Tail Behavior and Limit Distribution of Maximum of Logarithmic General Error Distribution

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1 Tail Behaior and Limit Distribution of Maximum of Logarithmic General Error Distribution Xin Liao, Zuoxiang Peng & Saralees Nadarajah First ersion: 8 December 0 Research Report No. 3, 0, Probability and Statistics Group School of Mathematics, The Uniersity of Manchester

2 Tail behaior and limit distribution of maximum of logarithmic general error distribution a Xin Liao a Zuoxiang Peng b Saralees Nadarajah a School of Mathematics and Statistics, Southwest Uniersity, Chongqing, China b School of Mathematics, Uniersity of Manchester, Manchester, United Kingdom Abstract Logarithmic general error distribution, an extension of the log-normal distribution, is proposed. Some interesting properties of the logarithmic general error distribution are deried. These properties are applied to establish the asymptotic behaior of the ratio of probability densities and the ratio of the tails of the logarithmic general error and log-normal distributions, and to derie the asymptotic distribution of the partial maximum of an independent and identically distributed sequence obeying the logarithmic general error distribution. Keywords Extreme alue distribution; Logarithmic general error distribution; Mills ratio; Partial maximum; Subonentiality. Mathematics subject classification Primary 6E0, 60E05; Secondary 60F5, 60G5. Running title Limit distribution of the maximum of log GED Introduction The normal and log-normal distributions hae played an important role in both the theory and applications of probability and statistics. Some fundamental properties of normal and log-normal distributions can be found in Johnson et al The log-normal distribution with heaier tail than that of the normal distribution has been widely used in risk management, ealuation of credit risk, aggregate claim distributions in insurance and operational risk, see Mikosch and Nagae 998, Asmussen et al. 0, Kortschak and Hashora 0 and references therein. The general error distribution denoted by GED is an extension of the normal distribution. The GED has been used as a substitute for the normal distribution in many areas. Some recent examples of this are: identification of nonwoen uniformity Liu et al., 00; modeling of human longeity Robertson and Allison, 0; analysis of bit-plane probability for audio coding Shu et al., 0; The probability density function pdf of the standardized GED is Corresponding author. pzx@swu.edu.cn g x = / x/λ +/ λγ/

3 for > 0 and x R, where λ = [ ] / / Γ/ Γ3/ and Γ denotes the gamma function. Let G x = x g sds denote the distribution function df of the standardized GED. The following inequality and Mills ratio of the standard GED were deried by Peng et al. 009: λ x + for > and all x > 0, and for > 0 λ x < G x < λ g x x. G x g x λ x. as x. Notice that. and. for = reduce to the well-known inequality and Mills ratio cf. Mills 96 of the standard normal distribution, i.e., for x > 0 and x + x φx < Φx < x φx Φx φx x as x, where Φx and φx denote the df and pdf of the standard normal distribution. For Mills ratio of the multiariate normal distribution, see Hashora and Hüsler 003. Peng et al. 009, 00a considered the tail behaior, extreme alue distribution and its associated conergence rate for standard GED random ariables. In this note, we propose the logarithmic general error distribution written as log GED, a natural extension of the log-normal distribution. The motiation for proposing the log GED is the same as the need to use the GED oer the normal distribution. That is, the log GED can be ected to be a better model for areas to which the log-normal distribution can be applied. We now gie an illustratie example showing that the log GED can be a better model than the log-normal distribution. The latter distribution is a popular model for extreme rainfall see, for example, Tie et al So, we fitted the log-normal distribution and the log GED to two real rainfall data sets: annual maximum rainfall from Bartow, Florida and annual maximum rainfall from Orlando, Florida for the years 907 to 000. For practical reasons, we fitted locationscale ariants of the two distributions with µ denoting the location parameter and σ denoting the scale parameter. Both distributions were fitted by the method of maximum likelihood. For the first data set, we obtained the estimates µ = , σ = , log L = for the log-normal distribution, and µ = , σ = , = , log L = for the log GED, where log L denotes the maximized log-likelihood alue and the numbers in brackets are standard errors obtained by inerting the obsered information matrices. For the second data set, we obtained the estimates µ = , σ = , log L = for the log-normal distribution, and µ = , σ = , = , log L = for the log GED. It follows by the standard likelihood ratio test that the log GED proides a significantly better fit than the log-normal distribution. This is supported by the fitted density plots and probability plots shown in Figure. We can see that plotted points for the log GED are closer to the diagonal line in the probability plots. We can also see that fitted pdfs for the log GED better capture the general pattern of the empirical histograms.

