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Approximation of the conditional number of exceedances Han Liang Gan University of Melbourne July 9, 2013 Joint work with Aihua Xia Han Liang Gan Approximation of the conditional number of exceedances 1 / 16 Conditional exceedances Han

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Approximation of the conditional number of exceedances

Han Liang Gan

University of Melbourne

July 9, 2013 Joint work with Aihua Xia

Han Liang Gan Approximation of the conditional number of exceedances 1 / 16

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Conditional exceedances

Han Liang Gan Approximation of the conditional number of exceedances 2 / 16

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Conditional exceedances

Given a sequence of identical but not independent random variables X1, . . . , Xn, define Ns,n =

n

  • i=1

1{Xi>s}.

Han Liang Gan Approximation of the conditional number of exceedances 2 / 16

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Conditional exceedances

Given a sequence of identical but not independent random variables X1, . . . , Xn, define Ns,n =

n

  • i=1

1{Xi>s}. The fragility index of order m, denoted FIn(m) is defined as FIn(m) = lim

sր E(Ns,n|Ns,n ≥ m).

Han Liang Gan Approximation of the conditional number of exceedances 2 / 16

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Conditional exceedances

Given a sequence of identical but not independent random variables X1, . . . , Xn, define Ns,n =

n

  • i=1

1{Xi>s}. The fragility index of order m, denoted FIn(m) is defined as FIn(m) = lim

sր E(Ns,n|Ns,n ≥ m).

Applications: Insurance.

Han Liang Gan Approximation of the conditional number of exceedances 2 / 16

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Compound Poisson limit

Under certain conditions, Tsing et. al. (1988) showed that the exceedance process converges to a Compound Poisson limit.

Han Liang Gan Approximation of the conditional number of exceedances 3 / 16

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Assumptions

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Assumptions

Existence of the limits.

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Assumptions

Existence of the limits. Tail behaviour is nice enough.

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Assumptions

Existence of the limits. Tail behaviour is nice enough. Mixing condition.

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Assumptions

Existence of the limits. Tail behaviour is nice enough. Mixing condition. Uniform convergence.

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Assumptions

Existence of the limits. Tail behaviour is nice enough. Mixing condition. Uniform convergence. Uniform integrability.

Han Liang Gan Approximation of the conditional number of exceedances 4 / 16

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Alternatives

In contrast to limit theorems, we can use approximation theory. Our tool

  • f choice will be Stein’s method.

Han Liang Gan Approximation of the conditional number of exceedances 5 / 16

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Alternatives

In contrast to limit theorems, we can use approximation theory. Our tool

  • f choice will be Stein’s method.

For the next section of the talk, we shall stick to conditioning on just one exceedance, Poisson approximation, and total variation distance.

Han Liang Gan Approximation of the conditional number of exceedances 5 / 16

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A simple example

Let X1, . . . , Xn be a sequence of i.i.d. exponential random variables, p = ps = P(X1 > s) and Zλ ∼ Po(λ).

Han Liang Gan Approximation of the conditional number of exceedances 6 / 16

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A simple example

Let X1, . . . , Xn be a sequence of i.i.d. exponential random variables, p = ps = P(X1 > s) and Zλ ∼ Po(λ). dTV (L(N1), Po1(λ)) = 1 2

∞

  • j=1
  • P(N1 = j) − P(Z 1

λ = j)

  • .

Han Liang Gan Approximation of the conditional number of exceedances 6 / 16

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A simple example (2)

Set λ such that e−λ = (1 − p)n.

Han Liang Gan Approximation of the conditional number of exceedances 7 / 16

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A simple example (2)

Set λ such that e−λ = (1 − p)n. So now we have dTV (L(N1), Po1(λ)) = 1 2(1 − P(N = 0))

∞

  • j=1

|P(N = j) − P(Zλ = j)| .

Han Liang Gan Approximation of the conditional number of exceedances 7 / 16

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A simple example (3)

Set λ∗ = np, and we can reduce the problem to known results. dTV (L(N1), Po1(λ)) = 1 2(1 − P(N = 0))

∞

  • j=1

[|P(N = j) − P(Zλ∗ = j)| + |P(Zλ∗ = j) − P(Zλ = j)|].

Han Liang Gan Approximation of the conditional number of exceedances 8 / 16

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A simple example (3)

Set λ∗ = np, and we can reduce the problem to known results. dTV (L(N1), Po1(λ)) = 1 2(1 − P(N = 0))

∞

  • j=1

[|P(N = j) − P(Zλ∗ = j)| + |P(Zλ∗ = j) − P(Zλ = j)|]. Using results from Barbour, Holst and Janson (1992), it can be shown that: dTV (L(N1), Po1(λ)) = p + o(p).

Han Liang Gan Approximation of the conditional number of exceedances 8 / 16

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The conditional Poisson Stein Identity

Lemma

W ∼ Po1(λ) if and only if for all bounded functions g : Z+ → R, E

  • λg(W + 1) − Wg(W ) · 1{W ≥2}
  • = 0.

