correctly fitted two-parameter lognormal distribution has the smallest BIAS,

followed by the parsimony- inclined SIC, and lastly by the overfitted three-parameter

lognormal distribution, for the maximum quantity; for the 98% quantity, however,

SIC has BIAS equal to that of the estimated two-parameter distribution, followed by

the LR method and lastly by the overfitted three-parameter lognormal distribution.

Even when the shape parameter is reduced to 0.9, these rankings are not sustained.

While the estimated three-parameter lognormal distribution has the largest BIAS for

both the maximum and 98% quantities, in the former case SIC is best, followed by

the estimated two-parameter lognormal distribution, and in the latter case the LR

method is best, followed by SIC. Interesting results arise when the shape parameter is

reduced to 0.5 or 0.4. In the former case, the LR method has lowest BIAS, followed

by SIC and lastly by the overfitted three-parameter distribution for the maximum

quantity; for the 98% quantity, AIC is best, followed by the overfitted three-parameter

distribution and lastly by the correctly fitted two-parameter distribution. The results

for BIAS are especially striking when the shape parameter is reduced to 0.4. The LR

method has lowest BIAS, followed by SIC, for the maximum quantity, while AIC is best,

followed by the LR method, for the 98% quantity. However, the correctly estimated

two-parameter distribution has the largest BIAS in each case, even larger than those

of the overfitted three-parameter variant. Thus, overfitting the lognormal distribution