Assessing Dependence in Extreme Values

Transcription

1 02/09/2016

2 1 Motivation Motivation 2 Comparison 3 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations 4 Idea Results

3 Motivation Given historical flood levels, how high should we build our defences? How extreme was the earthquake in central Italy? Was it a once every 10 year year quake or a once every 1,000 year phenomenon? Does the Rt Hon Theresa May PM need extra immigration staff at Dover if there is high demand at Poole?

4 To answer these problems we wish to make inference about the distribution of extremal events, M n = max(x 1,..., X n ) m n = min(x 1,..., X n ) for historical data X 1, X 2,..., X n. Note that m n = max( X 1,..., X n ) so WLOG we focus on maxima.

5 For i.i.d data P(M n z) = (F (z)) n where F is the c.d.f for X. Problems Slight errors in estimating the margins of X lead to large errors in the margins of M n. By definition, inference for M n relies on the part of F that we know the least about. Hence we require a general framework to deal with these extreme values.

6 Comparison We wish to determine the distribution of M n as n but how should we decide what counts as an extreme value?

7 Comparison Generalised Extreme Value Theorem If a sequence of constants {a n > 0} and {b n > 0} and non-degenerate function G s.t. ( ) Mn b n P z G(z) then G is a member of the GEV family ( [ ( )] 1 ) z µ ζ G(z) = exp 1 + ζ σ Which is defined on {z : 1 + ζ(z µ)/σ > 0} with < µ, ζ < and σ > 0. a n

8 Comparison Accidental American Deaths

9 Comparison Return Levels Definition: Return Levels If G(z p ) = 1 p then z p is the return level for return period 1 p. Expect z p to be exceeded every 1 p years. z p exceeded w.p. p If ζ < 0 then G has a finite upper end point whereas for ζ 0 any value of z is feasibly attainable. [ ] {ˆµ 1 ( log(1 p)) z p = ˆσˆζ ˆζ for ˆζ 0 ˆµ ˆσlog( log(1 p)) otherwise

10 Comparison Example: Fastest Annual 100m Times Since 1972 (ˆµ, ˆσ, ˆζ) = (9.935, 0.225, 0.203),

11 Comparison Alternatively choose a threshold u, anything above is classified as an extreme event. We then have threshold excesses y i = X i u. Theorem Suppose we have X i such that M n has margins from the GEV family, then for large enough u, the distribution of (X u), conditional on X > u, is approximately a member of the Pareto family H(y) = 1 (1 + ζỹ σ ) 1/ζ defined on {y : y > 0 and (1 + ζy/ σ) > 0}, where σ = σ + ζ(u µ).

12 Comparison

13 Comparison Choosing Thresholds Mean residual plot { (u, 1 } Σ nu (x n (i) u)) : u < xmax u where x (1),..., x (nu) are the n u observations that exceed u. For u 0 satisfying a GPD E(X u X > u) = σ u 0 + ζu 1 ζ for all u > u 0. Hence valid choices of u match an approximately linear plot from u onwards.

14 Comparison Return Levels The N year return level z N solves { [ u + σ ζ (Nny p u ) ζ 1 ] for ˆζ 0 z N = u + σlog(nn y p u ) otherwise for n y as the number of observations a year and p u = P(X > u). Estimate using maximum likelihood as before. Note that ˆp m = nu n, the proportion of points above u in the sample.

15 Comparison Application: 2777 Best Ever 100m Times (a) 2777 Best ever 100m Times (b) Mean Residual Life Plot (ˆσ, ˆξ) = (0.067, 0.144), 95% C.I. for ˆξ = [ 0.293, 0.005] and ẑ 0 =

16 Comparison Model Checking (c) Fastest Annual 100m (d) 2777 Best Ever 100m

17 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Extremal Dependence (e) 2012 Olympians (f) O2 and NO3 In the bivariate case our variables have extremes that either occur simultaneously (Asymptotic Dependence) or independently (Asymptotic Independence).

18 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Component-wise Maxima Theorem Let Mn = If then ( ) max {X i } n, max {Y i } n. P(M n,1 x, M n,2 y) d G(x, y) G(x, y) = exp{ V (x, y)}, where V is in a large un-parametrisable family. In practice we try to fit a subset of this family e.g. Logistic G(x, y) = exp { (x } 1 1 α + y α ) α for α (0, 1].

19 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Measures Two common measures of asymptotic dependence are chi and the coefficient of tail dependence η. 1 χ = lim z z + P(Y > z, X > z) If χ = 0 then asymptotically independent. χ increases with the strength of dependence at extreme levels 2 P(X > x, Y > x) L(x)x 1 η If η = 1 then asymptotically dependent. If η = 1 2 then asymptotically independent. as x (for slowly varying function L)

20 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Estimators of Extremal Dependence Measures Hill: Let T i = min(x i, Y i ) then with T 1,n... T n k,n as the order statistics, ˆη h = 1 k [log(t n k+i,m ) log(t n k,n )]. k i=1 Peng:Let S n (k) = n i=1 I {X i >X n k,n,y i >Y n k,n}, where X i,n as the i th order statistic, then log2 ˆη p = log {S n (2k)/S n (k)} Draisma: ηˆ d = k j=1 S n(j) ks n (k) k j=1 S n(j)

21 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Limitations Figure: Hill (1), Peng (2), Draisma (3) for collections of 100,000 logistic(α) data points from Fréchet margins.

22 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations Figure: Hill (1), Peng (2), Draisma (3) for collections of 100,000 inverted logistic(α) data points from Fréchet margins with the true values in red.

23 Idea Results Idea There is also the problem that individual estimates of chi and η often give contradictory information. We consider estimating both simultaneously (shotgun estimation) P(Z > z) = χz 1 + Cz 1/δ There are three different cases to consider: 1 χ = 0, C > 0, δ χ > 0, C > 0, δ χ > 0, C = 0, and we will choose between them by likelihood-ratio tests. for δ ( 1 2, 1].

24 Idea Results Results Figure: MLE estimates for Logistic(α) in Pareto margins, with true values in red.