rare-event simulation for maxima of dependent random variables · 2020-03-11 · rare-event...

Rare-eventsimulation formaxima ofdependentrandomvariables

Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Rare-event simulation for maxima ofdependent random variables

Patrick J. Laub

BE(Software)/BSc, BSc(Hons. I, Math)University of Queensland, Brisbane Australia

February 1, 2016

1 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Overview

What is a rare event?

Classes of rare-event estimators

Importance sampling and exponential tilting

Cross-entropy method

Multi-level splitting

Our estimator

Application to maxima of dependent r.v.s

2 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

What is a rare event?

Consider some random X ∼ F (·), and imagine estimatingp(`) := P(X ≥ `). As `� 0 then p(`)→ 0, and roughlyspeaking we call it a rare event if p(`) ≤ 10−3.

Consider crude Monte Carlo estimation of p(`). We would take

X1, . . . ,XRi.i.d.∼ F (·)

then

pCMC(`) :=1

R

R∑r=1

1{Xr ≥ `} .

As `� 0 then pCMC(`) = 0.

3 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Evaluating CMC for rare events

Consider the relative error of the CMC estimator,

RE(pCMC(`)) :=

√Var(pCMC(`))

p(`).

As R p(`) ∼ Bin(R, p(`)),

RE(p(`)) =

√p(`)(1− p(`))√

Rp(`)=

√1− p(`)

Rp(`)→∞

as `→∞.

4 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Classes of rare-event estimators

We say p has vanishing relative error if

lim sup`→∞

Var(p(`))

p(`)2= 0 ,

or displays bounded relative error if

lim sup`→∞

Var(p(`))

p(`)2<∞ ,

or lastly exhibits logarithmic efficiency if for all ε > 0

lim sup`→∞

Var(p(`))

p(`)2−ε= 0 .

5 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Importance sampling

Standard solution: If normally X ∼ F (·), replace F with aproposal distribution G ,1 and rely on

p(`) =

∫ ∞`

dF (x)

1F absolutely continuous w.r.t. G , i.e., Supp(G) ⊂ Supp(F ).6 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Importance sampling


p(`) =

∫ ∞`

dF (x)

dG (x)dG (x) = EG

[dF

dG(X )1{X ≥ `}

].



Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Importance sampling


p(`) =

∫ ∞`

dF (x)

dG (x)dG (x) = EG

[dF

dG(X )1{X ≥ `}

].

Assume densities f (·) and g(·) exist, then

pIS(`) =1

R

R∑r=1

f (Xr )

g(Xr )1{Xr > `} , X1, . . . ,XR

i.i.d.∼ G (·) .



Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Importance sampling example

Example: say X ∼ N (0, 1), and ` = 10. Propose importancedistribution of N (µ, 1), where µ ∈ {0, . . . , 20}. The resultingpIS(`) for R = 104 is

5 10 15 20μ

2.×10-24

4.×10-24

6.×10-24

8.×10-24

1.×10-23

p

7 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Optimal proposal density

Is there a “best” proposal density?

Yes,

g∗(x) :=1{x ≥ `}f (x)

p(`).

Zero variance, but . . .

8 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Optimal proposal density

Is there a “best” proposal density? Yes,

g∗(x) :=1{x ≥ `}f (x)

p(`).

Zero variance, but . . .

8 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Exponential tilting

Which proposal?: The exponentially tilted density of f (·) is

fθ(x) :=eθx

E eθXf (x) .

Example: f (x) = N (x ;µ, σ2)⇒ fθ(x) = N (x ;µ+ θσ2, σ2).

Which θ?: Set θ such that EXθ = `.

9 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Exponential tilting


fθ(x) :=eθx

E eθXf (x) .


Which θ?:

Set θ such that EXθ = `.

9 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Exponential tilting


fθ(x) :=eθx

E eθXf (x) .


Which θ?: Set θ such that EXθ = `.

9 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy idea

Find the best proposal density from the same family. Say

X ∼ f (·; u) , p(`) := P(S(X ) ≥ `) .

Find v∗ such that

v∗ = arg minv

Var[p IS(`; v)] .

Idea: probably have f (·; v∗) ≈ g∗(·).Solve instead

v∗ = arg minvD(g∗(·), f (·; v))

Kullback–Leibler divergence

KL(g∗ || f ) :=

∫g∗(x) log

(g∗(x)

f (x ; v)

)dx .

10 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy idea


X ∼ f (·; u) , p(`) := P(S(X ) ≥ `) .

