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The International Journal Of Engineering And Science (IJES) || Volume || 3 || Issue || 4 || Pages || 29-35 || 2014 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805

A note on estimation of parameters of multinormal distribution with constraints 1,

Dr. Parag B. Shah, 2, C. D. Bhavsar

Dept. of Statistics, H.L. College of Commerce, Ahmedabad-380 009. INDIA Dept. of Statistics, School of Science, Gujarat University, Ahmedabad-380 009. INDIA

-------------------------------------------------------ABSTRACT--------------------------------------------------Mardia et al (1989) considered the problem of estimating the parameters of nonsingular multivariate normal distribution with certain constraints. Nagmur (2003) considered the problem of estimating the mean sub-vector of non-sigular multivariate normal distribution with certain constraints. In this paper we try to estimate mean sub-vector under some different constraints and submatrix of ∑ with certain constraints for a nonsingular multivariate normal distribution.

Keywords: Likelihood Function, Maximum Likelihood Estimator, Non-singular Multivariate Normal Distribution, Constraints. ---------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 31 March 2014 Date of Publication: 15 April 2014

----------------------------------------------------------------------------------------------------------------I. INTRODUCTION Mardia et al (1989) considered the estimation of parameters of non-singular multivariate normal distribution with and without constraints. Nagnur (2003) tried to obtain the Mle of sub mean vector of the distribution which can be useful in some practical problems. Mardia et al (1989) considered two type of constraints on the mean vector  . = k i.e.  is known to be proportional to a known vector  . For example, the elements of x could represent a sample of repeated measurements, in which case  = k1 ii. Another type of constraint is R = r , where R and r are pre-specified The first type of constraint was considered by Nagnur (2003) for sub mean vector of the distribution. In this paper, we consider type of constraint for sub mean vector and give the explicit expression for the estimators. Mardia et al (1989) also considered constraint on variance-covariance matrix , viz =k0 where 0 is known. We consider constraint on sub matrix of  and obtain its estimator with constraints. i.

2. ML Estimator of µ :

Let x1 , x 2 ,…, x n be n iid observation from Np , population, where  is positive definite matrix. Suppose that x ,  and  are partitioned as follows :

 x   x   1  ,  x 2 

11 : r x r, 22

       1  ,  2 

      11 12      21 22 

n

: s x s with r + s = p. Let

x and S    x i  x  xi  x i1

vector and SSP matrix based on the sample observations

'

be the mean

x1 , x2 ,...........xn . The sample mean vector and

SSP matrix is partitioned as

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A note on estimation of parameters of multinormal distribution with constraints

 x1  x     x 2 

 S11 S12  S   S S   21 22 

where S11 : r x r , S 22 : s x s ,with r + s = p. Our problem is to estimate  : r  1 under the constraint R = r , where R and r

are pre-specified.

Maximizing the log likelihood subject to this constraint may be achieved by augmenting the log likelihood with a Lagrangian expression, i.e. we maximize L+ = L - n ' (Rµ - r)

(2.1)

where  is a vector of Lagrangian multipliers and L is given by

'

L   2 log 2  n2 log   12 tr1 S  n2  x   1  x   np

Case-1 ( Known ) With  assumed to be known, to find m.l.e.'s of

L

 1

 1 we are required to find  for which the solution to

 0 satisfies the constraint R  1 = r. 

Observe that L

L  

(2.2)

can be expressed as

np n n log 2  log   12 tr  1 S  2 2 2

{ x     '

1

11

 x   } 1

'  ' n   2 x 1  1 12  x 2   2    x 2   2  22  x 2   2   n ' R1  r  (2.3) 2  

Now

L  n11 x    n12 x    nR ' l 0  2 2 1 1  1

(2.4)

and

L  n22 x    n21 x    0  2 2  1 1  2

(2.5)

From (2.5) we get

x

2



Substituting for

2

    

x

22

2

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1

 21 x 1   1

(2.6)

  2 in (2.4), we get

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A note on estimation of parameters of multinormal distribution with constraints

 

11 12 22 n  x     n  1  1 

Since

  11

we get n

12

1

11

x

Thus

1

x

1

 

22 1

21

x

1

  1  nR '   0

1

  .11 . 21

.,

  1  nR '  ..

(2.7)

  1 =  11 R '  ,

Pre-multiplying by R gives we take,

1

  R11 R '

 Rx

1

 

 r  R11 R '  if the constraint R  1 = r is to be satisfied. Thus,

  Rx1  r  ,so 1

1  x1  11 R ' 

(2.8)

From (6), the ML estimator of  2 is

 

 2  x2  

22 1

21

 x    1

Case 2 ( ∑ unknown) : When the covariance matrix ∑

(2.9)

1

is not known, we have to estimate  ,  and 1 2

∑ using the likelihood

function (2.3) The ML estimator of ∑ is      x    x    '   i   i  

*

n   S , where S * =   n i 1

(2.10)

 is the ML estimator of  under the restriction R 1 = r. and  To estimate (  1 ,  2 ) the likelihood equations are given by (2.4) and (2.5). Now we have



*      S  S + x  x  n n

'

(2.11)

From (2.11), we have 1

1

 S   I  n where

 x   x  '

(2.12)

  ij         ij .   1 ,  2  is the solution of (2.4) and (2.5) after replacing  by   

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A note on estimation of parameters of multinormal distribution with constraints Since  and  1 2

have to satisfy these equations, from (2.12) we get the equations

 11 S / n   '   12 S / n  '  1 '  11 1 21

(2.13)

 21 S / n    22 S O 11

(2.14)

21

/n

From above equations it follows that

  1 1   11  12  22   21   S11   '   '         n  1 1        =

' 1

1  11 

(2.15)

