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 Liang Gan Approximation of the conditional number of exceedances 2 / 16
Conditional exceedances Given a sequence of identical but not independent random variables X 1 , . . . , X n , define n � N s , n = 1 { X i > s } . i =1 Han Liang Gan Approximation of the conditional number of exceedances 2 / 16
Conditional exceedances Given a sequence of identical but not independent random variables X 1 , . . . , X n , define n � N s , n = 1 { X i > s } . i =1 The fragility index of order m , denoted FI n ( m ) is defined as FI n ( m ) = lim s ր E ( N s , n | N s , n ≥ m ) . Han Liang Gan Approximation of the conditional number of exceedances 2 / 16
Conditional exceedances Given a sequence of identical but not independent random variables X 1 , . . . , X n , define n � N s , n = 1 { X i > s } . i =1 The fragility index of order m , denoted FI n ( m ) is defined as FI n ( m ) = lim s ր E ( N s , n | N s , n ≥ m ) . Applications: Insurance. Han Liang Gan Approximation of the conditional number of exceedances 2 / 16
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
Assumptions Han Liang Gan Approximation of the conditional number of exceedances 4 / 16
Assumptions Existence of the limits. Han Liang Gan Approximation of the conditional number of exceedances 4 / 16
Assumptions Existence of the limits. Tail behaviour is nice enough. Han Liang Gan Approximation of the conditional number of exceedances 4 / 16
Assumptions Existence of the limits. Tail behaviour is nice enough. Mixing condition. Han Liang Gan Approximation of the conditional number of exceedances 4 / 16
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
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
Alternatives In contrast to limit theorems, we can use approximation theory. Our tool of choice will be Stein’s method. Han Liang Gan Approximation of the conditional number of exceedances 5 / 16
Alternatives In contrast to limit theorems, we can use approximation theory. Our tool of 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
A simple example Let X 1 , . . . , X n be a sequence of i.i.d. exponential random variables, p = p s = P ( X 1 > s ) and Z λ ∼ Po ( λ ). Han Liang Gan Approximation of the conditional number of exceedances 6 / 16
A simple example Let X 1 , . . . , X n be a sequence of i.i.d. exponential random variables, p = p s = P ( X 1 > s ) and Z λ ∼ Po ( λ ). ∞ d TV ( L ( N 1 ) , Po 1 ( λ )) = 1 � P ( N 1 = j ) − P ( Z 1 � � � � . λ = j ) 2 j =1 Han Liang Gan Approximation of the conditional number of exceedances 6 / 16
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
A simple example (2) Set λ such that e − λ = (1 − p ) n . So now we have ∞ 1 d TV ( L ( N 1 ) , Po 1 ( λ )) = � | P ( N = j ) − P ( Z λ = j ) | . 2(1 − P ( N = 0)) j =1 Han Liang Gan Approximation of the conditional number of exceedances 7 / 16
A simple example (3) Set λ ∗ = np , and we can reduce the problem to known results. ∞ 1 d TV ( L ( N 1 ) , Po 1 ( λ )) = � [ | P ( N = j ) − P ( Z λ ∗ = j ) | 2(1 − P ( N = 0)) j =1 + | P ( Z λ ∗ = j ) − P ( Z λ = j ) | ] . Han Liang Gan Approximation of the conditional number of exceedances 8 / 16
A simple example (3) Set λ ∗ = np , and we can reduce the problem to known results. ∞ 1 d TV ( L ( N 1 ) , Po 1 ( λ )) = � [ | P ( N = j ) − P ( Z λ ∗ = j ) | 2(1 − P ( N = 0)) j =1 + | P ( Z λ ∗ = j ) − P ( Z λ = j ) | ] . Using results from Barbour, Holst and Janson (1992), it can be shown that: d TV ( L ( N 1 ) , Po 1 ( λ )) = p + o ( p ) . Han Liang Gan Approximation of the conditional number of exceedances 8 / 16
