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View Article Fire Research Note No. 837 SOME POSSIBLE APPLICATIONS OF THE THEORY OF EXTREME VALUES FOR THE ANALYSIS OF FIRE LOSS DATA . .' by G. RAMACHANDRAN September 1970 '. FIRE "RESEARCH STATION © BRE Trust (UK) Permission is granted for personal noncommercial research use. Citation of the work is allowed and encouraged. " '.. .~ t . i\~ . f .." ~ , I "I- I Fire Research Station, Borehamwood, He-rts. Tel. 81-953-6177 F.R.. Note No.837. t".~~. ~ . ~~:_\": -~: SOME POSSIBLE APPLIC AT 10m OF THE THEORY OF EXTREME VALUES ,'c'\ .r: FOR THE ANALYSIS OF FIRE LOSS DATA by G. Ramachandran SUMMARY This paper discusses the possibili~ of applying the statistical theory of extreme values to data on monetary losses due to large fires in buildings • ." The theory is surveyed in order to impart the necessary background picture. With the logarithm of loss as the variate, an initial distribution of the exponential type is assumed. Hence use of the first asymptotic distribution of largest values is illustrated. Extreme order statistics other than the largest are also discussed. Uses of these statistics are briefly outlined. Suggestions for further research are also made. I KEY WORDS: Large fires, loss, fire statistics. i ' ~ Crown copyright .,. ~.~ This repor-t has not been published and should be considerec as confidential odvonce information. No rClferClnce should be made to it in any publication without the written consent of the DirClctor of Fire Research. MINISTRY OF TECHNOLOGY AND FIRE OFFICES' COMMITTEE JOINT FIRE RESEARCH ORGANIZATION F.R. Note ·No.837. SOME POSSIBLE APPLICATIONS OF TliE THEORY OF EXTREME VALUES FOR THE ANALYSIS OF FIRE LOSS DATA by G. Ramachandran INTRODUCTION Large fire losses are of economic importance as their total cost is over 60 per cent of the total cost in all fires. However, they represent only one ." per cent (overall) of the total number of fires. But these values .. are at .the upper tail of the fire loss distribution and hence, in the statistical· sense, have to be regarded as extreme (largest) values. The olqest problems connected with extreme values arise from floods. An inundation happens when water flows where it ought not to flow. Floods. destroy life and property. Hence engineers are confronted with the need to forecast floods in order to tackle flood control problems. Floods are defined as the largest daily discharges of a stream during a year. The smallest daily discharges are droughts, a study of which is of essential value in treating problems arising from stream pollution, sewage disposal and water supply. Although the theory is comparatively recent, it has a~ted the attention of scientists working in different fields. Comparable meteorological phenomena are the extreme pressures, temperatures, rainfalls etc. Other examples occur in fracturing of metals, textiles and other materials (breaking strength) under applied force, which are problems arising from materials testing and. qu~lity, .control. Gusts are the largest strengths of wind and gust..loads play an important role in aeronautics. An example in the actuarial field is 'oldest ages', i.e. the longest duration of life. The extreme span of human life is linked to the study of extremes. Recently Epstein1 considered bacterial extinction time as an extreme value phenomenon. Once it is recognised that these phenomena are of a statistical nature, it is possible to obtain solutions .to some of the problems connected with them by applying the asymptotic theory of extreme values. The most comprehensive treatise26n the subject is the book by Gumbe12 This book also contains a discussion of applications and a bibliography of papers on the subject. The latest work on the subject appears ·to be the thesis of Harris3. Fire in a particular bUilding is a rare event. However, in a group of identical risks fires are frequent. If a fire spreads beyond the material or 'room of origin it might cause widespread destruction. In the monetary sense a loss of £10,000 or more may be of economic importance. Such a phenomenon is similar to floods in rivers. The largest loss in a year may be regarded as equivalent to an annual 'flood'. The statistical problems being similar to those of flood cor-trol, the feasibility of applying the theory of extreme values to large fire losses is eXamined in this paper. THE VARIABLE ~. The variable (x) considered is the direct monetary loss to a building and its contents due to fire. The values available are preliminary estimates made by the British Insurance Association. It is not possible to include consequential losses (due to loss of production etc) as estimates for such losses are not available. The operational variable is the logarithm of x and is designated by z. If a minimum loss of one pound is assumed, the variable z is non negative. (One could always consider the variable x1 = x + 1 