Large Quantile Estimation for Distributions in the Domain of - - PowerPoint PPT Presentation

▶

large quantile estimation for distributions in the domain

Large Quantile Estimation for Distributions in the Domain of - - PowerPoint PPT Presentation

May 21, 2023 128 likes •522 views

Large Quantile Estimation for Distributions in the Domain of Attraction of a Max-Semistable Law Lusa Canto e Castro 1 Sandra Dias 2 1 Department of Statistics University of Lisbon, Portugal 2 Department of Mathematics University of

slide-1

SLIDE 1

Large Quantile Estimation for Distributions in the Domain of Attraction of a Max-Semistable Law

Luísa Canto e Castro1 Sandra Dias2

1Department of Statistics

University of Lisbon, Portugal

2Department of Mathematics

University of Trás-os-Montes e Alto Douro, Portugal

4th Conference on Extreme Value Analysis Gothenburg, August 15-19, 2005

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 1 / 30

slide-2

SLIDE 2

Introduction

F is in the domain of attraction of a max-stable d.f. G, F ∈ MS(G), if and only if there exist an > 0 and bn such that lim

n→+∞ F n(anx + bn) = G(x).

(1) F is in the domain of attraction of a max-semistable d.f. G, F ∈ MSS(G), if and only if there exist an > 0 and bn such that lim

n→+∞ F kn(anx + bn) = G(x), ∀ x ∈ CG

(2) where kn verifies lim

n→+∞ kn+1/kn = r ≥ 1 (r < ∞).

(3) Remark: Max-stable laws are a particular case of the max-semistable laws when r = 1.

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 2 / 30

slide-3

SLIDE 3

Introduction (continuation)

Pioneers in max-semistable laws: Grinevich (1992, 1993) Pancheva (1992) (Grinevich 1992) G is a max-semistable d.f. if and only if there exist r > 1, a > 0 and b ∈ R such that G is solution of the functional equation G(x) = Gr(ax + b). (4)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 3 / 30

slide-4

SLIDE 4

First characterization (Grinevich 1992)

Unifying standard expressions for max-semistable d.f.’s For γ = 0 Gγ,ν(x) = exp

−(1 + γx)−1/γν(log(1 + γx))
, x ∈ R, 1 + γx > 0

a = 1 and p = log a = γ log r For γ = 0 G0,ν(x) = exp{−e−xν(x)}, x ∈ R a = 1 and p = b = log r where ν is a positive, bounded and periodic function with period p.

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 4 / 30

slide-5

SLIDE 5

Some graphical features of max-semistable laws

1 2 3 4 5 6 7 8 9 10 11 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure: D.f.’s G0.5(x) = exp(−x−2) and G0.5,ν(x) = exp(−x−2(8 + cos(4π log x)))

5 10 15 20 25 30 1 2 3 4 5 6 7 8 9 10 11

Figure: QQ-Plot of G0.5,ν against G0.5

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 5 / 30

slide-6

SLIDE 6

Second characterization (Canto e Castro et al. 2000)

General to location and scale, for any d.f. G max-semistable we have: − log(− log G(sm + amx)) = m log r + y(x), ∀ x ∈ [0, 1], m ∈ Z (5) where y : [0, 1] → [0, log r] is non decreasing, right continuous and continuous at x = 1 sm = m if a = 1 sm = (am − 1)/(a − 1) if a = 1 and a > 0

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 6 / 30

slide-7

SLIDE 7

Graph of the function − log(− log(G))

1 log r

y Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

slide-8

SLIDE 8

Graph of the function − log(− log(G))

1 1+a log r 2log r

y Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

slide-9

SLIDE 9

Graph of the function − log(− log(G))

1 1+a log r 2log r 3log r 1+a+a2

y Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

slide-10

SLIDE 10

Graph of the function − log(− log(G))

1 1+a −log r log r 2log r 3log r 1+a+a2 −a−1

y Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

slide-11

SLIDE 11

Graph of the function − log(− log(G))

