On the Maximum and Minimum of Multivariate Pareto Random Variables

Transcription

1 ISSN X (rint) ISSN X (online) INFORMATION TECHNOLOGY AND CONTROL 213 Vol.42 No.3 On the Maximum Minimum of Multivariate Pareto Rom Variables Jungsoo Woo Deartment of Statistics Yeungnam University Gyongsan South Korea Saralees Nadarajah School of Mathematics University of Manchester Manchester M13 9PL UK htt://dx.doi.org/1.5755/j1.itc Abstract. Aksomaitis Burauskaite-Harju [Information Technology Control ] studied the distribution of max(x 1 X 2... X ) when (X 1 X 2... X ) follows the multivariate normal distribution. Here we study the moments of min(x 1 X 2... X ) max(x 1 X 2... X ) when (X 1 X 2... X ) follows the most commonly known multivariate Pareto distribution. Multivariate Pareto distributions are most relevant for modeling extreme values. Keywords: maximum; minimum; multivariate Pareto distribution. 1. Introduction Let (X 1 X 2... X ) be a continuous rom vector with means μ 1 μ 2... μ variances σ 1 2 σ σ 2 correlation coefficient ρ. The extreme values of ( X 1 X 2... X ) are M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). It is often of interest to know how E(M) E(m) vary with resect to the means variances the correlation coefficient. For examle Smith Sardeshmukh (2) find that "The change of variance is associated both with altered skewness a change in high low extremes". Griffiths et al. (25) use the change in mean temerature as a redictor of extreme temerature change in the Asia-Pacific region. Karl et al. (28) observe that "A relatively small shift in the mean roduces a larger change in the number of extremes for both temerature reciitation". Nicholls (28) argues the need for careful statistical analysis "since the likelihood of individual extremes such as a late sring frost could change due to changes in variability as well as changes in the mean climate". Burton Allso (29) observe that "In recent years there has been an increasing tendency to use mean seeds to redict extremes". In the extreme value literature only four aers have studied the distributions of M m with resect to the means variances the correlation coefficient: Ker (21) Lien (25) Aksomaitis Burauskaite- Harju (29) Hakamiour et al. (211). Ker (21) suoses that (X 1 X 2 ) has a bivariate normal distribution. Lien (25) suoses that (X 1 X 2 ) has a bivariate lognormal distribution. Aksomaitis Burauskaite-Harju (29) suose that (X 1 X 2... X ) has a multivariate normal distribution. Hakamiour et al. (211) suose that ( X 1 X 2... X ) follows a multivariate Pareto distribution due to Muliere Scarsini (1987). As we can see the first two of these aers are limited to the case = 2. The distributions considered by the first three aers (bivariate normal bivariate lognormal multivariate normal) are not the most aroriate ones for modeling extreme values. Muliere Scarsini s (1987) multivariate Pareto distribution the distribution considered by the fourth aer suffers from discontinuity has limited alicability. Bivariate multivariate Pareto distributions have been the most oular distributions for modeling extremes. Among these the most commonly known distribution due to Arnold (1983 Chater 6) has the joint survivor function secified by x i F (x 1 x 2... x ) = 1 + (1.1) 242

