Estimation of Parameters of Truncated Binormal · PDF fileEstimation of Parameters of...

Estimation of Parameters of Truncated Binormal Distribution

By

Dr. C. D. Bhavasar & Mr. Parag B. Shah Dept. of Statistics, Dept. of Statistics, Gujarat University, H.L.College of Commerce Ahmedabad-380009. Ahmedabad-380008 GUJARAT. GUJARAT INDIA INDIA [email protected] [email protected]

Abstract : In this paper we define doubly truncated binormal distribution. We have estimated the parameters of

this distribution by method of moments. Mean deviation from mean in general form and recurrence

relation has been obtained. Results regarding singly truncated binormal distribution have been shown

as a particular case of doubly truncated binormal distribution.

Keywords :

Doubly truncated binormal distribution, Method of moments, Recurrence relation.

International Journal of Advancements in Research & Technology, Volume 3, Issue 4, April-2014 ISSN 2278-7763

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1 Introduction :

The binormal distribution was first introduced as the joint half Gaussian distribution by Gibbons

and Mylorie (1973), in their study of experimental impurity profiles in ion-implanted amorphous

targets. It has also been used by Toth and Szentimrey (1990) to model temperature distribution in

climatological studies.

Johnson and Kotz (1970) have discussed about estimation of Truncated Normal distribution.

S.B.Nabar and S.P. Barpande (2001) have given a note on the m.l.e. of the binormal

distribution BIN (σ µ κσ, , ). Cohen, A.C. (1949) has discussed about the estimation of mean and

standard deviation of truncated normal distribution.

We have defined doubly truncated binormal distribution and have obtained it’s mean and variance

by method of moments in section – 2. In this section we have also obtained Mean deviation from

mean. In section – 3 a general recurrence relation for the moments of distribution has been derived.

Singly truncated binormal distribution and it’s estimation has been shown as a particular case of

doubly truncated binormal distribution in section – 4.

2. Density function and estimation of parameters

The density function of binormal distribution BIN ( µ σ σ, ,1 2 ) is given as :

2 1 -1 - exp ,

( 21 =

2 1 -1 - exp , ( 21

2x

x)2 1

f ( x )2

xx

)2 2

µ µπ σ σ σ

µ µπ σ σ σ

� � �� ≤� � � +� � �� >� � � +� � ��


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Now, the density function of a doubly truncated binormal distribution is given as :

-12 22 1 -1 - 2 1 -1 -

exp 2 exp ( ) 2 ( ) 2

1 2 1 1 2 1

( )

x xdx

A

f x

µµ µπ σ σ σ π σ σ σ

�+ +

=

� � � � � ��

2 22 1 -1 - 2 1 -1 -

exp 2 exp ( ) 2 ( ) 2

1 2 2 1 2 2

A x

x xdx

µ

µ µπ σ σ σ π σ σ σ

< ≤

+ +

� � � ��

-1

B

x B

µ

µ

�

< ≤

��

� ��

�

��

………….(2.1)

(1) can be rewritten as

2-2 1 -11 exp 1 2( ) 11 2

A< ( ) = 2

-2 1 -11 exp 2 2( ) 21 2

xC

xf x

xC

µπ σσ σ

µ

µπ σσ σ

� ��

� � � � � ��

�

� ��

� � � � ��

�

−+

≤

−+

< x Bµ

��

��

��

≤

………….(2.2)

where

1

21 -2 -1 2 exp ( ) 2 11 2

4 11 ( )( ) 21 2

xC dxA

A

µ µπ σσ σ

σσ σ

� ��

� ��

= �+

∗= − Φ+

and

2

21 -2 -1 2 exp ( ) 2 21 2

4 12 ( ) ( ) 21 2

B xC dx

B

µπ σσ σ µ

σσ σ

� ��

� ��

= �+

∗= Φ −+

with

21

21* , * ( )

2-1 2

xxA BA B and x e dx

µ µσ σ π

−− −= = Φ = �

∞ .......... (2.3)


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Estimation of parameters

Mean and variance of the distribution has been obtained using method of moments.

1 11 2

2 2

( ) ( ) ( ) ( )

( *) ( *)

B B

A A

E x f x dx x f x dx x f x dx

g A g B

µ

µ

µ

σ σµ

= = = +

= + +

� � �

.......... (2.4)

and

2 2

2 2 2 22 1 2 1 2

1 1 2 1 2 22 22

( ) ( )

( *) ( *) ( *) ( *)

B

A

E x x f x dx

g A g B g A g Bσ σ σ σ

σ µ σ µµ

=

+= + + + + +

�

.......... (2.5)

Hence, Variance of x is given by

( ) ( )2 2 2

1 2 1 2 1 22 1 2 1 1 1

2

( ) 2 ( *) ( *) 2 ( *) ( *) ( *) ( *) 2 4 4 2

V x g A g A g B g B g A g Bσ σ σ σ σ σ+ +

= + − + − −

.......... (2.6)

where

1 2

12

2( ) -1 2 ( ) 1( ) , ( ) and ( )

