Some statistical aspects of the estimation of fire losses.
This study arose from a problem in which the author \.,ras required to
estimate the overall mean fire loss using a sample of censored data.
The OIUY information available was the total number of fires and the
average large loss. A method, which can be described as a HomentQuantile
method, was developed to solve the original problem. The H~
method was convenient to use as the estimates could be read from tables,
and the method appeared to give good results.
This study \o,Jas undertaken to investigate why the W~ method had worked
so well. The chaxacteristics of the l"lq estimator are examined hereunder
a Hide range of conditions - different degrees of skewness, different
aL'10unts of censorship and assuming either a Lognormal or Gamma parent
distribution. The HQ estimator is compared with the Haximum Likelihood
(hL) estim:':-ltor. The two estimators are examined al.gebraically and these
results are supplemented by a Nonte Carlo simulation.
It is found that the IvlQ and HL estimators of th'2 Lognormal mean perform
very differently in different circumstances. In the area of the original
study - highly skewed data,high censorship and small sa.mple sizes - the
}·tS':':: of the H~ estimate is much less than that of the V;L estimate. For
large sarnples and very highly skewed data the HL estimate is more efficient.
In the case of the Gamma mean the HQ and I'lL methods produce very similar
estimates. The reasons for the different bebAviour of the estimators are
It is concluded that for estimating overall fire losses the simple ML
estimator is unsatisfactory if the Lognorma]. is assumed. The NQ
Lognormal estimator and HL or MQ Gamma estimator appear to be acceptable.
A kno\dedge of the distribution of small fire losses (about which the
author cannot obtain any information) is necessary to choose the best