2 On the Maximum Minimum of Multivariate Pareto Rom Variables for x i > i = > i = 12 >. This distribution has received widesread attention: Yeh (24) studies extreme order statistics of (1.1); Li (26) investigates tail deendence roerties of (1.1); Cai Tan (27) roose (1.1) as a model for deendent risks; Vernic (211) uses (1.1) to estimate tail conditional exectation a oular measure of risk. For the multivariate Pareto distribution given by (1.1) stard calculations show that μ i = > 1 (1.2) 1 σ i 2 = 2 ( 1) 2 ( 1) > 2 (1.3) ρ = 1 (1.4) The aim of this short note is to study how E(M) E(m) vary with resect to the means variances the correlation coefficient. The main results are Theorem 2.1 Theorem 2.2 Theorem 2.3. The level of mathematics required in Section 2 is not high but not less elementary than the mathematics used in Ker (21) Lien (25) Aksomaitis Burauskaite-Harju (29) Hakamiour et al. (211). We feel that the given results are imortant because of the rominence of multivariate Pareto distributions because of the rominence of (1.1) in modeling extreme values. Besides the results known on this toic have so far been limited for the bivariate case or distributions which are not most significant in modeling extreme values. 2. Main results The main results in this section need the following lemma. Lemma 2.1. Define I(k a ) = x k 1 (1 + aa). Then I(k a ) = a k B( k k) for a > < k < where B(a b) denotes the beta function defined by 1 B(a b) = t a 1 (1 t) b 1. Proof. Setting y = 1/(1 + ax) we can write I(k a ) = x k 1 (1 + aa) = 1 = a k y k 1 (1 y) k 1 = a k B( k k) where the final equality follows from the definition of the beta function. Theorem 2.1 derives exlicit exressions for the cumulative distribution functions of M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Theorem 2.2 derives exlicit exressions for the nth moment of M = mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Theorem 2.3 shows how E(M n ) vary with resect to the means variances the correlation coefficient in (1.2)-(1.4). Theorem 2.1. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Then Pr (M x) = x x 1 i<j 1 + x + x + x θ j θ k 1 i<j<k + + ( 1) 1 + x Pr (m x) = x for x >. Proof. We have Pr (M x) = Pr maxx 1 X x = Pr (X i x) = 1 Pr (X i > x) = 1 Pr (X i > x) + Pr X i > x X j > x 1 i<j Pr X i > x X j > x X k > x 1 i<j<k + + ( 1) Pr (X i > x) = x x 1 i<j 243

3 J. Woo S. Nadarajah 1 + x + x + x θ j θ k 1 i<j<k + + ( 1) 1 + x Pr (m x) = 1 Pr (m > x) = 1 Pr (min(x 1 X n ) > x) = 1 Pr (X i > x) = x The roof is comlete.. Theorem 2.2. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Let k A i 1 ik = 1 l=1 for 1 i 1 < < i k. Then l E(M n ) = mb( n n) A i A ij 1 i<j + A ijk 1 i<j<k = mb( n n)a 1 for n <. Proof. Using Theorem 2.1 we can write E(M n ) = n x n 1 PP(M > x) = n x n x n x n x + x θ j 1 i<j +n x n x + x θ k 1 i<j<k n( 1) ( 1) A 1 x n x = n I(n A i ) n I(n A i ) + n In A ijk n( 1) In A 1 1 i<j 1 i<j<k = n A i B( n n) A ij B( n n) +n A ijk B( n n) 1 i<j<k 1 i<j n( 1) A 1 1 B( n n) where the last ste follows from Lemma 2.1. Also using Theorem 2.1 we can write = n x n 1 Pr(m > x) = n x n x = min A 1 = na 1 B( n n) where the last ste follows from Lemma 2.1. The roof is comlete. Theorem 2.3. Let ( X 1 X 2... X ) be distributed mmm(x 1 X 2... X ) m = mmm(x 1 X 2... X ). Then (a) E(M n ) is an increasing function of ρ for ρ < 1/n; (b) is an increasing function of ρ for ρ < 1/n; (c) is an increasing function of µ i ; 244

