Tail Properties and Asymptotic Expansions for the Maximum of Logarithmic Skew-Normal Distribution

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1 Tail Properties and Asymptotic Epansions for the Maimum of Logarithmic Skew-Normal Distribution Xin Liao, Zuoiang Peng & Saralees Nadarajah First version: 8 December Research Report No. 4,, Probability and Statistics Group School of Mathematics, The University of Manchester

2 Tail properties and asymptotic epansions for the maimum of logarithmic skew-normal distribution a Xin Liao a Zuoiang Peng b Saralees Nadarajah a School of Mathematics and Statistics, Southwest University, 475 Chongqing, China b School of Mathematics, University of Manchester, Manchester, United Kingdom Abstract We discuss tail behaviors, subeponentiality and etreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic epansion of the distribution of the normalized maimum of logarithmic skew-normal random variables is derived. It shows that the convergence rate of the distribution of the normalized maimum to the Gumbel etreme value distribution is proportional to /log n /. Key words Etreme value distribution; Logarithmic skew normal distribution; Maimum; Pointwise convergence rate; Subeponentiality. AMS subject classification Primary 6E, 6G7; Secondary 6F5, 6F5. Introduction The major weakness of the normal distribution is its inability to model skewed data. Several skewed etensions of the normal distribution have been proposed in the literature. The most popular and the most widely used of these is the skew-normal distribution due to Azzalini 985. The probability density function pdf of this distribution is given by g λ = φφλ, R,. where λ R, φ is the standard normal pdf, and Φ is the standard normal cumulative distribution function cdf. Let G λ = g λtdt denote the cdf corresponding to.. If a random variable, say X, has the pdf. then we write X SNλ. Clearly, SN is a standard normal variable. Liao et al. studied the tail behavior of the skew-normal distribution, establishing its etreme value distribution and associated convergence rates. The following epansion for the distribution of the normalized maimum of SNλ random variables was derived by Liao et al. : [ ] b n b n G n λ an + b n Λ κλ ω + κ Λ as n, where Λ = ep{ ep } denotes the Gumbel cdf and κ = / + e,

3 ω = 4 /8 + 3 / + + e with for λ ; and G λ = n, a n = b n κ = + λ / + e, ω = λ + λ λ 4 /8 + λ 3 + 3λ + + 3λ e with G λ = n, a n = + λ b n for λ <. The skew-normal distribution applies to data on the real line. Its version for positive data can be obtained by setting X = epξ, where ξ SNλ. Then, we say that X follows the logarithmic skew-normal distribution, written X LSNλ. The pdf of LSNλ is given by f λ = φlog Φλ log, >.. Let F λ denote the cdf corresponding to.. Clearly, LSN is standard log-normal random variable. The logarithmic skew-normal distribution is relative more recent compared to the skew-normal distribution. But it has already received wide spread applications. Some selected applications and application areas have been: modeling of income data Azzilini et al., 3; analysis of auto insurance claim costs Bolance et al., 8; analysis of continuous data in a two-part stochastic model Chai and Bailey, 8; wireless communications Wu et al., 9, Li et al., ; model for particle size Huang and Ku, ; cohort studies of paediatric respiratory symptoms Mahmud et al., ; modeling of precipitation data Marchenko and Genton,. Some probabilistic properties of LSNλ have been studied by Lin and Stoyanov 9. The aim of this short note is to consider some further probabilistic properties of the logarithmic skew-normal distribution. The contents are organized as follows. Section presents some preliminary results, including the tail behavior, the subeponentiality and the etreme value distribution of LSNλ. Distributional epansions for the normalized maimum of LSNλ random variables are derived in Section 3. To the best of our knowledge, all of the properties presented are new. Preliminary results In this section, we derive Mills inequalities, Mills ratios, and an eact decomposition of the tail of LSNλ. We also prove that LSNλ is strongly subeponential, denoted by F λ S. For LSNλ and SNλ, note that F λ = G λ log and F λ f λ = G λlog. g λ log So, by Proposition in Liao et al. and by Mills inequality and Mills ratio of the standard normal distribution, we have the following two results.