Han Liang Gan Approximation of the conditional number of exceedances 9 / 16

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Stein’s method in one slide

We construct a function g for any set A, that satisfies: 1{j∈A} − Po1(λ){A} = λg(j + 1) − jg(j) · 1{j≥2}.

Han Liang Gan Approximation of the conditional number of exceedances 10 / 16

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Stein’s method in one slide

We construct a function g for any set A, that satisfies: 1{j∈A} − Po1(λ){A} = λg(j + 1) − jg(j) · 1{j≥2}. It now follows that, P(W ∈ A) − Po(λ){A} = E

  • λg(W + 1) − Wg(W ) · 1{W ≥2}
  • .

Han Liang Gan Approximation of the conditional number of exceedances 10 / 16

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Stein’s method in one slide

We construct a function g for any set A, that satisfies: 1{j∈A} − Po1(λ){A} = λg(j + 1) − jg(j) · 1{j≥2}. It now follows that, P(W ∈ A) − Po(λ){A} = E

  • λg(W + 1) − Wg(W ) · 1{W ≥2}
  • .

We then only need to bound the right hand side.

Han Liang Gan Approximation of the conditional number of exceedances 10 / 16

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Generator interpretation

If we set g(w) = h(w) − h(w − 1), the equation becomes:

Han Liang Gan Approximation of the conditional number of exceedances 11 / 16

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Generator interpretation

If we set g(w) = h(w) − h(w − 1), the equation becomes: E

  • λ (h(W + 1) − h(W )) − W (h(W ) − h(W − 1))1{W ≥2}
  • .

Han Liang Gan Approximation of the conditional number of exceedances 11 / 16

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Generator interpretation

If we set g(w) = h(w) − h(w − 1), the equation becomes: E

  • λ (h(W + 1) − h(W )) − W (h(W ) − h(W − 1))1{W ≥2}
  • .

This looks like the generator for a immigration death process.

Han Liang Gan Approximation of the conditional number of exceedances 11 / 16

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Generator interpretation

If we set g(w) = h(w) − h(w − 1), the equation becomes: E

  • λ (h(W + 1) − h(W )) − W (h(W ) − h(W − 1))1{W ≥2}
  • .

This looks like the generator for a immigration death process. Moreover, the stationary distribution for this generator is Po1(λ).

Han Liang Gan Approximation of the conditional number of exceedances 11 / 16

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Stein Factors

We can use theorem 2.10 from Brown & Xia (2001) to get:

Han Liang Gan Approximation of the conditional number of exceedances 12 / 16

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Stein Factors

We can use theorem 2.10 from Brown & Xia (2001) to get:

Theorem

The solution g that satisfies the Stein Equation for the total variation distance satisfies: ||∆g|| ≤ 1 − e−λ − λe−λ λ(1 − e−λ) .

Han Liang Gan Approximation of the conditional number of exceedances 12 / 16

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The example revisited

Han Liang Gan Approximation of the conditional number of exceedances 13 / 16

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The example revisited

Recall that from before: dTV (L(N1), Po1(λ)) = p + o(p).

Han Liang Gan Approximation of the conditional number of exceedances 13 / 16

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The example revisited

Recall that from before: dTV (L(N1), Po1(λ)) = p + o(p). Using Stein’s method directly for conditional Poisson approximation, it can be shown that: dTV (L(N1), Po1(λ∗)) = p 2 + o(p).

Han Liang Gan Approximation of the conditional number of exceedances 13 / 16

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Generalisations

The conditional Poisson Stein Identity generalises for conditioning on multiple exceedances:

Han Liang Gan Approximation of the conditional number of exceedances 14 / 16

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Generalisations

The conditional Poisson Stein Identity generalises for conditioning on multiple exceedances:

Lemma

W ∼ Poa(λ) if and only if for all bounded functions g : {a, a + 1, . . .} → R, E

  • λg(W + 1) − Wg(W ) · 1{W >a}
  • = 0.

Han Liang Gan Approximation of the conditional number of exceedances 14 / 16

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Negative Binomial Approximation

We can do exactly the same for negative binomial random variables.

Lemma

W ∼ Nba(r, p) if and only if for all bounded functions g : {a, a + 1, . . .} → R, E

  • (1 − p)(r + W )g(W ) − Wg(W )) · 1{W >a}
  • = 0.

Han Liang Gan Approximation of the conditional number of exceedances 15 / 16

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Different metrics

Have we solved our original problem? Can we calculate errors when estimating fragility indicies?

Han Liang Gan Approximation of the conditional number of exceedances 16 / 16

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Different metrics

Have we solved our original problem? Can we calculate errors when estimating fragility indicies? No, we still have the uniform integrability problem.

Han Liang Gan Approximation of the conditional number of exceedances 16 / 16

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Different metrics

Have we solved our original problem? Can we calculate errors when estimating fragility indicies? No, we still have the uniform integrability problem. We need to use a stronger metric, such as the Wasserstein distance.

Han Liang Gan Approximation of the conditional number of exceedances 16 / 16