Find v∗ such that

v∗ = arg minv

Var[p IS(`; v)] .

Idea: probably have f (·; v∗) ≈ g∗(·).

Solve instead

v∗ = arg minvD(g∗(·), f (·; v))


KL(g∗ || f ) :=

∫g∗(x) log

(g∗(x)

f (x ; v)

)dx .

10 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy idea


X ∼ f (·; u) , p(`) := P(S(X ) ≥ `) .

Find v∗ such that

v∗ = arg minv

Var[p IS(`; v)] .

Idea: probably have f (·; v∗) ≈ g∗(·).Solve instead

v∗ = arg minvD(g∗(·), f (·; v))


KL(g∗ || f ) :=

∫g∗(x) log

(g∗(x)

f (x ; v)

)dx .

10 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 0

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 1

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 2

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 3

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 4

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 5

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 6

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 7

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 8

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 9

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 10

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 11

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 12

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 13

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Example: P(N (0, 1) ≥ 5) ≈ . . .

-2 2 4 6 8x

1

2

3

4

5

f(x)Iteration 14

11 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy outline

So construct {v0, v1, . . . } s.t. vn → v∗ by

1 Sample X1, . . . , XNi.i.d.∼ f (·; vt).

2 Select elite samples, judged by large S(Xi ) values.

3 Set vt+1 as the (weighted) MLE estimate of the elites.

4 Repeat until convergence.

Then finally estimate P(S(X ) ≥ `) by using IS with f (·; vT ).

12 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

First-year problem

Consider (Wt)t≥0 as standard Brownian motion. What is

P

(1

10

∫ 10

0W (dt) ≥ 10

)?

13 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy example

2 4 6 8 10

-8

-6

-4

-2

2

4

6

Iteration 1 Drift 0

14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

-6

-4

-2

2

4

6

8

Iteration 2 Drift 0.301829

14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

-5

5

10


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

-2

2

4

6

8

10


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10

15


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10

15

20


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10

15


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10

15

20


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

5

10

15

20

25


14 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

First-year problem solution

The cross-entropy estimate is

p(10) = 2.1302×10−8 , 95% CI is (9.4891×10−9, 7.1043×10−8) .

Actual solution is

P

(1

10

∫ 10

0W (dt) ≥ 10

)

= P

N (0,

∫ 10

0(10− s)2 ds

)≥ 100

= P

(N(

0,1000

3

)≤ −100

)= Φ (−5.4772) = 2.1602× 10−8 .

15 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Second-year problem

Consider (Wt)t≥0 as standard Brownian motion. What is

P

(1

10

∫ 10

0exp{W (dt)} ≥ 1000

)?

Using the same code as before, but adding Exp in two lines. . .

16 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy example 2

2 4 6 8 10

100

200

300

400

500

600

Iteration 1 Drift 0

17 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

500

1000

1500

2000

2500

3000

3500


17 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

50000

100000

150000


17 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima


2 4 6 8 10

20000

40000

60000

80000

100000

120000

140000


17 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Second-year problem solution

The cross-entropy estimate is

p(1000) = 3.6039× 10−3

with a 95% CI of (3.3623× 10−3, 3.8968× 10−3).

Thus we have (more-or-less) just priced a heavilyout-of-the-money Asian option.

18 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Third-year problem

Consider the Markov process (Xt)t≥0 defined by

X0 = 1 , dXt = −(1 + cos(t))Xt dt + σ dWt .

What is the probability that this thing reaches the level ` = 10before it drifts down to 0?

19 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Multilevel splitting

Take n levels, 1 < l1 < · · · < ln = `, and consider hitting timesTi := inf{Xt ≥ li}, and also T/ := inf{Xt ≤ 0}.

Say we have the events Di := {Ti < T/}, and definepk := P(Dk |Dk−1) for k > 1, and p1 = P(D1).

Now we have

P(Tn < T/) = P(Dn) = P(Dn |Dn−1)P(Dn−1)

= · · · =n∏

i=1

pi .

20 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Multilevel splitting example: level one

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.5

1.0

1.5

2.0

21 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Multilevel splitting example: level one

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.5

1.0

1.5

2.0

4 ↑ 16 ↓ so p1 = 420 .

21 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Multilevel splitting example: level two

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

22 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Multilevel splitting example: level two

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

2.5

6 ↑ 14 ↓ so p2 = 620 , et cetera. . .

22 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Rare maxima

Say X = (X1, . . . ,Xn) ∼ F (·) with marginals (F1, . . . ,Fn) andcopula C (·).

