Statistical Parameter Estimation - Advanced Hydrology - Lecture Slides, Slides of Aeronautical Engineering

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Lecture 7: Statistical parameter estimation

Module 7

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Parameter Estimation

Methods of Parameter Estimation

1. Method of Matching Points

2. Method of Moments

3. Maximum Likelihood method

θ

θ

θ

θ θ

Population Parameter

Sample Parameter

Unbiased estimation of parameter:An estimate of a parameter

is said to be unbiased estimate, if E( )

i

i

i

i i

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2) Method of Moments

θ θ θ θ i j i j

1 2 n

Given a function f( ,......., ,x) and values ,.......,

we need to find x ,x ,..........x

Generate number of equations by taking moments of the distribution

Take, any distribution, like f(x

θ

θ

θ α α

− −

−

2

1

2

2

( x )

1 2 2 2

2

) (2 ) e - < x < +

Take the 1st moment Mean about the origin

θ α

θ

α

θ α

θ

α

μ θ

θ

− −

−

−

− −

−

−

= ∏

∏

∫ ∫

∫

2 1 2 2 2 1 2 2 ( )

1 2 2 2

2

( )

1 2 2 2

2

( ) (2 )

(2 )

x

x

x fx dx x e dx

xe dx

E(X) = =

=

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2) Method of Moments Contd...

α

α

−

−

∫

2

2

2 1 2

2

y

E(X) = y e dx

1

2

2 1

2

substituting, y = ( )

x =

dx =

x

y

dy

α α

α α

α

α

− −

− −

−

−

∫ ∫

∫

2 2

2

2 2

2 1

2

1

y y

y

e ydy e dy

e dy

E(X) =

E(X) = +

1

1

1

E x ( )

[As odd multiplier, h(-y) = -h(y)

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3) Maximum Likelihood method

1 1

1 2 3

We have the following, ( ; ) ( ; ) ( ; )

i i

Sample

x

f x f x f x

x

1 3

Product of ( ; ) ( ; ) is "likelihood " L

If L( ; ) ( ; ), then is the estimate preferred,

which maxmizes the likelihood function.

θ θ

θ θ θ

× × ≈

∗

∗



i i

f x f x

x f x

( ; θ ) → evaluated at x = x i i

f x pdf

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3) Maximum Likelihood method Contd…

{ }

1 2

1 2 ) 1

1 2

1 2 3

(

( ) ; 0 is a parameter

, , , Sample available;

L = ( , ) ( , ) ( , ) ( , )

= = (formulation of like

n

n

i

n i

x

n

n

x x x

x

n x^ x^ x n

f x e x

x x x

f x f x f x f x

e e e

e e

β

β β β

β

β

β β

β β β β

β β β

β β

=

−

− − −

−

− + + +

∴ × × × ×

× × ×



lihood function)

Because ln(L) is an increasing function of L, it reaches maximum value

ln( )

at the same pt., as ln(L) does, 0

(When there is no other method feasible, this method is best one)

θ

L

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3) Maximum Likelihood method Contd…

=

 ^  

2

2

1 2

2

2

( ) exp

[Take, as parameter not S.D. and also]

L = ( , , ) ( , , ) ( , , ) = exp

ln( ) ln(2 )

i

n (^) n

i

i

x

f x

x

f x f x f x

n x

or L

=

1

2

1

ln( ) ln( )

0 set & ,Now

ln( )

n

n

i

i

i

L L

or

L x

X

1

n

i

i

x

or x

n

μ

=

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3) Maximum Likelihood method Contd…

2

2

1 2

2

2

1 1 ( ) exp

2 2

1 1 ( , , ) ( , , ) ( , , ) exp

2

2

1 ln( ) ln(2 )

2 2

μ

σ σ

σ μ

μ

μ σ μ σ μ σ

σ

σ

μ

σ

σ

   −  − =    

∏      

 

   −    − ∴ (^)      

 ^   ∏      

 −  − = − ∏ −  

 

[Take, as parameter not S.D. and also]

L =   =

i

n (^) n

i

x

f x

x

f x f x f x

n x

or L

1

2

1

ln( ) ln( )

0

ln( )

0

( ) 0

μ σ

μ σ

μ

μ σ

μ

=

=

∂ ∂

= =

∂ ∂

∂  − 

= =  

∂  

∴ − =

∑

∑

∑

set & ,Now

n i n i i i

L L

or

L x

X

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 In hydrology, most of the phenomena are random in nature.

E.g. rainfall-runoff model

 Random variables involved in a hydrological process may be dependent or

independent.

 The ‘random variables’ X & Y are ‘stochastically independent’ if and only if

their ‘joint density’ is equal to the product of ‘marginal density functions’.

 Joint density function : Simultaneous occurrence

 Marginal density function : Distribution of one variable irrespective of the value

of the other variables

 Conditioned distribution: Distribution of one variable conditioned on the other

variable.

Highlights in the Module

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 Measures of Central Tendency:

Mean

 Arithmetic average (for sample)

Mode

Median

 Measures of Spread or Dispersion:

Range [(xmax-xmin)]

Relative Range [=(range/mean)]

Variance

Highlights in the Module Contd…

Standard deviation,

Coefficient of variation

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 Methods of statistical parameter estimation

1. Method of Matching Points

2. Method of Moments

3. Maximum Likelihood method

Highlights in the Module Contd…

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