4 On the Maximum Minimum of Multivariate Pareto Rom Variables (d) is an increasing function of σ i ; (e) V ar(m) is an increasing function of ρ for ρ < 1/2; (f ) V ar(m) is an increasing function of µ i ; (g) V ar(m) is an increasing function of σ i. Proof. Since E(M n ) = mb( n n) A i A ij Γ( n)γ(n) = n Γ() = 1 i<j A i A ij n! ( n) ( 1) A i 1 i<j A ij 1 i<j + A ijk 1 i<j<k + A ijk 1 i<j<k + A ijk 1 i<j<k ( 1) A 1 ( 1) A 1 ( 1) A 1 we see that E(M n ) is a decreasing function of for > n hence an increasing function of ρ for ρ < 1/n. Since = mb( n n)a 1 P Γ( n)γ(n) = NA 1 P Γ() = A 1 P n! ( n) ( 1) we see that is a decreasing function of for > n hence an increasing function of ρ for ρ < 1/n. Since = mb( n n)a 1 P = NN( n n) 1 = mb( n n)( 1) n 1 μ i where the last equality follows by (1.2) we see that is an increasing function of μ i. Since = mb( n n) 1 = mb( n n) ( 1)n ( 2) n/2 n/2 1 σ i where the last equality follows by (1.3) we see that is an increasing function of σ i. Since V aa(m) = E(m 2 ) E 2 (m) = A 1 [2B( 22) B 2 ( 11)] = A 2Γ( 2) 1 Γ() = A 1 ( 1) 2 ( 2) Γ2 ( 2) Γ 2 () we see that V aa(m) is a decreasing function of for > 2 hence an increasing function of ρ for ρ < 1/2. Since V aa(m) = A 1 = 1 μ i ( 1) 2 ( 2) 2 where the last equality follows by (1.2) we see that V aa(m) is an increasing function of μ i. Since V aa(m) = A 1 ( 1) 2 ( 2) = 1 σ i where the last equality follows by (1.3) we see that V aa(m) is an increasing function of σ i. Acknowledgments The authors would like to thank the Editor the referee for careful reading for their comments which greatly imroved the aer. References [1] A. Aksomaitis A. Burauskaite-Harju. The moments of the maximum of normally distributed deendent values. Information Technology Control 29 Vol [2] B. C. Arnold. Pareto Distributions. International Cooerative Publishing House: Maryl [3] M. D. Burton A. C. Allso. Predicting design wind seeds from Anemometer records: Some interesting findings. In: Proceedings of the 11th Americas Conference on Wind Engineering 29. [4] J. Cai K. S. Tan. Otimal retention for a sto-loss reinsurance under the VaR CTE risk measures. ASTIN Bulletin 27 Vol [5] G. M. Griffiths et al. Change in mean temerature as a redictor of extreme temerature change in the Asia- Pacific region. International Journal of Climatology 25 Vol

5 J. Woo S. Nadarajah [6] N. Hakamiour A. Mohammadour S. Nadarajah. Extremes of a bivariate Pareto distribution. Information Technology Control 211 Vol [7] T. R. Karl et al. Weather climate extremes in a changing climate: Regions of focus: North America Hawaii Caribbean U.S. Pacific Isls. Reort by the U.S. Climate Change Science Program 28. [8] A. P. Ker. On the maximum of bivariate normal rom variables. Extremes 21 Vol [9] H. Li. Tail deendence of multivariate Pareto distributions. Technical Reort 26-6 Deartment of Mathematics Washington State University Pullman WA USA 26. [1] D. Lien. On the minimum maximum of bivariate lognormal rom variables. Extremes 25 Vol [11] P. Muliere M. Scarsini. Characterization of a Marshall-Olkin tye class of distributions. Annals of the Institute of Statistical Mathematics 1987 Vol [12] N. Nicholls. Australian climate weather extremes: Past resent future. A Reort for the Deartment of Climate Change Australia 28. [13] C. A. Smith P. D. Sardeshmukh. The effect of ENSO on the intraseasonal variance of surface temeratures in winter. International Journal of Climatology 2 Vol [14] R. Vernic. Tail conditional exectation for the multivariate Pareto distribution of the second kind: Another aroach. Methodology Comuting in Alied Probability 211 Vol [15] H.-C. Yeh. Some roerties characterizations for generalized multivariate Pareto distributions. Journal of Multivariate Analysis 24 Vol Received August