Computers & Geosciences 27 (2001) 261–262

Comment

Reply to comment on ‘‘A FORTRAN program for fittingWeibull distribution and generating samples’’ by H. Tsai$

Amitava Ghosh

Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute, 6220 Culebra Road, San Antonio, TX 78238, USA

Before I respond to Dr. Heng Tsai’s comment, Iwould like to correct two errors in his letter to avoid any

confusion. First, the data set of annual drought of theIroquois river was published by Ang and Tang (1984),not by Hahn and Shapiro (1967). Second, figure 2 of

Ghosh (1999), not figure 1, shows the fit of a Weibulldistribution to this data set.Dr. Tsai indicated that the Weibull distribution is not

appropriate for the annual drought data of the Iroquoisriver, as figure 2 of Ghosh (1999), shows an unsatisfac-tory fit, especially at extreme values. He concluded,

based on Q–Q plots, that the Weibull distribution tail istoo short for the data set. As stated in Ghosh (1999), theobjective of the paper was to describe the programWEIBUL, which can either estimate the Weibull

parameters to describe a data set or generate randomnumbers following a given Weibull distribution. Twopublished data sets, annual drought of the Iroquois river

and number of cycles to failure of carbide inserts, wereused to verify the program. Ang and Tang (1984) fittedthe Weibull distribution to the Iroquois river data using

two methods: (i) graphically and (ii) using method ofmoments. Program WEIBUL fits the Weibull distribu-tion to the data using the (i) log-transform method and(ii) maximum-likelihood method. Table 2 of Ghosh

(1999) compares the Weibull parameters calculated byWEIBUL with those by Ang and Tang (1984). Theobjective of this fitting exercise to the data set by Ang

and Tang (1984) is to demonstrate that the programWEIBUL is working and produces acceptable resultswhen compared with other methods. Results given in

table 2 of Ghosh (1999) demonstrate that. However, theprogram WEIBUL cannot infer whether the Weibulldistribution is the best distribution that can be fitted to a

given data set. It can only estimate parameters if theWeibull distribution is fitted to the given data. Aprogram or an analysis that fits several distributions to

the data set and compares the fitting through differentgoodness-of-fit tests could give such an inference.

However this was not the objective of Ghosh (1999). Ifthe given data does not follow the Weibull distribution,the fitting will be poor. It is up to the user to check

whether the Weibull distribution is appropriate for agiven data set or not.Dr. Tsai has fitted four distributions to the annual

drought data. The goodness-of-fit test (table 1 of Tsai)

accepts both Weibull and lognormal distributions at

95% confidence level. Dr. Tsai pointed out that in

addition to quantitative statistics, a user should also

look at the probability distribution plots especially the

tail region for extreme value distributions. A user may

use this criterion to decide whether a particular

distribution out of a few possible distributions, which

have passed the quantitative statistical tests, would be

the appropriate distribution for a given data set.

However, this aspect was neither the focus nor the

scope of Ghosh (1999). Dr. Tsai rightly pointed out that

the error due to incorrect statistical assumptions

decreases as the sample size increases. It may be possible

that with more data on the annual drought of the

Iroquois river, the Weibull distribution may be more

appropriate for the data or at least at par with the

lognormal distribution even in the tail region. However,

this is beyond the scope of Ghosh (1999).

In summary, Dr. Tsai recommended the user to lookbeyond the quantitative statistics while fitting different

probability distributions to a given data set to select the

‘best-fit’ distribution. However, with increased sample

size the error due to an incorrect assumption of the

probability distribution decreases. Although the recom-

mendation to the user has some merit, it is no way

related with the program WEIBUL and its verification

analyses presented in Ghosh (1999). It seems that it

would have been more appropriate that the letter of

Dr. Tsai had been directed to Ang and Tang (1984)

instead of Ghosh (1999).

$PII of article referred to: PII S0098-3004(00)00080-7.

E-mail address: aghosh@swri.org (A. Ghosh).

0098-3004/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 3 0 0 4 ( 0 0 ) 0 0 0 8 1 - 9

References

Ang, A.H.-S., Tang, W.H., 1984. Probability concepts in

engineering planning and design. In: Decision, Risk, and

Reliability, Vol. II. John Wiley and Sons, New York.

Ghosh, A., 1999. A FORTRAN program for fitting Weibull

distribution and generating samples. Computers & Geo-

sciences 25, 729–738.

Hahn, G.J., Shapiro, S.S., 1967. Statistical Models in Engineer-

ing. Wiley, New York.

A. Ghosh / Computers & Geosciences 27 (2001) 261–262262