4 Proposition. Let F λ and f λ denote the cdf and the pdf of LSNλ. For all >, we have i. if λ >, + log F λ < < log f λ log φ λ log ; λ log ii. if λ =, + log F < < log f log ; iii. if λ <, + log λ log < F λ f λ + λ < λ log + λ + + λ log + λ log. Proposition. Let F λ and f λ denote the cdf and the pdf of LSNλ. For λ, we have F λ f λ log. as. For λ <, we have as. F λ f λ + λ log. The following result shows that LSNλ is strongly subeponential. Corollary. F λ S, so F λ S, the class of subeponential distributions. Proof. By Proposition, the hazard rate function m Fλ = f λ F λ is ultimately decreasing to zero as. If ep m Fλ F λ is integrable over R +, where F λ = F λ, Theorem 3.3 in Foss et al. shows that F λ S. Combining with Theorem 3.7 in Foss et al., we have F λ S. So, we just need to check that ep m Fλ F λ is integrable over R +. Consider the case of λ. By., we know for arbitrary ε > that there eist a sufficiently large A > such that ε log < F λ f λ < + ε log. Hence, for > A, we have ep m Fλ F λ < + ε f λ log ep < + ε φlog ep log A 3 ε log ε log

5 = + ε log A ep ε φ log. ε So, one can check that lim k ep m Fλ F λ = for any k >, implying ep m Fλ F λ is integrable over R +. The same can be shown for the case of λ < by using.. The arguments are similar and are omitted here. The desired result follows. In order to derive epansions for the distribution of the normalized maimum of LSNλ random variables, we need the following tail decomposition of LSNλ. Proposition 3. Let F λ denote the cdf of LSNλ. Then, for large, if λ we have If λ <, we have F λ = F λ = f λlog log + 3log 4 + O log 6 log = πe Φλ log log + 3log 4 = Proof. Follows by integration by parts. +O log 6 ep e log s s ep +λ log λπ + λ log + 3λ λ + λ log + 5λ4 + λ log s ds..3 λ 4 + λ log 4 + O log 6 ep +λ λπ + λ + 3λ λ + λ log + 5λ4 + λ + 3 λ 4 + λ log 4 +O log 6 + λ log s ep + s + λ log s e ds. Using Proposition 3, we can now derive the distributional tail representation of LSNλ. Proposition 4. For large, F λ = c ep gt e ft dt, where c, g and f depend on λ as follows: In the case of λ, c as, πe f = log > with f = log log as 4

6 and In the case of λ <, g = + as ; log ep +λ c λπ + λ as, and f = + λ log > with f log = + λ log as g = + + λ as. log In fact, Proposition 4 can also be obtained from Mills ratio of LSNλ. By Corollary.7 in Resnick 987, we have F λ DΛ and the norming constants a n and b n are given by such that n = F λ b n, a n = f b n.4 lim F n λ a n + b n = Λ. Remark. The tail representation of LSNλ can be rewritten as: F λ = c ep f t dt with f = ft/gt eventually nondecreasing, where c, ft and gt are those given by Proposition 4. By Corollary.5 in Goldie and Resnick 988, we can easily check that F λ S DΛ since lim f h/f = h for any constant h >. e 3 Epansion for the distribution of maimum In this section, we derive an eact epansion for the distribution of the maimum of LSNλ random variables. This epansion is used to show that the convergence rate of F n λ a n + b n to Λ is of the order of O log n /. Theorem. For norming constants a n and b n given by.4, we have Fλ n a n + b n Λ κλ ω + κ as n, where κ and ω depend on λ as follows: in the case of λ, in the case of λ <, κ = e, ω = e ; κ = + λ e, Λ ω = 4 + λ λ 48 + λ e. 5

7 To prove Theorem, we need the following auiliary result. Lemma. Let H λ b n ; = F λ a n + b n and h λ b n ; = n log H λ b n ; + e, where the norming constants a n and b n are given by.4. Then lim h λ b n ; κ = ω, where κ and ω are those given by Theorem. Proof. First, consider the case of λ. It is easy to check the following two facts by. and F λ DΛ: lim n F λ + b n = e 3. and Setting A λ b n = [ Φ λ lim F λ + b n =. 3. +O 6 ][ Φ +3 log + b n λ log + b n log 4 + O 6 ], + b n we have lim A λ b n = and So, by.3, we have F λ b n e F λ + b n + log log s = A λ b n ep b n s = A λ b n ep = A λ b n o lim A λ b n = t + log + t + t t + log + t + t t + log + t + t log s ds + dt + t + log + t + dt + t + log + t + dt. + t + log + t 6