1/2 - ( ) 1/2 - ( ) 2

xz x x z xg x g x z x

x xe

π

π

−= = =

Φ Φ

Mean deviation from mean : In this section, we have obtained mean deviation from mean for the doubly truncated binormal distribution

Mean deviation from mean ( )µ is given as

1 21 1 1

12 1

*2 1 1

22 2

2 2

2 2

( ) ( )

( *, *) * ( *, *)

+ ( *, *) ( *, *)

B

A

M E x x f x dx

g A g B

g A g B

µ µ

σ σµ µ µ µ

σσµ µ

= − = −

= +

+

�

.......... (2.7)


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where

1 2 1 21 2

( ) ( ) ( ) ( )( , ) , ( , ) and * , *

1/ 2 ( ) 1/ 2 ( )

x y z x z yg x y g x y

y y

µ µ µ µµ µσ σ

Φ − Φ − − −= = = =

− Φ − Φ

3. Recurrence relation The general recurrence relation for the moments of doubly truncated binormal distribution is

given as

-1

1 2

1 2

21

1( 2)1

2(

111 ( 1)

2

1 ( 1) = ( *)

2 2 2

= 2

( ) ( ) + ( )

where

B Br r r

rA A

r r

rr

r r

r

rA z A

x f x dx x f x dx x f x dx

r

and

µ

µ

σ σµµ µ σ µπ π π

µµ µπ

µ

µ µ

µ−

−−

− −′ ′ + − −

′ ′

′ = =

′ ′= +

′

� � �

21 12 2

1) 2 2( 2)

( 1)( *)

2 2r r

r rB z B

rσ σσ µ

π πµ− −

− −

−− + − ′

11 1

1 1 2 1 2

2 21 2

1( 2) 2( 2)

( ) ( ) z( )2 2

( 1) ( 1)

2 2

rr r

r r

r r

A z A B B

r r

µ µµ σ σ σ σπ π

σ σµ µ

π π

µ−

− −−

− −

∗ ∗′∴ = − − + −

− −′ ′− −

′

.......... (3.1)

4. Singly truncated binormal distribution as a particular case. In this section we obtain the results of signly truncated binormal distribution as a particular case of doubly truncated binormal distribution.

In (2.1) by taking B = ∞ , lower truncated binormal distribution can be obtained as a particular case of doubly truncated binormal distribution. Its p.d.f is given by

1

2-2 1 -1 exp , A <

2( ) 11 2( ) =

2-1 -1 exp , <

22 22

C 1- x x

f x

x x

µ µπ σσ σ

µ µσπσ

��

� ��

� ��

≤+

≤ ∞��

.......... (4.1)


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The Mean and Variance of the above distribution are given by

( )

1 21

2 2 2 21 2 1 2 1 2

2 1

*

* *1

2

2 4 2

( ) 2

2 ( *) ( ) ( )2 2

Mean g A

Variance g A g A g A

σ σµ

πσ σ σ σ σ σ

π π

= + +

+= + − − −

.......... (4.2)

Similarly in (2.2) by taking A = − ∞ , upper truncated binormal distribution can be obtained as a particular case of doubly truncated binormal distribution. Its p.d.f is given by

2

11

2

12

1 2 2

1 -1 - exp , -

22( )

2 1 -1 - exp , <

2

xx

f x

xC x B

µµ

σπσ

µµ

π σ σ σ−

∞ < <

=

<+

� ��

� ��

��

.......... (4.3)

The mean and variance of the above distribution are given as

1

2

22

2

*2 = + + ( )2 1

2 2 2* *1 2 1 1 2 Variance 2 ( *) ( ) ( )2 1 12 4 22

Mean g B

g B g B g B

π

σµ

σ σ σ σ σσππ

σ

� � � �

+= − − −+

.......... (4.4)

Note : We have also written a paper on “Estimation of Parameters of Truncated Bivariate Binormal distribution and have send it for publication.

REFERENCES:- 1. Cohen, A. C. (1949). On estimating the mean and standard deviation of truncated normal

distributions, Journal of the American Statistical Association, 44, 518-525.

2. Gibbons, J.B. and Mylorie, S. (1973). Estimation of impurity profiles in ion-implanted amorphus targets using joined half gaussian distributions Appli. Phys. Lett., 22, 568-569.

3. Johnson and Kotz (1970). "Continuous Univariate distributions". John Wiley & Sons, Inc.

4. S.B.Nabar and S.P. Barpande (2001). A note on the maximum likelihood estimator of the binormal distribution Bin (�, �, ��), Journal of the Indian Statistical Association, 39, 189-175.

5. Toth and Szentimrey (1990). The binormal distribution: a distribution for representing asymmetrical but normal like weather elements, J. Clim., 3, 128-136.


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