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Extreme Value Type-I Distribution
(Gumbel s Extreme Value Distribution)
{ }
( ) exp exp
f x x x
x
m m
- for maximum values and + for minimum values
Y = (X – β)/ α → transformation
[ ]
{ }
f ( ) y = exp m y −exp m y
( ) { }
( ) { }
( ) exp exp
1 exp exp
F y y
y
(maximum)
(minimum)
σ
ˆ
β μ σ
μ σ
=
= +
(maximum)
(minimum)
(Double Exponential
Distribution)
The annual peak flood of a stream exceeds 2000m
3
/s
with a probability of 0.02 and exceeds 2250m
3
/s with a
probability of 0.
- Obtain the probability that annual peak flood exceeds
2500m
3
/s
Solution:
The parameters α and β are obtained from the given data
as follows
P[X > 2000] = 0.
P[X < 2000] = 0.
i.e., = 0.
Example-
(Gumbel s Extreme Value distribution)
4
( ) e
y
e
F y
−
−
e
y
e
−
−
Solving (1) and (2),
α = 358 and β = 603
Now P[X > 2500] = 1 – P[X < 2500]
= 1 – exp{-exp(-y)}
y = (x – β)/ α
P[X > 2500] = 1 – exp{-exp(-5.299)}
Example-1 (contd.)
6
Extreme Value Type-III Distribution
- Referred as Weibull distribution for minimum values
- pdf is given by (for minimum values)
- CDF is given by
- Mean and variance of the distribution are
μ = E[X] = β Γ(1+1/α)
σ
2
= Var(X) = β
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
7
( )
{ }
1
f ( ) x x exp x x 0; , 0
α
α α
α β β α β
− −
( ) { }
F x ( ) 1 exp x x 0; , 0
α
= − − β ≥ α β >
Weibull Distribution
α
→ transformation
- Mean and variance of the 3-parameter Weibull
distribution are
μ = E[X] = ε + (β - ε) Γ(1+1/α)
σ
2
= Var(X) = (β- ε)
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
9
Weibull Distribution
10
α → Shape parameter
β → Scale parameter
α=
β=
x
f(x)
α=
β=
α=
β=
α=
β=
α=
β=
Example-2 (contd.)
12
σ
2
= Var(X) = β
2
{Γ(1+2/α) – Γ
2
(1+1/α)}
2
2 2
2
2 2
2 2 2
×
β
α α
μ
α
μ = β Γ(1+1/α)
μ /β = Γ(1+1/α)
Substituting the
sample moment
values
P[X < 0.1] = F(0.1)
P[X < 0.1] = 0.
Example-2 (contd.)
13
( )
{ }
( )
{ }
( ) 1 exp
(0.1) 1 exp 0.1 0.
F x x
F
α
Parameter Estimation
- f(x) : pdf and F(x) : CDF
- In general, f(x) and F(x) are also functions of parameters
f(x; θ
1
; θ
2
………… θ
m
) or F(x; θ
1
; θ
2
………… θ
m
1
, x
2
, …… x
n
is available
: Estimate of θ
i
: a function of the sample
- : random variable since it is a function of the random
sample.
- Two important properties of the estimators
§ Unbiasedness
§ Consistency
15
i
i
Parameter Estimation
Unbiased estimate:
- An estimate of a parameter θ is said to be unbiased if
E( ) = θ.
- The bias is given by E( ) – θ
- An estimator is unbiased does not guarantee that an
individual is equal to θ or even close to θ
- The average of many independent estimates of θ will be
equal to θ.
16
Parameter Estimation
Methods of estimating parameters from samples of data:
- Method of matching points:
- Method of moments
- Method of maximum likelihood
18
Method of matching points
- Not a commonly used method
- Can produce reasonable first approximations to the
parameters
- Use the data set to obtain probabilities and estimate the
parameters
- Simple and approximate method
19
Method of Moments (MoM)
- One of the most common used methods for estimating
the parameters
- Equate the first m moments of the population to the
sample estimates of the first m moments
- Results in m equations; solve to get the m
unknown parameters of the distribution.
21
Example-
Obtain the parameter λ using method of moments for
the pdf
The first moment is
22
x
f x e x
λ
−
0
x
x e dx
λ
∞
−
∫
0 0
2
0
0
x x
x x
x
xe e
dx
xe e
e x
λ λ
λ λ
λ
∞
∞ − −
∞
− −
∞
−
