Tests for extreme value assumptions

joint work with Deyuan Li

Extreme value analysis:

Basic ingredients: X_{i}, i ≤ n, iid. with X_{i} ∼ F. Consider M_{n} = max{X_{i}, i ≤ n}

Limit distributions with normalisation: P {(M_{n} − b_{n})/a_{n} ≤ x} → G(x) as n → ∞. G(x) = G_{γ}(x) = exp(−(1 + γx)^{−1/γ} if 1 + γx > 0, γ ∈ ℝ, where G_{0}(X) = exp(− exp(−x)). standard form, usually with a location and scale parameter with density g_{γ}(x) = G_{γ}(x)(1 + γx)^{−1−1/γ} for γ ≠ 0, and g_{0}(x) = G_{0}(x) exp(−x).

Notation: F ∈ MDA(G_{γ}).

Not every function F is in MDA(G_{γ}). Dependent on the tail behaviour of F as x → x_{0}, x_{0} upper endpoint of support In applications how to know whether F ∈ MDA(G_{γ}), for some γ?

Test: goodness-of-fit as CvM, AD or similar tests.

Test 1: Dietrich, de Haan and Hüsler (2002) H_{0}: if F ∈ MDA(G_{γ}) for some γ ∈ ℝ with some additional (second order) conditions, then for each η > 0, the test statistic

E_{n} := k ∫_{0}^{1} (\frac{log X_{n−[kt],n} − log X_{n−k,n}}{γ̂_{+}} − \frac{t^{−γ̂_{−}} − 1}{γ̂_{−}}(1 − γ̂_{−}))^{2}t^{η} dt (1)

converges in distribution to E_{γ} (explicit expression) (as k, n → ∞ with k/n → 0), where moment estimators γ̂, γ̂_{−} and γ̂_{+}(Dekkers, de Haan and Einmahl (1989)). E_{γ} depends on γ and η, only. Derive the quantiles for the test E_{n} for large n by simulations of E_{γ}.

This test is based on the quantile functions of F and G_{γ}, uses of the upper order statistics of the sample as information of the tail of the quantile functions.

Particular case: If γ ≥ 0,

P E_{n} := k ∫_{0}^{1} (\frac{log X_{n−[kt],n} − log X_{n−k,n}}{γ̂_{+}} + log t)^{2}t^{η} dt →^{d} P E (n → ∞)

Test 2: Drees, de Haan and Li (2005) if F ∈ MDA(G_{γ}) for some γ > −1/2 with some additional (second order) conditions, then for each η > 0, the test statistic

T_{n} := k ∫_{0}^{1} (\frac{n}{k}F̄_{n}(â_{n/k} \frac{x^{−γ̂} − 1}{γ̂} + b̂_{n/k}) − x)^{2}x^{η−2} dx  (2)

converges in distribution to T_{γ}, (explicit expression), (F̄ = 1 − F) (as k, n → ∞ with k/n → 0), where γ̂_{n}, â_{n}, b̂_{n} are mle. Limiting distribution of T_{γ} depends on η and γ, only. Derive the quantiles of the test T_{n} for large n by simulations of T_{γ}.

Test is based on the tail of the estimated distribution F^{n} by nF̄, with location and scale estimation, uses also the upper order statistics.

Application: Estimate γ, use the asymptotic quantiles with γ̂. Tabulated for certain γ, interpolate for other values γ. Compare the test values T_{n} or E_{n}, with the selected (limiting) quantiles to decide, for n large and k/n small. If we think γ ≥ 0, then compare P E_{n} with the (limiting ) quantiles.

Properties of the tests? Which η should be selected?

Extensive simulation study showed a good picture of the behaviour.

Questions:

1) Convergence rates of the quantiles Q_{p,n,η} of the test for fixed n to the limiting ones Q_{p,∞,γ}?

2) Accuracy of the test level α for finite n, for F ∈ MDA(G_{γ}) with use of Q_{p,∞,γ}?

3) How, if second order properties are not satisfied?

4) Power if F ∉ MDA(G_{γ})?

1) Convergence rates

A good choice of η (w.r.t. convergence rates) for the test E_{n}: η = 2 for the test T_{n}: η = 1 PE is no such a good choice

Choice of k plays also a role, select for E_{n} smaller k’s (with k/n ≤ 0.05) for T_{n} larger proportion ≤ 0.1.

Used criteria: T RE(F) := ∑_{Case j} ∑_{n_{i}(k_{i}/n_{i})} |Q̂_{0.95,n_{i},γ} − Q_{0.95,γ}|/Q_{0.95,γ}.