4 Fitted PDFs Log normal Log GED Fitted PDFs Log normal Log GED Annual Maximum Rainfall Annual Maximum Rainfall Obsered Probability Log normal Log GED Obsered Probability Log normal Log GED Expected Probability Expected Probability Figure Fitted pdfs of log-normal and log GED for Bartow top left; Fitted pdfs of log-normal and log GED for Orlando top right; Probability plots for the fits of log-normal and log GED for Bartow bottom left; Probability plots for the fits of log-normal and log GED for Orlando bottom right. Extreme rainfall modeling is only one area where the log GED can be useful. Other potential areas could include: enumeration and absence/presence data, neuronal ariability during hand- 3

5 writing, digg online social network, failure time analysis, cellular uptake of radioactiity, modeling length of stay in cardiac critical care, world container port throughput, and the model of cutoff grade impurities. In this note, some fundamental properties like the Mills type inequality, Mills-type ratio, subonentiality and the tail distribution representation of the log GED are studied. Two applications are proided. The first application considers the asymptotic behaiors of the ratio of the pdfs and the ratio of the tails of the log GED and the log-normal distribution. The second application deries the limiting distribution of the maximum of independent and identical log GED random ariables. To the best of our knowledge, there has been no study about the tail behaior or the extreme alue distribution of the log GED. Firstly we gie the definition of log GED. Definition.. Let ξ denote a random ariable following the standard general error distribution with parameter > 0. Let η = ξ. Then we say that η follows the logarithmic general error distribution with parameter, written as η log GED. It is easy to check that f x, the pdf of η log GED, is x log x λ f x = +/ λγ/ for x > 0, where λ = λ = [ / Γ//Γ3/ ] /. Let F x denote the cdf of η, i.e. F x = x 0 f t dt for x > 0. Note that the log GED reduces to the logarithmic Laplace distribution for =. It reduces to the log-normal distribution for =. The contents of this note are organized as follows: Section proides some interesting properties of log GEDs. In Section 3, we consider the asymptotic tail behaiors of the log GED and the log-normal distribution. In Section 4, the limiting distribution of the partial maximum of an independent and identically distributed sequence from the log GED is established. We find suitable norming constants such that the asymptotic distribution of the normalized partial maximum is the Gumbel extreme alue distribution. The result is also extended to the case of a finite mixture of log GEDs. Preliminary results In this section, we derie some interesting properties of the log GED. The first assertion is about the Mills type inequality and Mills ratio of the log GED. Let ξ denote a standard GED random ariable with pdf g and df G. Then η = ξ follows the log GED with pdf f and df F. By.,. and noting that F x f x = x G log x, g log x we can derie the following Mills type inequality and Mills ratio of the log GED. 4

6 Proposition.. Let F denote the df of log GED with parameter > 0. i. for >, we hae λ log x x + λ log x < F x < λ f x log x x for all x >. ii. for fixed > 0, we hae as x. F x f x λ log x x. Remark.. For =, the log GED is just the standard log-normal distribution with λ =. Hence, Proposition. reduces to the inequality about the log-normal distribution which is log x x + log x F x < < log x x f x for x >. The Mills-type ratio of the log-normal distribution is F x f x log x x for large x. Remark.. For >, Proposition. shows that F DΛ, i.e. there exist normalized constants a n > 0 and b n R such that F n a nx + b n Λx = x as n. Since d/dxf x f x = x + log x λ, we hae by Proposition. that F x d/dxf x f x f x as x. So, it follows by on Mises conditions for domains of attraction cf. Proposition. in Resnick 987 that F DΛ. The choice of the constants a n and b n will be discussed in Theorem 4.. By Proposition.ii, it is easy to check that lim F xe sx = x for all s > 0, which means F is heay-tailed, where F x = F x. Another interesting result is the following. 5