1 1+a −2log r −log r log r 2log r 3log r 4log r 1+a+a2 −a−1 1+a+a2+a3 −a−2−a−1

y Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

slide-12

SLIDE 12

Statistical inference in max-semistable models

Temido (2000) proposed that, in the estimation of the parameters r, p and γ, appropriated functions of the following sequence of statistics should be used Zs(m) := X(m/s2) − X(m/s) X(m/s) − X(m) where X(m) := XN−[m]+1:N are order statistics of a sample of size N from the random variable X m := mN is an intermediate sequence (that is, mN is an integer sequence verifying mN → +∞ and mN/N → 0)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 8 / 30

slide-13

SLIDE 13

Behaviour of the sequence of statistics Zs

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1 2 3 4 5 6 7 8 s = 1.9 ≠ r 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 1 2 3 4 5 6 7 8 s = e 0.5 = r

Figure: Sample trajectories of Zs(m) for s = 1.9 and s = e0.5 ≈ 1.65 F(x) = 1 − x−1(14 + cos(4π log x)), r = e0.5

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 9 / 30

slide-14

SLIDE 14

Behaviour of the sequence of statistics Zs (continuation)

Dias and Canto e Castro (2004) proved that Zs(m)

P

− →

n→+∞ ac if and only if s = rc, c ∈ N.

Furthermore, if s = rc then Zs(m) as an oscillatory behaviour. Using these result we have that if s = rc, c ∈ N, then Rs(m) :=

Zs2(m) (Zs(m))2 P

− → 1, n → +∞

Ps(m) := log (Zs(m))

P

− → log ac = cp, n → +∞, γ = 0

γs(m) := log(Zs(m))

log s P

− →γ, n → +∞

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 10 / 30

slide-15

SLIDE 15

Behaviour of the sequence of statistics Zs (continuation)

Dias and Canto e Castro (2004) proved that Zs(m)

P

− →

n→+∞ ac if and only if s = rc, c ∈ N.

Furthermore, if s = rc then Zs(m) as an oscillatory behaviour. Using these result we have that if s = rc, c ∈ N, then Rs(m) :=

Zs2(m) (Zs(m))2 P

− → 1, n → +∞

Ps(m) := log (Zs(m))

P

− → log ac = cp, n → +∞, γ = 0

γs(m) := log(Zs(m))

log s P

− →γ, n → +∞

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 10 / 30

slide-16

SLIDE 16

Proposed estimators for the parameters r, p and γ

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5 10 15 20 25 30 35 s = 1.5 s = 1.7 s = 1.9 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s = 1.5 s = 1.7 s = 1.9

Figure: Left: Sample trajectories of Rs(m) for s = 1.5, s = 1.7 and s = 1.9 Right: Magnified version F(x) = 1 − x−1(14 + cos(4π log x)), r = e0.5 ≈ 1.65

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 11 / 30

slide-17

SLIDE 17

Proposed estimators for the parameters r, p and γ

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5 10 15 20 25 30 35 s = 1.5 s = 1.7 s = 1.9 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 s = 1.5 s = 1.7 s = 1.9

Figure: Left: Sample trajectories of Rs(m) for s = 1.5, s = 1.7 and s = 1.9 Right: Magnified version F(x) = 1 − x−1(14 + cos(4π log x)), r = e0.5 ≈ 1.65

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 11 / 30

slide-18

SLIDE 18

Proposed estimators for the parameters r, p and γ

(continuation)

An estimate of r

r = mode
arg max

s=1.1,(0.1),3.0

Bs(ǫ), ǫ = 0.01, (0.01), 0.1

where

◮ Bs(ǫ) := 1

k k

P

i=1

1 I{m(i)∈An:|Rs(m(i))−1|<ǫ}(m(i)) (percentage of time that the sequence of statistics Rs spends in a ǫ-neighborhood of 1)

◮ An is a set of suitable values of m

An = {m(1), m(2), ..., m(k)}

◮ k = #An Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 12 / 30

slide-19

SLIDE 19

Proposed estimators for the parameters r, p and γ

(continuation)

An estimate of p

◮ if γ = 0

b p = 1 k X

m∈An

b Pb

r(m) = 1

k X

m∈An

log(Zb

r(m))

◮ if γ = 0

b p = log b r

An estimate of γ

γ =

p/ log r (γ = 0)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 13 / 30

slide-20

SLIDE 20

Simulation study

Selected d.f.’s 1 − F(x) = (1 − F0(x))θ(x) where F0 ∈ MS(G) for some max-stable d.f. G. In particular

◮ Generalized Pareto d.f.