Write Mn := max{X1, . . . ,Xn}. What is

p(`) := P(Mn ≥ `) = ?

23 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Rare maxima



p(`) := P(Mn ≥ `) = P(X1 ≥ ` ∪ · · · ∪ Xn ≥ `) .

Use inclusion–exclusion principle:

23 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Rare maxima



p(`) := P(Mn ≥ `) = P(X1 ≥ ` ∪ · · · ∪ Xn ≥ `) .

Use inclusion–exclusion principle:

p(`) =n∑

i=1

P(Xi ≥ `)−∑

1≤i<j≤nP(Xi ≥ ` ∩ Xj ≥ `) + . . .

23 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Rare maxima

Assume that ` becomes large, so p(`) is a rare event.

Idea: If {Xi ≥ `} is rare, then the chance of two or morecomponents ≥ ` must be near impossible.

P(Mn ≥ `) ≈n∑

i=1

P(Xi ≥ `) .

The Boole–Frechet inequalities tell us that

maxi{P(Xi ≥ `)} ≤ P(Mn ≥ `) ≤

n∑i=1

P(Xi ≥ `) .

So we use MC to estimate

P(Mn ≥ `)−n∑

i=1

P(Xi ≥ `) .

24 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Our estimator

Say that

K (ω) :=n∑

i=1

1{Xi (ω) ≥ `} .

We will soon see that

P(Mn ≥ `) =n∑

i=1

P(Xi ≥ `)− E[(K − 1)1{K ≥ 2}] .

Simulate X ∼ F (·), find K =∑

1{Xi ≥ `}, and estimate

pLP(`) :=n∑

i=1

P(Xi ≥ `)− (K − 1)1{K ≥ 2} .

25 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Proof of unbiasedness

Inclusion–exclusion

p(`) =n∑

i=1

P(Xi ≥ `)

+n∑

i=2

(−1)i−1

∑j1<···<ji

P(Xj1 ≥ `, . . . ,Xji ≥ `)

Define

Ci (ω) :=∑

j1<···<ji

1{Xj1 ≥ `, . . . ,Xji ≥ `}(ω) ,

so

p(`) =n∑

i=1

P(Xi ≥ `) + E

n∑i=2

(−1)i−1Ci

.26 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Proof of unbiasedness 2

p(`) =n∑

i=1

P(Xi ≥ `) + E

n∑i=2

(−1)i−1Ci

.The Ci can be rewritten as

Ci =

{(Ki

), 1 ≤ i ≤ K ,

0 , K < i ≤ n .

so

p(`) =n∑

i=1

P(Xi ≥ `) + E

K∑

i=2

(−1)i−1(K

i

)1{K ≥ 2}

.

27 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Proof of unbiasedness 3

Simplifying . . . K∑i=2

(−1)i−1(K

i

)1{K ≥ 2}

=

1− K +K∑i=0

(−1)i−1(K

i

)1{K ≥ 2}

= −(K − 1)1{K ≥ 2}

Therefore

p(`) =n∑

i=1

P(Xi ≥ `)− E[(K − 1)1{K ≥ 2}] = E pLP(`) .

28 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Efficiency of our estimator

The estimator again:

pLP(`) :=n∑

i=1

P(Xi ≥ `)− (K − 1)1{K ≥ 2} .

Want lim sup`→∞Var(p(`))

p(`)2<∞ for BRE.

Var pLP(`) ≤ E [(K − 1)1{K ≥ 2}]2

=n∑

k=2

(k − 1)2 P(K = k)

< (n − 1)2 P(K ≥ 2)

≤ (n − 1)2∑

1≤i<j≤nP(Xi ≥ `,Xj ≥ `) .

29 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Efficiency of our estimator 2

lim`→∞

Var pLP(`)

p(`)2< (n − 1)2 lim

`→∞

∑1≤i<j≤n P(Xi ≥ `,Xj ≥ `)

maxi{P(Xi ≥ `)}2

≤(n

2

)(n − 1)2 lim

`→∞

maxi 6=j P(Xi ≥ `,Xj ≥ `)maxi{P(Xi ≥ `)}2

?<∞ .

Proposition

The estimator pLP(`) has bounded relative error iff

maxi 6=j

P(Xi ≥ `,Xj ≥ `) = O(maxi{P(Xi ≥ `)}2)

as `→∞.

30 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Correlation in the limit

Most common measure is the coefficient of asymptotic uppertail dependence

λ(Xi |Xj) := lim`→∞

P(Xi ≥ ` |Xj ≥ `) .