7 Proposition.. Let F log GED for > 0. Then F is a subonential df written as F S, i.e., F F x lim =. x F x Proof. Firstly we check that F L. By Proposition.ii, we hae for all y > 0, i.e. F L. For 0 <, by Proposition., we hae F x y lim = x F x F x/ lim, x F x implying F is a dominatedly arying distribution for 0 < written as F D. Hence, F L D implies F S for 0 <, cf. Embrechts et al To proe F S for >, we only need to check the conditions of Theorem in Teugels 975. Define a df F # x such that F # x = F # x = λ λ where fx and gx are gien by fx = / Γ/ x e gs/fs ds,. xλ log x, gx = + λ log x.3 for large x. Define ψx = log F # x. Obiously ψx is asymptotically concae since ψ x = log x λ x for large x. By. we can proe + log x λ λ log x log x + < 0 F # x sx lim = x F # x for any 0 < sx = x δ with 0 < δ <. Certainly, sx and x sx as x. By using. once again, we hae log x ψx ψsx 4λδ Γ / δ log x 0 λ as x. So, by Theorem in Teugels 975, we hae F # S. Combining with Theorem 3 in Teugels 975 and Theorem 3. to follow, we hae F S. The desired result follows. 6

8 3 Tail behaior of the log GED In this section, we establish the asymptotic behaior of the ratio of pdfs and the ratio of the tails of the log GED and the log-normal distribution. Similar topics hae been discussed by Finner et al. 008 for the asymptotic behaior of the ratio of the pdfs of the Student s t and normal distributions and by Peng et al. 009 for the general error and normal distributions. For the asymptotic behaior of interest, we first consider the case of. Secondly, we consider the case of x for fixed. Then we derie an licit form for F x. The asymptotic behaior for the first case has obious practical appeal. For instance, high quantiles or high critical alues of the log GED can be approximated by the same for the log-normal distribution. Note that the log GED reduces to the log-normal distribution for =. Theorem 3.. Let F denote the df of log GED with parameter.. let x = x be such that = γ log x log log x + log λ log log x for some γ R. Then and f x γ lim f x = F x γ lim F x =.. for fixed, we hae f x = f λlog x +/ λγ/ λlog x 3. πx and lim x log x F x F λlog x = / Γ/ π. 3. Proof.. Note that λ and λ+/ Γ/ π f x lim f x = lim = lim = lim as, so log x λ log x log x λ log x 7 log log x λ γ log x

9 γ =. The conditions of the theorem imply log x as. By Proposition. and Remark., we hae F x lim F x = lim log x f x γ λ f x =.. Note that 3. follows by elementary calculus. By 3., Proposition. and Remark., one can check that 3. holds. The claimed result follows. By Remark. we can obtain the tail distribution representation of the log GED. By Corollary.7 of Resnick 987, the limiting distribution of the maximum of independent and identical log GED random ariables is Λx. Corollary.7 of Resnick 987 can also help to find suitable normalized constants α n and β n. Theorem 3.. Let F denote the df of log GED with parameter >. Then for large x we hae x gt F x = cx ft dt, where cx λ λ / Γ/ and fx and gx are those gien by.3. Proof. By Proposition., for sufficiently large x, we hae F x = + θ x f x λ log x x = / Γ/ + θ x = λ λ / Γ/ = λ λ / Γ/ = cx x e + θ x { + θ x gt ft dt, where θ x 0 as x. The proof is complete. e log log x log x x e as x, λ log x λ + t log t tλ λ t dt} + λ log t dt 4 Asymptotic distribution of the maximum In this section, we consider the limiting distribution of the partial maximum of an independent and identically distributed sequence from the log GED. Remarks. and Theorem 3. showed that F DΛ for >. Hence, the main work here is to find the norming constants. We hae the following result. 8

10 Theorem 4.. Let {X n,n } be independent and identical log GED random ariables with common df F x, >. Let M n = maxx k, k n denote the partial maximum. Then lim P Mn β n x = x, n α n where α n = λ / λlog n / / λ / λlog n / + log log n / log log n + log Γ/ / λ log n / log log n + log Γ/ and β n = / λlog n / / λ / λlog n / log n / log log n + log Γ/. Proof. Remarks. and Theorem 3. showed that F DΛ for >. So, there exist a n > 0, b n R such that lim P Mn b n x = x. n a n By Proposition. of Resnick 987 and Theorem 3., the norming constants can be chosen as b n = F n, a n = fb n. Since F x is continuous, F b n = /n. By using., we hae λ nb n f b n log b n as n, i.e. / λ Γ /n log b n log n λ as n. Taking logarithms, we hae log log Γ/ + log λ + log n log n λ + log log b n 0 4. as n, which implies as n, so log b n λ log n log log b n log log λ log log n 0 as n, so log log b n = log + log λ + log log n + o. 9