Hence, from (2.7), we have

x

1



1

  S R'  n

(2.16)

11

Pre multiplying (2.16) by R we get

 Rx

1

 r =  RS11 R '   n

(2.17)

provided the constraints R = r is to be satisfied. Thus we have

  n RS11 R '

1

 Rx  r 

(2.18)

1

1   1  x  S11 R ' RS11 R ' Rx  r

1

The ML estimator of  is

1     2  x 2   S 22  S 21 x1   1

1

(2.19)

3. ML Estimators of ∑ : According to Mardia et al (1989), the likelihood function of p-variate normal distribution with constraints   k  0 , where  0 is known, is

2n 1l  x,  , k    p log k  log 2 0  k 1 1

where   tr0 S  x  

0

 '  x    is independent of k. 1 0

0

Our problem is to obtain estimate of k for the constraint

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(3.1)

 11  k 110 , where  110 is known and

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A note on estimation of parameters of multinormal distribution with constraints

 11 :

r

x

0  k

r

is

a

submatrix

.

of

  1 1 0  2 2  2 1  1110  1 2  2 21   

r

j0

we

 1  C j    k 

let

s j

 k 110  0     21

  

 12  ,  22 

then

(3.2)

1 1

1

where C j  tr j  21 110  12  22 Further, by considering

s

If

 01 as

1 1  1 11012 22.10  k  1   22.10 

 1 110.2  0     1  1  22 21 110.2 1

where

1 1 1  110.2  k 110   12  22  21 , 22.10  22  21110 12 ; we get k 1 1 1 1 1 1 tr  01 S  tr  221 S 22  tr  110 S11  110 12  221 S 21   221  21110 S12   22  21110 S 22   O  k 2  k (3.3) Since,

1 110.2

1 1  110  O  k 2  k

(3.4)

and 1  22.10   221 

1 1 1  22  21110  O  k 2  , k

(3.5)

We have

 x   '   x    1 0

x

2

 

 

 

 

1 1  2 221 x  2 ' 1k x  1 ' 11 x  1  x  2 ' 221 21 110 2x1  1  x 2  2   Ok 2 

2

1

1

2

(3.6) Using results (3.2), (3.3) and (3.6) in (3.1) we get

 

l  = 2n 1l x,  , k  Const.   p  r  log k  1  * O k 2 k

(3.7)

where 1 1 Const   p log 2  log  110  log  22  2 log  21 110  12  22 , 1

*  b

s 1

and

bs 1

1

 tr  22 S 22  x 2   2 '  22 x 2   2

1 1 cs 1 trs 1   21110 12  22    1 1 1 cs 12  22  21110 

Thus, if  is known, then the mle of

1

(3.8)

k is

 kˆ  pr *

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(3.9)

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A note on estimation of parameters of multinormal distribution with constraints If  is unknown and unconstrained, then the mle's of  is x and that of 1 b  tr  22 S22 kˆ  s 1 pr

Note that for the constraint  22 and that of for unknown  is

k is  * p  r , together these gives (3.10)

 k 220 , where  220 is known, the mle's of kˆ for known  is   p  s 

 *  p  s  where

1 1   dr  1  tr11 S11   x1  1  ' 11  x1  1  ,  *  dr 1  tr111S11 with

1 1 1 1 d r 1   12  22  21 11 trr 1  12  22  21 11

4. Illustrative Example : 1. Estimation of sub mean vector with constraint R 

1

r.

For 47 female cats the body weights (kgs), heart weights (gms), lungs weights (gms) and Kidney weights (gms) were recorded. The sample mean vector and covariance matrix are

 x1   2.36        x 9.20   x2  1 x         x 3   x 2   3.56    x   2.76   4

S S   11 S  21

 0.0735   0.1937 12    S   0.2156 22    0.3072

S

0.1937 0.2156 0.3072   1.8040 0.1969 0.1057  0.1969 2.1420 0.1525   0.1057 0.1525 2.0136 

Note that here x and S are unconstrained mle's of  and . However, from other information we know that

 1   2 .5      2    1   10 . 0   = =   3    2   3 . 5  2 .5       4

      

and

0.15620 0.21745 0.31033   1.56200 0.17767 0.07322  0.17767 2.15436 0.17338   0.07322 0.17338 2.04885  With above given information we estimate sub-mean vector  1 = 2  1 under the constraint R 1  r where   11    21

 0.07810    0.15620 12   22   0.21745   0.31033

R : 2 x 2 and r : 2 x 1 are pre-specified as follows :

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A note on estimation of parameters of multinormal distribution with constraints

 0.45000 R    0.12500

0.12500  and 0.91875 

For unknown , we have

  n RS11 R'

1

 2.37500  r    9.50000 

2.25821   Rx  r    0.056431   1

 

 2.49993  1  x  S11 R '     1 10.00096   

and

 

2  x  S 2

22 1

 

21

S

   x   1 1   

  

=

x  S S111  x  1  2 21 1

=

 3.84642     3.10968 

For known , we have

 2.0804    2.4999    3.9219  ,      ,  1    2   0.1706   3.1205  10.0006  2. Estimation of sub-matrix 11 of  with constraint 11 = k110. 

   0.1 0.2  For 110 =   , we have for known  k = 0.49426 and for unknown , we have k = 0.4886  0.2 2.0 

REFERENCES [1]. [2]. [3].

Anderson , T.W. (2003) “An introduction to multivariate Statistical Analysis”. (3rd edition), Wiley Series in probability and Statistics. Mardia, K.V., Kent, J.T. and Bibby, J.M. (1989). “Multivariate Analysis”. Academic Press, London. Nagnur, B.N. (2003) : A short note on estimating the mean vector of a multivariate non-singular normal distribution. ISPS Vol-7 83-86.

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