Wang YaoDepartment of Statistics

Rutgers Universitywyao@eden.rutgers.edu

Mentor: Professor Regina Y. Liu

DIMACS -- July 17, 2008

Extreme Value Theory (EVT):Application to Runway SafetyExtreme Value Theory (EVT):Application to Runway Safety

Motivation Motivation

X

s

Q: How to determine s such that: P(X> s) .0000001=

allow multiple runway usage to ease air traffic congestion!

Cut-off point: Require all landings to be completed before the cut-off point with certain “guarantee”

Task:

(Extremely small!)

*

Difficulty (why Extreme Value Theory)Difficulty (why Extreme Value Theory)

• Extremely small tail probability e.g. p= 0.0000001

• Few or no occurrences (observations) in reality e.g. Even with sample size=2000

2000 0.0000001 0.0002 1

Possible Solution: Extreme Value Theory (EVT)

Difficulty: No observations!

Overview of EVTOverview of EVT

1: 2: : :n n n nX X X

1 2, , , :nX X X Random sample from unknown distr. fun. F

Order statistics

{ ; 1} { 0; 1}, . .n nb n and a n s t :( )

( ),n n nn

n

X bP x G x

a

Fréchet distribution heavy tail

Weibull distribution finite end point, e.g. uniform dist.

: Tail index ↔ Characterizes tail thickness of F

1

( ) ( ) exp( (1 ) ), ,1 0G x G x x x

1

(1 ) 0xx e for

0 :

0 :

0 :

Gumbel distribution in between, e.g. normal dist.

Extreme QuantileExtreme Quantile

1

(1 ( )) log ( ) (1 )t tt F a x b G x x

Take with , then n

tk

k n

( ) ( )nn nF a x b G x ( )F D G

11 1

1 ( ) log ( ) (1 )t t

t t

Y b Y bF Y G

t a t a

t tY a x b Let

1

p n n

k k

knp

x a b

For want to find s.t. ( )F D G px 1 ( )pF x p

ˆ

1ˆˆ ˆ

ˆp n n

k k

knp

x a b

Estimated by

Learning from Some Known DistributionsLearning from Some Known Distributions

• Generate random samples • For p= 0.001, estimate the p-th upper quantile

• Analysis: • Bootstrap Method:

a resampling technique for obtaining

limiting distribution of any estimator

e.g. Normal, Exponential, Chi-square,…

100n

P=0.001 Distribution Estimated True Error

Case A N(0,1) 4.37951 3.09023 1.28928

Case B Exp(1) 5.99578 6.907755 0.911975

Case C Chi-square(3) 14.1312 16.2662 2.135

pxpx

Small sample size Method of moments

P

Real DataReal Data

1k2k

*

underling distribution/model unknown!

px

Task: Applying Bootstrap Method to find a proper k(Bootstrap method: completely nonparametric approach and does not need to know the underlying distribution)

e.g. Landing distance:

** * * *** ** * ***X

• Analyze landing data collected from airport runways

• Apply bootstrap method with proper choice of k

• Determine the suitable cut-off point --

estimate the tail index , and extreme quantile

Remarks:

pxˆ

1ˆˆ ˆ

ˆp n n

k k

knp

x a b

Yet to be completedYet to be completed

Important project with real application.

Well motivated and requires new interesting statistical methodology

I learned some interesting new subjects, e.g. EVT, bootstrap method.

Statistics is a practical field and theoretically challenging.

Questions?

Questions?

Acknowledgment: Thanks to DIMACS REU!