Towards a multivariate Extreme Value Theory Samuel Hugueny Institute of Biomedical Engineering Life Sciences Interface Doctoral Training Centre University of Oxford March 20, 2009 Contents 1 Classical EVT results 2 1.1 Fisher-Tippett Theorem . 2 1.2 Maximum Domains of Attraction . 3 1.2.1 Tail-equivalence . 4 1.2.2 Maximum domain of attraction of the Fr´echet distribution . 4 1.2.3 Maximum domain of attraction of the Weibull distribution . 4 1.2.4 Maximum domain of attraction of the Gumbel distribution . 4 2 Univariate Gaussian distribution 7 3 Univariate one-sided Gaussian distribution 13 4 Probability of probabilities 19 4.1 Sampling in the data space is equivalent to sampling in the image probability space . 19 4.2 Univariate standard Gaussian distribution . 19 4.3 Multivariate standard Gaussian distribution . 20 4.4 Gaussian Distributions with Generic Mean and Covariance Matrix . 24 5 Extreme Value Distribution for the standard bivariate Gaussian distribution 26 5.1 EVD of minima for G ........................................ 26 5.2 EVD for minima of F ........................................ 27 6 Extreme Value Distribution for a generic bivariate Gaussian distribution 31 7 Extreme Value Distribution for the standard n-dimensional Gaussian distribution 37 7.1 Cumulative distribution function for the standard n-dimensional Gaussian distribution . 37 8 Notations 42 1 1 Classical EVT results Useful classical EVT results taken from [1] and adapted so that the notations are consistent throughout this document. In particular, the number of samples from which an extremum is drawn is called n throughout [1]. Here, n will be the dimension of the data space and the number of samples from which extrema are drawn will be m. Furthermore, in [1], Embrechts notes cn and dn the scale and location parameters of extreme value distri- butions, whereas Roberts notes them σm and µm, respectively, in [2] and [3]. We choose to note them cm and dm, respectively. 1.1 Fisher-Tippett Theorem With our notations: Theorem 1. (Fisher-Tippett theorem - Theorem 3.2.3 in [1], p.121) Let (Xm) be a sequence of iid rvs. If there exist norming constants dm 2 R, cm > 0 and some non degenerate distribution function H such that −1 d cm (Mm − dm) ! H; (1.1) then H belongs to the type of one of the following three distribution functions : −x Type I (Gumbel): Λ(x) = exp {−e g, x 2 R. 8 <> 0; x ≤ 0 Type II (Fr´echet): Φα(x) = α < 0 :> exp {−x−αg ; x > 0 8 <> exp {−(−x)αg ; x ≤ 0 Type III (Weibull): Ψα(x) = α < 0 :> 1; x > 0 Definition 1. (Extreme Value distribution and extremal random variables - Definition 3.2.6 in [1], p.124) The distribution functions Λ, Φα, Ψα as presented in Theorem 1 are called standard extreme value dis- tributions the corresponding random variables standard extremal random variables. Distribution functions of the types of Λ, Φα, Ψα are extreme value distributions; the corresponding random variables extremal random variables. 2 d Gumbel: Mm = X + log m d 1/α Fr´echet: Mm = m X d −1/α Weibull: Mm = m Notes • An extreme value distribution is only be dependent on m, the number of samples from which the extrema is taken, and the parameters of the generative distribution. • Embrechts refers to theorem 1 as being `the basis of classical extreme value theory'. • The Weibull distribution in theorem 1 is sometimes referred to as the `inverse Weibull distribution'. It is obtained from the other Weibull distribution by reversing the direction of the x-axis for the probability density function . 1.2 Maximum Domains of Attraction Definition 2. (Maximum domain of attraction - Definition 3.3.1 in [1], p. 128) We say that the random variable X (the distribution function F of X, the distribution of X) belongs to the maximum domain of attraction of the extreme value distribution H if there exist constants dm 2 R and cm > 0, such that: 1 d (Mm − dm) ! H: (1.2) cm We write X 2 MDA (H) (F 2 MDA (H)). Proposition 1. (Characterisation of MDA (H) - Proposition 3.3.2 in [1], p. 129) The distribution function F belongs to the maximum domain of attraction of the extreme value distribution H with norming constants dm 2 R, cm > 0, if and only if lim mF (cmx + dm) = − ln(H(x)); x 2 : (1.3) m!1 R When H(x) = 0, the limit is interpreted as 1. 