which does not alter the picture materially especially when dealing with large values of x). Theoretically the maximum loss that could occur in a building is the total value at risk in that building. However, there is a finite non-vanishing probability that fire could spread beyond the building of origin. Also, in a group of buildings, the vable of the costliest building could be high. In these circumstances the variable z could assume values extending to infinity. Estimates of fire losses rounded to the nearest ten pounds (or even to the nearest thousand pounds in the case of large fires) would suffice for practical purposes. Theoretically, however, the damage could assume all values in terms of infinitesimal units. Hence the variate x or z is conceptually of a continuous nature. THE INITIAL DISTRIBUTION In order to apply extreme value theory it is essential to identify the nature of the initial or parent distribution. This is the distribution of the probabilities with which the cumulative monetary damage reaches various amounts in a single fire. Probability distributions which have been suggested are the pareto and the logarithmic normal. In a recent paper4 the author discussed the possible relevance of distributions with an increasing failure rate and of - 2 - mixtures of such distributions. For the variable x he suggested a distribution of the form the density fUnction being The density function of z is of the form ·" if z = k log x. The function •... .. (4) (where - P(-z.) = f(z.) ) is known as failure rate 0 r hazard func tion. This is the conditional probability of extinction in the interval (1, ':L.+d.sJ .given that the fire has survived till the value z has been reached. In..theiearly . stages of growth, fire has a greater tendency to spread so that r( z ). is decreasing. However, for large values of z , f.(z) is likely to be increasing due to the commencement of fire fighting and the exhaustion of natural forces contributing to fire spread. The structure of expression (4) could be generalised ", so that ft- (a ) is a p:>lynomial in z or an exponential function. Application of L'Hopital's rule gives the critical quotient ~ (z) as - 3 - with ~ [(2.) z: 0 Hence according to Gumbel's terminology f~is a distribution of the exponential type and belongs to the 'first' class. The probability density converges towards zero for Z-7 otJ faster t.han· an exponential distribution. Therefore all moments exist for this distribution (expression (3». For the application of extreme value theory it would suffice to note that the probability distribution of z is of exponential type; the .va.luea of the parameters 0< f and ~' are not required. The well known distributions logistic , gamma, chisquare, normal and log normal all belong to distri~ution this type, but pareto does not. The , , (3) is for all risks taken . together. However it is reasonable to assume that the distribution for any industry is of exponential type though with different values of the parameters. ORDER STATISTICS Consider the variable z , The cumulative distribution function F (:zl) denotes the probability that the loss is less than or equal to z. It is given by F(;l-.) = 1- cp(Zjwhere it is assumed that -Z-{o<.!-T- (!JI -z-) . •..•.• (7) 9 (?-J :=. e the corresponding density function f ('Z-J being expresaion (3). The function .. 9'(2}~~the survivo! function giving the probability that the loss is greater than z , In reliability theory 1J (z) is lmown as the r~liability function. Now consider independent random samples z1' z2' z3' •..•••. zn drawn from the population with distribution function f:(z..). Rearranging them in increasing order of their magnitudes we may write .... .• (8) These are known as order statistics wher~ z (1) is the smallest and z (n) the largest. A collection of useful papers' on order statistics and their applications is contained in the book edited by Sarhan and Greenberg5. The mt h (m = 1, 2, ••...• n) among n observations taken in increasing order, viz. Z(m) has the probability density functio~ ~h(Zm) which depends upon the initial density f (z), the sample size n and the order m. The value of f')'y (zm:) is given by rvv-I ."')t,-'M- ~ ~,..) (~_:: (~) .. •. (9) = (......I)! [ F(';z.) J pCZr f - 4 - The generating function of the mt h order statistics can be written as an integral which will facilitate the calculation of mean, yariance and; higher moments of these statistics. However, for most of the iilltial '.' " distributions the integration 'is difficult. The book by Sarhan' and Greenberg' contains tables of expected values and standard deviations of order statistics from populations which .are normal, exponential, gamma and rectangular•. The tables give the. values for a small. number n of sample sizes. ·Calculation of moments of order statistics for the distribution fn(Zm) is equally difficult.
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