F0(x) = ( 1 − (1 + γx)−1/γ x ∈ R, 1 + γx > 0 and γ = 0 1 − e−x x ∈ R and γ = 0

◮ Burr d.f.

F0(x) = 1 − (1 + x−ρ/γ)−1/ρ, x ≥ 0, ρ < 0 and γ > 0

θ positive and periodic function

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 14 / 30

slide-21

SLIDE 21

Simulation study (continuation)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 70 N = 1000 N = 2000 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 70 N = 5000 N = 10000

Figure: Empirical distribution of the estimates of r F(x) = 1 − x−2(8 + cos(4π log x)), r = e ≈ 2.71

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 15 / 30

slide-22

SLIDE 22

Simulation study (continuation)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 70 80 90 N = 1000 N = 2000 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 10 20 30 40 50 60 70 80 90 N = 5000 N = 10000

Figure: Empirical distribution of the estimates of r F(x) = 1 − x−1(27 + cos(8π log x)), r = e0.25 ≈ 1.28, r2 ≈ 1.65, r3 ≈ 2.12 and r4 ≈ 2.72

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 16 / 30

slide-23

SLIDE 23

Simulation study (continuation)

20 40 60 80 100 120 140 160 180 200 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 N = 1000 20 40 60 80 100 120 140 160 180 200 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 N = 5000

Figure: Estimates of the period p for each replica (sample sizes 1000 and 5000) F(x) = 1 − x−1(27 + cos(8π log x)), p = 0.25

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 17 / 30

slide-24

SLIDE 24

Results from the simulation study

the root global mean square error, RGMSE RGMSE =

n1

n MSE1 (θ(1))2 + n2 n MSE2 (θ(2))2 + ... MSEj = 1 nj

nj

i=1
θ(j)

i

− θ(j)2 the global bias, Gbias Gbias = n1 n bias1 θ(1) + n2 n bias2 θ(2) + ... biasj = 1 nj

nj

i=1
θ(j)

i

− θ(j) where θ(j)

i

are the estimates of θ which are nearer θ(j) than from any other θ(k), k = j and nj is the number of values θ(j)

i , j = 1, 2, ...

Remark: In the estimation of r: θ(j) = rj. In the estimation of p : θ(j) = jp.

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 18 / 30

slide-25

SLIDE 25

Estimates of r

◮ GRMSE ⋆ less than 0.13 ⋆ decreases when n increases ⋆ slightly less in the Generalized Pareto models ⋆ increases when r increases ◮ Gbias ⋆ less than 0.07 in terms of absolute value ⋆ does not have a monotonous behaviour with n

Estimates of p

◮ GRMSE ⋆ less than 0.18 ⋆ decreases when n increases ⋆ slightly less in the Generalized Pareto models ⋆ increases when r increases ◮ Gbias ⋆ less than 0.10 in terms of absolute value ⋆ does not have a monotonous behaviour with n

Estimates of γ

◮ RMSE/γ ⋆ less than 0.14 ⋆ decreases when n increases ◮ bias/γ ⋆ less than 0.09 in terms of absolute value ⋆ decreases when n increases ⋆ usually preserves the signal Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 19 / 30

slide-26

SLIDE 26

Estimation of the function y

log r log r log r

X(k/r) X(k/r

2)

X(k/r

3)

X(k)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 y(1) y(2) y(3)

Figure: Left: Empirical function −log(−log( b Fn)) Right: Empirical versions of y F(x) = 1 − x−1(14 + cos(4π log x))

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 20 / 30

slide-27

SLIDE 27

Estimation of the function y

log r log r log r

X(k/r) X(k/r

2)

X(k/r

3)

X(k)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 y(1) y(2) y(3) y estimated

Figure: Left: Empirical function −log(−log( b Fn)) Right: Empirical versions of y F(x) = 1 − x−1(14 + cos(4π log x))