11 Putting this into 4., we hae log b n = λ log log n + log n log Γ/ + o, which implies that log b n = / λlog n / + log log n log n log Γ/ + o, log n log n so b n with = / λlog n / / λ log n / = β n + o / λlog n / log n / β n = Define α n = fβ n = / λlog n / / λ λ / λlog n / log n / / λlog n / / λ / λlog n / + log By elementary calculus, one can check that both and log log n + log Γ/ + o log n log log n + log Γ/. log n / log log n + log Γ/ / λ log n / log log n + log Γ/. a n fb n lim = lim n α n n fβ n = b n β n lim n α n b n β n = lim n fβ n hold, so by Theorem..3 in Leadbetter et al. 983, the desired result follows. Remark 4.. Theorem 4. shows that the asymptotic distribution of the partial maximum from the standard log-normal distribution is the Gumbel extreme alue distribution with the norming constants log n log n log log n + log 4π α n =, log n + log log n log log n + log 4π and = 0 log n β n = log n log n log log n + log 4π. 0

12 To end this section, we extend Theorem 4. to the case of a finite mixture of log GED distributions. A random ariable X is said to hae a finite mixture distribution if its df F satisfies Fx = p F x + p F x + + p r F r x, where F i, i r are distinct dfs of the r mixture components or populations. The mixture weights satisfy p i > 0,i =,,...,r and r k= p k =. Finite mixed distributions hae been applied in many areas like medicine Venturini et al., 008 and meteorology Wan et al., 005. They hae also been studied in the extreme alue literature. For instance, Peng et al. 00b hae studied the extreme alue distribution and its associated conergence rate for a finite mixture of onential distributions. Now, we consider a finite mixture with component dfs F i following log GED i, where the parameter i > for i r and i j for i j. We write the df of the finite mixture as for x > 0. Fx = p F x + p F x + + p r F r x 4. Theorem 4.. Let Z n denote a sequence of independent random ariables with common df F defined by 4.. Let M n = max{z,z,...,z n } denote the partial maximum. Then lim P Mn β n x = x n α n with the normalized constants α n and β n gien by λ / λlog n / / λ log n α n = / log log n + log Γ/ / λlog n / + log / λ log n / log log n + log Γ/, and β n = α n log p + / λlog n / / λ log n / log log n + log Γ/, [ where = min{,,..., r }, λ = ] Γ/ Γ3/, and p = ps, where s is the one such that s =. Proof. Obiously, by 4., By using??, r i= p i λ i i log x i xf i x i Fx = + λ i r p i F i x. i= i i log x i < Fx < i r i= p i λ i i log x i xf i x i for all x >, so by the definition of f we hae log x pλ λ log x Γ + λ log x + H x

13 where < Fx < pλ log x H x = i log x λ Γ + H x, 4.3 p i λ i i Γ + λ log x pλ Γ i + λ i i i i log x i i log x i log x log x i λ λ i and H x = i p i λ i i Γ i log x i pλ Γ i log x log x i λ λ i as x since = min{,,..., r }. Combining and. together, for sufficiently large x, we hae Fx p F x 4.6 as x, where F denotes the df of log GED, and and p are those defined before. We can conclude F DΛ by Proposition.9 in Resnick 987. The norming constants can be deried by Theorem 4. and 4.7. The proof is complete. By 4. and Proposition., for sufficiently large x, we hae Fx = r p i F i x i= log x λ pλ log x / Γ/ p F x 4.7 as x, where F is the df of log GED, and and p are those defined before. We can conclude that F DΛ by Proposition.9 of Resnick 987. The norming constants can be deried by Theorem 4. and 4.7. The proof is complete. Acknowledgements. The authors would like to thank the Editor-in-Chief, the Associated Editor and the two referees for their careful reading and for their comments which greatly improed the paper. The authors also thank Professor Enkelejd Hashora for his kind help with improing the paper greatly. The first two authors research has been supported by the National Science Foundation of China no. 775, the Program for ET in Chongqing Higher Education Institutions No and the SWU grant for Statistics Ph.D.

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