3 1.2.1 Tail-equivalence Definition 3. (Tail-equivalence - Definition 3.3.3 in [1], p.129) Two distribution functions F and G are called tail-equivalent if they have the same right-end point, i.e. if xF = xG, and if there exists some constant 0 < c < 1, such that lim F (x)=G(x) = c: x"xF We note F ∼t G 1.2.2 Maximum domain of attraction of the Fr´echet distribution Theorem 2. (Maximum domain of attraction of Φα - Theorem 3.3.7 in [1], p.131) The distribution function F belongs to the maximum domain of attraction of Φα, α > 0, if and only if there exists some slowly varying function L such that F (x) = x−αL(x). If F 2 MDA (Φα), then 1 d (Mm − dm) ! Φα; (1.4) cm where the norming constants can be chosen as dm = 0 and cm = (1=F ) (m). 1.2.3 Maximum domain of attraction of the Weibull distribution Theorem 3. (Maximum domain of attraction of Ψα - Theorem 3.3.12 in [1], p.135) The distribution function F belongs to the maximum domain of attraction of Ψα, α > 0, if and only if −1 −α xF < 1 and there exists some slowly varying function L such that F (xF − x ) = x L(x). If F 2 MDA (Ψα), then 1 d (Mm − dm) ! Ψα; (1.5) cm −1 where the norming constants can be chosen as dm = xF and cm = xF − F (1 − m ). 1.2.4 Maximum domain of attraction of the Gumbel distribution Theorem 4. (Maximum domain of attraction of Λ - Theorem 3.3.26 in [1], p.142) The distribution function F with right endpoint xF ≤ 1 belongs to the maximum domain of attraction of 4 Λ if and only if there exists some z < xF such that F has representation Z x g(t) F (x) = c(x) exp − dt ; z < x < xF ; (1.6) z a(t) where c and g are measurable functions satisfying c(x) ! c > 0, g(x) ! 1 as x " xF , and a(x) is a positive, 0 0 absolutely continuous function (with respect to Lebesgue measure) with density a (x) having limx"xF a (x) = 0. For F with representation 1.6, we can choose the norming constants as −1 dm = F (1 − n ) and cm = a(dm): A possible choice for the function a is Z xF F (t) a(x) = dt; x < xF : (1.7) x F (x) Proposition 2. (Closure property of MDA (Λ) - Proposition 3.3.28 in [1], p.142) Let F and G be distribution functions with the same right endpoint xF = xG and assume that F 2 MDA (Λ) with norming constants cm > 0 and dm 2 R; i.i m lim F (cmx + dm) = Λ(x); x 2 : (1.8) m!1 R Then m lim G (cmx + dm) = Λ(x + b); x 2 ; (1.9) m!1 R if and only if f and G are tail-equivalent with lim F (x)=G(x) = eb: (1.10) x"xF Notes • `The maximum domain of attraction of the Gumbel distribution consists of distribution functions whose right tails decrease to zero faster than any power function' ([1], p.139). • Every maximum domain of attraction is closed with respect to tail-equivalence. Moreover, for any two tail-equivalent distributions, one can take the same norming constants ([1], p.139). • An F 2 MDA (Λ) can have either a finite or infinite endpoint: xF ≤ 1. 5 • Every F 2 MDA (ΦΦα ) has an infinite right endpoint: xF = 1. • Every F 2 MDA (ΨΨα ) has a finite right endpoint: xF < 1. • Proposition 2 is useful when searching for the parameters of a Gumbel distribution. If we can show that the distribution of interest is tail-equivalent to a distribution of reference, it becomes possible to deduce its parameters (see section 2 for an example.). 6 2 Univariate Gaussian distribution In this section, our aim is to find the parameters of the Gumbel distribution of maxima of univariate Gaussian distribution with arbitrary mean and variance. We do this as an exercise, whose aim is to show how one goes about identifying such parameters. We first show that the Gaussian distribution is in the Maximum Domain of Attraction of the Gumbel distribution and use the closure property of the MDA as well as a tail-equivalent distribution to find a formula for the location parameter. The scale parameter is easily deduced from the previous steps using theorem 4. Probability density function The probability density function of a Gaussian distribution with mean µ 2 R and standard deviation σ > 0 1 (x − µ)2 f(x) = p exp − (2.1) 2πσ2 2σ2 Cumulative distribution function 1 x − µ F (x) = 1 + erf p (2.2) 2 σ 2 where x 2 Z 2 erf(x) = p e−t dt (2.3) π 0 is the so-called error function. Mill's ratio p 1 − F (x) 2πσ2 x − µ (x − µ)2 = 1 − erf p exp f(x) 2 σ 2 2σ2 p 2πσ2 x − µ (x − µ)2 = erfc p exp (2.4) 2 σ 2 2σ2 where 1 2 Z 2 erfc(x) = p e−t dt (2.5) π x 7 is the complementary error function.