Consider a m-vector linear process

where are i.i.d. m-vector random variables with p.d.f. on , are matrices depending on a parameter vector .

Set

Assume the following conditions are satisfied

**A1** i) For some

where denotes the sum of the absolute values of the entries of .

ii) Every is continuously two times differentiable with respect to , and the derivatives satisfy

for where .

iii) for and can be expanded as follows:

where , satisfy

iv) Every is continuously two times differentiable with respect to , and the derivatives satisfy

for

**A2** satisfies

**A3** The continuous derivative of exists on .

**A4**

where .

From A1 the linear process can be expressed as

and hence

where

From A1 it can be seen that

Let and be the probability distributions of and , respectively. Then

For two different values , the likelihood-ration is

where

Define

and

Then the following theorem follows

**Theorem**

The sequence of experiments

is asymptotically equicontinuous on compact subset of . That is

(i) , the log-likelihood ratio admits, under the hypothesis (i.e. ), as , the asymptotic representation

where

and

with

(note that by Taylor expansion )

(ii) Under ,

(iii) and all , the mapping

is continuous w.r.t. the variational distance

A proof of the theorem will follow in a future post.

————————————————

References:

L. Le Cam, G.L. Yang (2000). Asymptotics in Statistics. Springer-Verlag, New York

B. Garel and M. Hallin (1995). Local asymptotic normality of multivariate ARMA processes with linear trend. Ann. Inst. Statist. Math. 47 551–579.

J.P. Kreiss (1990b). Local asymptotic normality for autoregression with infinite order. J. Statist. Plann. Inference 26 185–219.

A.R. Swensen (1985). The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend, J.Multivariate Anal., 16, 54-70.

A.W. van der Vaart (2000). Asymptotic Statistics. Cambridge University Press

M. Taniguchi, Y. Kakizawa (1998). Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag, New York

(main source)

### Like this:

Like Loading...

*Related*