8 3.4 Combining., 3., 3., 3.3 and 3.4, we obtain lim h λ b n ; = lim n = lim n = lim = lim = e lim = e lim = e lim = e lim n log H λ b n ; + e log F λ + b n + F λ b n e F λ + b n F λ + b n + o + Fλ b n e n F λ + b n F λ + b n + o + + F λ b n F e λ log + + A λ b n + A λ b n + A λ b n = e := κ, t + log + t t+log + t log + t + +t + +t +log t+log + t log +t + +t +log + t + log + t log F λ b n F e λ log + + t log + t dt dt + o dt + o where the final step follows by the dominated convergence theorem. Similarly, one can show that lim h λ b n ; κ = ω. The same results hold for λ > by. and Proposition 4. The arguments are similar and are omitted here. The proof is complete. Proof of Theorem. Note that lim h λ b n ; = by Lemma. Using Lemma again, we have F λ a n + b n Λ κλ = ep h λ b n ; κ Λ 7

9 = = h λ b n ; + h λ b n; + h3 λ b n; 3! h λ b n ; κ + h λ b n; ω + κ Λ as n. The desired result follows. + o κ + h λ b n ; + o 3! Λ Remark. By the definition of b n, it is easy to check that / = O /log n /. So, Theorem shows that the pointwise convergence rate of F n λ a n + b n to its limit is proportional to /log n /. Further, the pointwise convergence rate of F n λ a n + b n Λ to its limit is also proportional to /log n / by Theorem. Acknowledgements The first two authors research has been supported by the National Science Foundation of China no. 775 and the SWU grant for Statistics PH.D Program. All of the authors would like to thank the Editor and the referee for careful reading and for their comments which greatly improved the paper. References [] Azzalini, A A class of distributions which includes the normal ones. Scandinavian Journal of Statistics,, [] Azzalini, A., dal Cappello, T. and Kotz, S. 3. Log-skew-normal and log-skew-t distributions as models for family income data. Journal of Income Distribution,, -. [3] Bolance, C., Guillen, M., Pelican, E. and Vernic, R. 8. Skewed bivariate models and nonparametric estimation for the CTE risk measure. Insurance Mathematics and Economics, 43, [4] Chai, H. S. and Bailey, K. R. 8. Use of log-skew-normal distribution in analysis of continuous data with a discrete component at zero. Statistics in Medicine, 7, [5] Foss, S., Korshunov, D. and Zachary, S.. An Introduction to Heavy-tailed and Subeponential Distributions. Springer Verlag, New York. [6] Goldie, C. M. and Resnick, S Distributions that are both subeponential and in the domain of attraction of an etreme-value distribution. Journal of Applied Probability,, [7] Huang, C. Y. and Ku, M. S.. Asymmetry effect of particle size distribution on content uniformity and over-potency risk in low-dose solid drugs. Journal of Pharmaceutical Sciences, 99, [8] Li, X., Wu, Z. J., Chakravarthy, V. D. and Wu, Z. Q.. A low-compleity approimation to lognormal sum distributions via transformed log skew normal distribution. IEEE Transactions on Vehicular Technology, 6,

10 [9] Liao, X., Peng, Z., Nadarajah, S. and Wang, X.. Rates of convergence of etremes from skew normal samples. Available on-line at [] Lin, D. G. and Stoyanov, J. 9. The logarithmic skew-normal distributions are momentindeterminate. Journal of Applied Probability, 46, [] Mahmud, S., Lou, W. Y. W. and Johnston, N. W.. A probit- log- skew-normal miture model for repeated measures data with ecess zeros, with application to a cohort study of paediatric respiratory symptoms. BMC Medical Research Methodology,, Article Number 55. [] Marchenko, Y. N. and Genton, M. G.. Multivariate log-skew-elliptical distributions to precipitation data. Environmetrics,, [3] Resnick, S. I Etreme Values, Regular Variation and Point Processes. Springer Verlag, New York. [4] Wu, Z. J., Li, X., Husnay, R., Chakravarthy, V., Wang, B. and Wu, Z. Q. 9. A novel highly accurate log skew normal approimation method to lognormal sum distributions. In: Proceedings of the 9 IEEE Wireless Communications and Networking Conference, pp

Tail properties and asymptotic distribution for maximum of LGMD

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