Asymptotic Normality of Maximum Likelihood

Asymptotic Normality of Maximum Likelihood Estimators for Multiparameter Markov Chains

M. H. Al-Towaiq Jordan University of Science and Technology Mathematical Sciences Department Irbid - Jordan

Abstract

In this work, the asymptotic normality of Maximum likelihood Estimators (MLE) for multiparameter Markov chains is proved. The proof is based on the sketch of Rao(1973). Some details and gaps had to be lled out.

1. INTRODUCTION

Let fxk g, (k = 0; 1; :::) be a Markov chain with the state- space S = f1; 2; :::; mg with transition probability matrix P = [pij ]; i; j 2 S. Let xn+1 = (x0 ; x1 ; :::; xn ) be a sequence of observations of length n + 1 . The likelihood function based on this sample is given by : L =

x0 0

m Y

pij nij ;

i;j=1

1

i; j 2 S

(1)

where

0

=(

10 ;

20 ; :::;

m0 )

is the initial distribution (we assume

0

is non-

informative about the transition probabilities pij ), and nij is the frequency of one step transitions i ! j in the sample xn+1 .(c.f. Basawa and Prakasa,1980). The set of m2 transition frequencies fnij g forms a su cient statistic for the transition matrix P . The maximization of logL = log

x0 0

+

m X

nij log pij ;

(2)

i;j=1

with respect to pij 's subject to the constraints

Pm

i;j=1

pij = 1 , note that for

large n the rst term of eqn. (2) can be ignored. It is easily shown that the MLE of pij 's are p^ij = where ni =

Pm

j=1

nij ni

(3)

nij .

Suppose pij are known functions of unknown vector of parameters = ( 1; parameter

0 2 ; :::; k )

taking values in an open region in Rk . The MLE of the

can be obtaind by solving the system of equations @logL = 0 ; r = 1; 2; :::; k @ r or

m X

i;j=1

simultaneously for

nij

@pij ( ) 1 = 0 ; r = 1; 2; :::; k ; @ r pij ( )

1 ; 2 ; :::; k .

(4)

The existance of consistent roots of (4) is

discussed in (Al-Eideh et al.,1988).

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2. Asymptotic Normality of MLE Let ^ be a vector of MLE which can be found by solving the likelihood equation: m X

nij @pij ( ) = 0 ; r = 1; 2; :::; k: @ r i;j=1 pij ( ) To nd the asymptotic distribution of MLE, we use the concept of the rst order e ciency(c.f. Rao,1973,p.349). Let 0 0)

D0 n = ( ^

= ( ^1

^

10 ; :::; k

0 k0 ) :

^ is said to be rst order e cient if in probability p

n j Dn

BZn j! 0 as n ! 1 ;

where B is a k by k matrix of constants, which may depend on

(5) , and

0

Zn = (zn 1 ; zn 2 ; :::; zn k ) be a vector of derivatives, where zn r =

m 1 X i( n i;j=1 pij (

0)

@pij ( ) ) @ r 0) (

; r = 1; 2; :::; k: r = r0

Suppose that the following assumptions are satis ed(c.f. Rao,1973,p.360): (i) pij ( ) = pij ( ) for all i; j implies that

= .

(ii) pij ( ) admit the rst order partial derivatives which are continuous with respect to

at the true vector

0.

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(iii) Let F = [Irs ] be the Fisher information matrix which is assumed to be non-singular at

0

i.e. m X

i(

0)

@pij ( ) ) @ r i;j=1 pij ( 0 )

Irs =

(

(

r = r0

@pij ( ) ) @ s

(6) s = s0

Further denote by zr : zr =

m X

i(

0)

@pij ( ) ) @ r i;j=1 pij ( 0 ) (

; r = 1; 2; :::; k: r = r0

Then there exists a vector of MLE ^ = ( ^1 ; ^2 ; :::; ^k ) and in probability p

n j ( ^r

where F

1

I r1 z1

r0 )

I rk zk j! 0; r = 1; 2; :::; k as n ! 1;

:::

(7)

= [I rs ], which implies the MLE equation are rst order e cient

in the sense of eqn. (5). Similarly, eqn. (7) implies in probability: p

n j Dn

F

1

Zn j! 0 as n ! 1:

As a consequence of eqn.(5) and by central limit theorem (c.f., Rao,1973,p.349350), the distribution of

p

nBZn is a k-variate normal distribution with mean

0 and covariance matrix BF B 0 ,i.e. p

nBZn

Nk (0 ; BF B 0 ): 4

(8)

p

Also, by eqn.(5), the asymptotic distribution of If we apply F

1

nDn is the same as

p

nBZn .

instead of B in eqn. (8) , we obtain p

nF

1

Zn

So, the asymptotic distribution of

p

Nk (0 ; F n( ^

0)

1

):

is a k-variate normal distribu-

tion.i.e. p

n( ^

0)

Nk (0 ; F

1

):

References

Al-Eideh,B. , Abu-Salih, M., and Capar,U.,(1988), Consistency of Maximum Likelihood Estimators for Multiparameter Markov Chains, JIOS,vol.9,No.2, pp.391-404. Basawa, I.V. and Prakasa, R. (1980). Statistical Inference For Stochastic Processes , Academic Press, London. Rao,C.R.(1973), Linear Statistical Inference and Its Applications, 2nd edition, Wiley,New York.

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