Unformatted text preview: FD", col="blue") Background on Extreme Value Analysis (EVA)
Simulations
# Simulate maxima from samples of size 30 from
# the Unif(0,1) distribution.
Umax < matrix( NA, 30, 1000)
for( i in 1:1000) Umax[,i] < − runif( 30)
Umax < − apply( Umax, 2, max)
hist( Umax, breaks="FD", col="blue")
# Simulate 1000 maxima from samples of size 30 from
# the Fréchet distribution.
Fmax < − matrix( NA, 30, 1000)
for( i in 1:1000) Fmax[,i] < − rgev( 30,
loc=2, scale=2.5, shape=0.5)
Fmax < − apply( Fmax, 2, max, finite=TRUE, na.rm=TRUE)
hist( Fmax, breaks="FD", col="blue") Background on Extreme Value Analysis (EVA)
Simulations Last one
# Simulate 1000 maxima from samples of size 30 from
# the exponential distribution.
Emax < matrix( NA, 30, 1000)
for( i in 1:1000) Emax[,i] < rexp( 30)
Emax < apply( Emax, 2, max) # This time, compare with samples from the
# exponential distribution.
par( mfrow=c(1,2))
hist( Emax, breaks="FD", col="blue")
hist( rexp(1000), breaks="FD", col="blue") Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Let X1, . . . , Xn be a sequence of independent and identically distributed (iid) random variables with common distribution function,
F . Want to know the distribution of
Mn = max{X1, . . . , Xn}.
Example: X1, . . . , Xn could represent hourly precipitation, daily ozone
concentrations, daily average temperature, etc. Interest for now is in
maxima of these processes over particular blocks of time. Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
If interest is in the minimum over blocks of data (e.g., monthly minimum temperature), then note that
min{X1, . . . , Xn} = − max{−X1, . . . , −Xn}
Therefore, we can focus on the maxima. Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn ≤ z } =
indep.
= Pr{X1 ≤ z, . . . , Xn ≤ z }
Pr{X1 ≤ z } × · · · × Pr{Xn z } ident. dist.
=
{F (z )}n. Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn ≤ z } =
indep.
= Pr{X1 ≤ z, . . . , Xn ≤ z }
Pr{X1 ≤ z } × · · · × Pr{Xn ≤ z } ident. dist.
=
{F (z )}n.
But! If F is not known, this is not very helpful because small discrepancies in the estimate of F can lead to large discrepancies for
F n. Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
Could try to derive the distribution for Mn exactly for all n as follows.
Pr{Mn ≤ z } =
indep.
= Pr{X1 ≤ z, . . . , Xn ≤ z }
Pr{X1 ≤ z } × · · · × Pr{Xn ≤ z } ident. dist.
=
{F (z )}n.
But! If F is not known, this is not very helpful because small discrepancies in the estimate of F can lead to large discrepancies for
F n.
Need another strategy! Background on Extreme Value Analysis (EVA)
Extremal Types Theorem
If there exist sequences of constants {an > 0} and {bn} such that
Mn − bn
Pr
≤z
an −→ G(z ) as n −→ ∞, where G is a nondegenerate distribution function, then G belon...
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 Spring '08
 Staff
 Normal Distribution, Fort Collins, qq qq qq

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