The conditional Poisson Stein Identity Lemma W ∼ Po 1 ( λ ) if and only if for all bounded functions g : Z + → R , � � E λ g ( W + 1) − Wg ( W ) · 1 { W ≥ 2 } = 0 . Han Liang Gan Approximation of the conditional number of exceedances 9 / 16
Stein’s method in one slide We construct a function g for any set A , that satisfies: 1 { j ∈ A } − Po 1 ( λ ) { A } = λ g ( j + 1) − jg ( j ) · 1 { j ≥ 2 } . Han Liang Gan Approximation of the conditional number of exceedances 10 / 16
Stein’s method in one slide We construct a function g for any set A , that satisfies: 1 { j ∈ A } − Po 1 ( λ ) { A } = λ g ( j + 1) − jg ( j ) · 1 { j ≥ 2 } . It now follows that, � � P ( W ∈ A ) − Po ( λ ) { A } = E λ g ( W + 1) − Wg ( W ) · 1 { W ≥ 2 } . Han Liang Gan Approximation of the conditional number of exceedances 10 / 16
Stein’s method in one slide We construct a function g for any set A , that satisfies: 1 { j ∈ A } − Po 1 ( λ ) { A } = λ g ( j + 1) − jg ( j ) · 1 { j ≥ 2 } . It now follows that, � � P ( W ∈ A ) − Po ( λ ) { A } = E λ g ( 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
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
Generator interpretation If we set g ( w ) = h ( w ) − h ( w − 1), the equation becomes: � � λ ( h ( W + 1) − h ( W )) − W ( h ( W ) − h ( W − 1)) 1 { W ≥ 2 } E . Han Liang Gan Approximation of the conditional number of exceedances 11 / 16
Generator interpretation If we set g ( w ) = h ( w ) − h ( w − 1), the equation becomes: � � λ ( h ( W + 1) − h ( W )) − W ( h ( W ) − h ( W − 1)) 1 { W ≥ 2 } E . This looks like the generator for a immigration death process. Han Liang Gan Approximation of the conditional number of exceedances 11 / 16
Generator interpretation If we set g ( w ) = h ( w ) − h ( w − 1), the equation becomes: � � λ ( h ( W + 1) − h ( W )) − W ( h ( W ) − h ( W − 1)) 1 { W ≥ 2 } E . This looks like the generator for a immigration death process. Moreover, the stationary distribution for this generator is Po 1 ( λ ). Han Liang Gan Approximation of the conditional number of exceedances 11 / 16
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
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 − λ e − λ . λ (1 − e − λ ) Han Liang Gan Approximation of the conditional number of exceedances 12 / 16
The example revisited Han Liang Gan Approximation of the conditional number of exceedances 13 / 16
The example revisited Recall that from before: d TV ( L ( N 1 ) , Po 1 ( λ )) = p + o ( p ) . Han Liang Gan Approximation of the conditional number of exceedances 13 / 16
The example revisited Recall that from before: d TV ( L ( N 1 ) , Po 1 ( λ )) = p + o ( p ) . Using Stein’s method directly for conditional Poisson approximation, it can be shown that: d TV ( L ( N 1 ) , Po 1 ( λ ∗ )) = p 2 + o ( p ) . Han Liang Gan Approximation of the conditional number of exceedances 13 / 16
Generalisations The conditional Poisson Stein Identity generalises for conditioning on multiple exceedances: Han Liang Gan Approximation of the conditional number of exceedances 14 / 16
Generalisations The conditional Poisson Stein Identity generalises for conditioning on multiple exceedances: Lemma W ∼ Po a ( λ ) if and only if for all bounded functions g : { a , a + 1 , . . . } → R , � � E λ g ( W + 1) − Wg ( W ) · 1 { W > a } = 0 . Han Liang Gan Approximation of the conditional number of exceedances 14 / 16
Negative Binomial Approximation We can do exactly the same for negative binomial random variables. Lemma W ∼ Nb a ( 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
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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