Case 1 n =  500 1000 2000 5000 10000
k/n = 5% 4% 3% 2% 1.5%
Case 2: n =  500 1000 2000 5000 10000
k/n = 8% 6% 4% 3% 2%
Case 3: n =  500 1000 2000 5000 10000
k/n = 10% 8% 5% 3% 2%

Examples: • Cauchy: F ∈ MDA(G_{1}) + 2nd order fulfilled

• EV(0): F ∈ MDA(G_{0}) + 2nd order fulfilled

• Burr(1,2,2): F ∈ MDA(G_{1/4}) + 2nd order fulfilled

Burr(β, τ, λ) distribution with β, τ, λ > 0:

F (x) = 1 − (\frac{β}{β + x^{τ}} )^{λ} , x > 0,

(γ = 1/(τλ) = 1/4, ρ = −1/λ).

• R-Burr(1,4,1): F ∈ MDA(G_{−1/4}) + 2nd order fulfilled

Reversed Burr(β, τ, λ) distribution with β, τ, λ > 0 :

F (x) = 1 − (\frac{β}{β + (x_{0} − x)^{−τ}} )^{λ} , x < x_{0} = 1,

(γ = −1/(τλ) = −1/4, ρ = −1/λ).

Figure 1: Type I error for different η with sample size n = 2000.

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2. If F ∈ MDA(G_{γ}) but 2nd order cond. does not hold?

let U(t) := F^{←}(1 − 1/t), t ≥ 1, generalized inverse function

F ∈ MDA(G_{γ}) is equivalent to U ∈ RV(γ) in case of γ > 0 or equivalent to U(∞) − U(.) ∈ RV(γ) in case of γ < 0.

second order condition: roughly U(t) = ct^{γ}(1 + A(t)) in case of γ > 0 or U(∞) − U(t) = ct^{γ}(1 + A(t)) in case of γ < 0 for large t, where c > 0, A(t) → 0 as t → ∞ and |A| ∈ RV(ρ) with ρ ≤ 0.

Examples:

• N2Ex1(γ, λ) distribution with γ > 0 and 0 < λ ≤ 1/e:

U(t) = t^{γ}(γ^{−1} + λ t^{−1} exp(sin t)), t ≥ 1.

• N2Ex2(γ, λ) distribution with γ ∈ ℝ and 0 < λ ≤ 1: for t ≥ 1,

U(t) = \begin{cases}(t + λ sin t)^{γ}, & γ > 0, \\ 1 − (t + λ sin t)^{γ}, & γ < 0. \end{cases}

• N2Ex3(γ, λ) distribution with γ > 0 and 0 < λ ≤ 1:

U(t) = t^{γ} + λ sin(t^{γ}), t ≥ 1.

Figure 2: Type I errors for the N2Ex1(0.5, λ) distributions with λ = 1/e, 0.2, 0.1 and 0.05, U(t) = t^{γ}(γ^{−1} + λ t^{−1} exp(sin t)), smaller λ, less influence of the sin-term, with ’best’ η.

Figure 3: Type I errors for the N2Ex2(0.5, λ) distributions with λ = 1, 0.5, 0.2 and 0, U(t) = (t + λ sin t)^{γ}, with ’best’ η.

Figure 4: Type I errors for the N2Ex2(-0.2, λ) distributions with λ = 1, 0.5, 0.2 and 0.1, U(t) = 1 − (t + λ sin t)^{γ}, with ’best’ η.

Figure 5: Type I errors for the N2Ex3(0.5, λ) distributions with λ = 1, 0.2, 0.1 and 0.05, U(t) = t^{γ} + λ sin(t^{γ}), with ’best’ η.

4. Power of the tests:

If F ∉ MDA(G_{γ})

Examples: Poisson(λ)

• Poisson(λ)

• Geometric or truncated Exponential: X = exp[E]

• Exam1: F (x) = 1 − exp ( − \sqrt{− log(1 − x)} ), 0 ≤ x < 1.

• Exam2(β) with β > 0: F (x) = 1 − (log x)^{−β}, x ≥ e.

• Exam3(β, λ)with β > 0 and 0 < λ ≤ 1/\sqrt{2}:

U(t) = t^{β}(1 + λ sin(β log t)), t ≥ 1.

Figure 6: Powers for the distributions not belonging to the max-domain.

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Figure 7: Powers for Poisson distributions with λ = 10, 100, 1000 and 5000.

16 Figure 8: Log return of daily equity of ING bank (n = 3283).

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Some conclusions:

• Parameter η:

good choices η = 1 or 2 for test T and η = 2 for test E. k/n should be small, ≤ 5%. (It can be applied for other statistical purposes, also).

• Second order condition is not necessary in testing the null hypothesis H_{0}, small impact on the type I error if F close to a F^{∗} which satisfies second order conditions.

• Power to detect a distribution F not belonging to the max-domain depends on how ’far’ F is away from the max-domain. Monotone w.r.t. the distance.

• T and E have rather similar properties.

Test E seems to do sometimes better with k/n ≤ 5%, e.g. for γ ≤ 0. T has a better power for certain distribution F ∉ MDA(G_{γ}).

• use both test T and E for accepting H_{0}.