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 20 / 30

slide-28

SLIDE 28

Estimation of the function y (continuation)

y(x) =
i

ni nt y(i)(x), x ∈ [0, 1] where y(i)(x) =

j

y(i)

j

1 I[x(i)

j ,x(i) j+1[(x) is the ith empirical version of y

x(i)

j

are the jump points of the ith empirical version of y y(i)

j

are the correspondent points of x(i)

j

ni is the number of points x(i)

j

in the ith empirical version of y nt is the total number of points x(i)

j

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 21 / 30

slide-29

SLIDE 29

Large quantile estimation

−log(−log p)

X(k/r) X(k) xp

log r log r log r log r z 1 z log r 2log r mlog r mlog r+z (m+1)log r

Figure: Empirical function −log(−log( b G))

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 22 / 30

slide-30

SLIDE 30

Large quantile estimation

−log(−log p)

X(k/r) X(k) xp

log r log r log r log r z 1 z log r 2log r mlog r mlog r+z (m+1)log r

tp amtp+sm

Figure: Empirical function −log(−log( b G))

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 22 / 30

slide-31

SLIDE 31

Large quantile estimation (continuation)

xp = (

am tp + sm)( X(k/r) − X(k)) + X(k) where m ∈ N such that z = − log(− log p) − (− log(− log G( X(k/r)))) − m log r ∈ [0, 1]

tp =

y−1(z)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 23 / 30

slide-32

SLIDE 32

Application to real data

Data: Major earthquake inter-arrival times registered in the period between January 1st, 1973 and March 31st, 2005. (A earthquake is a Major earthquake if its magnitude is greater or equal to 6.5.) Sample Size: 1450 values

12 20 30 40 50 60 70 80 90 2 3 4 5 6 7 Inter−Arrival Times Quantiles Exponential Quantiles

Figure: QQ-Plot of the inter-arrival times against the exponential (290 highest values)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 24 / 30

slide-33

SLIDE 33

Modeling with MSS laws

200 400 600 800 1000 1200 1400 1 2 3 4 5 6 s = 1.6 s = 1.8 s = 1.9

Figure: Sample trajectories of Rs(m) for s = 1.6, s = 1.8 and s = 1.9

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 25 / 30

slide-34

SLIDE 34

Estimation of the parameters

200 400 600 800 1000 1200 1400 0.5 1 1.5 2 2.5 3 3.5 4

Figure: Sample trajectory of Zb

r(m), b

r = 1.8

Estimates for the parameters Parameters Estimates r 1.8 a 1.000443 γ 0.000754

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 26 / 30

slide-35

SLIDE 35

Estimated function y

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6

Figure: Estimated function y

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 27 / 30

slide-36

SLIDE 36

Empirical functions − log(− log G) and G

−1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −0.5 0.5 1 1.5 2 2.5 3 −1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure: Left: Empirical function − log(− log b G)) Right: Empirical function b G

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 28 / 30

slide-37

SLIDE 37

12 20 30 40 50 60 70 80 90 2 4 6 8 10 12 14 16 18 Inter−Arrival Times Quantiles MSS Quantiles

Figure: QQ-Plot of the inter-arrival times against the estimated max-semistable (290 highest values)

Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 29 / 30

slide-38

SLIDE 38

References

Canto e Castro, L., de Haan, L., and Temido, M. G. (2000). Rarely observed sample maxima. Theory of Probab. Appl., Vol. 45, p. 779-782. Dias, S., e Canto e Castro, L. (2004). Contribuições para a estimação em modelos max-semiestáveis. In Estatística com acaso e necessidade, p. 177-187. Edições SPE, Lisboa. Dias, S., e Canto e Castro, L. (2005). Modelação de máximos em domínios de atracção max-semiestáveis. In Estatística Jubilar, p. 223-234. Edições SPE, Lisboa. Grinevich, I.V. (1992). Max-semistable laws corresponding to linear and power normalizations. Theory of Probab. Appl., Vol. 37, p. 720-721. Grinevich, I.V. (1993). Domains of attraction of the max-semistable laws under linear and power normalizations. Theory of Probab. Appl., Vol. 38, p. 640-650. Pancheva, Z. (1992). Multivariate max-semistable distributions. Theory of Probab. Appl., Vol. 37, p. 731-732. Temido, M.G. (2000) Classes de Leis Limite em Teoria de Valores Extremos - Estabilidade e Semiestabilidade. Ph.D. Thesis, University of Coimbra. Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 30 / 30