Examples:

Gaussian copula has λ = 0, called asymptoticindependence.

t-copula has λ > 0, called asymptotic dependence.

comonotone copula (Xi = Xj a.s.) has λ = 1. Boring!

31 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Correlation in the limit 2

Another finer measure is also called coefficient of taildependence. It states that 2 as `→∞

P(Xi ≥ `,Xj ≥ `) ∼ L (`) `−1η

for some slowly-varying function L (·) and η ∈ [0, 2].

2Given unit Frechet marginal distributions . . .32 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Table of dependence coefficients

33 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Efficiency of our estimator 3

Must have ∀ i 6= j that λ(Xi |Xj) = 0, but this isn’t strongenough. Assume we’ve transformed the problem to have unitFrechet marginals, then

P(Xi ≥ `,Xj ≥ `)P(Xk ≥ `)2

∼ L (`) `−1η

`−2= L (`) `2−

1η .

Therefore our condition becomes

lim sup`→∞

L (`) `2−1η <∞⇔ η ∈ [0,

1

2) or (η =

1

2and L (`)→ 0) .

34 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Gaussian case

Proposition

Let X ∼ N (µ,Σ) where Σ is non-singular and X1 is adominant tail. Define, ∀ i 6= 1,

Ci := (σ1 − ρ1iσi )2 − σ2i (1− ρ21i ) .

If ∀ i 6= 1 we have

Ci > 1 or (Ci = 1 and µ1 ≥ µi )

then the estimator pLP has bounded relative error.

Some examples where this condition is satisfied: if for all i 6= 1,

ρ1i ≤ 0, or

2σ2i < σ21 or (2σ2i = σ21 and µi ≤ µ1).

35 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Numerical results

Example 1: Two Pareto(1,2) r.v.s with Clayton(0.9),p(100) = 2.00× 10−4.

pCMC p IS pLP

RE −0.50 −0.45 3.52× 10−4

Var 1.00× 10−4 8.28× 10−6 1.63× 10−14

Example 2: Two Normal(0, 10) r.v.s with Frank(0.9),p(35) = 4.65× 10−4.

pCMC p IS pLP

RE / −0.48 8.64× 10−6

Var / 2.17× 10−7 4.77× 10−14

36 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Numerical results 2

Example 3: Two Lognormal(0,1) r.v.s withAli–Mikhail–Haq(0.9), p(20) = 2.73× 10−3.

pCMC p IS pLP

RE 1.19 −0.49 1.66× 10−4

Var 5.97× 10−3 3.13× 10−5 5.78× 10−10

Example 4: A Pareto(1, 5), Gamma(1.5, 2.5), andLognormal(1, 0.5) with Clayton(0.5), p(25) ≈ 10−4.

pCMC p IS pLP

Var / 5.91× 10−14 7.56× 10−21

37 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Thank-you for your attention!

P.s. My PhD travel plans.

Leave Denmark on December 15 2015.

Return to Denmark July 2016 for 6 months.

Finish PhD in Australia in 2017.

38 / 37


Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy nuts and bolts

So

v∗ := arg minv

∫g∗(x) log

(g∗(x)

f (x ; v)

)dx

= arg maxv

∫g∗(x) log f (x ; v) dx

= arg maxv

∫1{S(x) ≥ `}f (x ; u)

p(`)log f (x ; v) dx

= arg maxv

Eu1{S(X ) ≥ `} log f (X ; v)

= arg maxv

Ew1{S(X ) ≥ `}L(X ; u,w) log f (X ; v)

where

L(x ; u,w) :=f (x ; u)

f (x ; v).

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Patrick J.Laub

Introduction

Background

Typicalsolutions

Maxima

Cross-entropy algorithm

So construct {v0, v1, . . . } s.t. vn → v∗ as such:

1 Set v0 ← u, and t ← 1.

2 Sample X1, . . . , XNi.i.d.∼ f (·; vt−1). Find the quantile

t ← Quantile0.75({S(X1), . . . ,S(XN)}) ∧ ` .

3 Set vt as the maximiser over v of

1

N

N∑i=1

1{S(Xi ) ≥ t}L(Xi ; u, vt−1) log f (Xi ; v)

4 If < ` got to Step 2, else return the estimate

p(`)← 1

N

N∑i=1

1{S(Xi ) ≥ `}L(Xi ; u, vt) .

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rare-event simulation for maxima of dependent random variables · 2020-03-11 · rare-event...

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