Austral. J. Statist., 34(1), 1992,99-104

ROBUST ESTIMATION AND HYPOTHESIS TESTS FOR FIRST-ORDER THRESHOLD AUTOREGRESSIVE MODELS

P.M. KULKARNI~ University of South Alabama

Summary

Results of Petrucelli & Woolford (1984) for a first-order threshold au- toregressive model are considered from a robust point of view. Robust es- timators of the threshold parameters of the model are obtained and their asymptotic normality is proved. Testing the equality of the threshold pa- rameters is considered using the robust analogues of Wald and score test statistics. Limiting distributions of these statistics are given under both null and alternative hypotheses.

Key words: Robust estimation; M-estimation; Wald and score statistics; non-linear models; TAR (1) model.

1. Introduction

Non-linear models have been studied extensively by Tong (1983). In partic- ular a threshold model of order one (TAR (1)) has been considered by Petrucelli & Woolford (1984). They obtain necessary and sufficient conditions for the er- godicity of the process, and study the properties of the maximum likelihood estimators of the threshold parameters. It is well known that when there are outliers in the observations the method of maximum likelihood may seriously be affected and with adverse effects on the maximum likelihood estimators. Ro- bust procedures for estimating the parameters in such cases have been given by Huber (1981), Gastwirth & Rubin (1975), Basawa, Huggins & Staudte (1985), Kulkarni & Heyde (1987) and Kulkarni (1990) (also see the references therein). These authors, however, concentrate mainly on linear time series models.

In this paper we consider the TAR (1) model and obtain robust versions of some of the results in Petrucelli et d. (1984). We give robust estimators of the threshold parameters using a method described in Basawa et al. (1985) and prove the strong consistency and asymptotic normality of these estimators. Robust analogues of Wald and score test statistics, in time series context, have been given by Basawa et d. (1985). Here we discuss these two statistics for the hypothesis of equality of the threshold parameters and obtain their limit distributions under both null and alternative hypotheses. These results however reduce to the results given in Petrucelli et al. (1984) for the case of no outliers

Received March 1987; revised December 1990. 'Dept. Mathematics and Statistics, University of South Alabama, Mobiie, AL, USA 36688. Acknowledgements. The author thanks the referee for constructive criticism.

100 P.M. KULKARNI

in the observations. In Section 2 we obtain robust estimators and derive their limit distribution. Also strong consistency of the estimators is achieved. Then in Section 3 we carry out the tests for equality of the threshold parameters.

2. Robust Estimators and Their Limit Distributions

Let us consider the model

where Zr = max(Zi,O) and 2; = min(Zi,O), < 1, 92 < 1, 8182 < 1, 0 < u < 00, and the E S are independent identically distributed r.v.8 with zero mean and unit variance. Also, assume that the E S have a symmetric distribution with E ( E ~ + ~ ) < M.

For model (1) with u = 1 Petrucelli et al. (1984) have obtained the limiting distribution of the maximum likelihood estimators of 81 and 8,. It is known that when there are outliers in the observations maximum likelihood estimation can be seriously affected and have adverse results on the properties of the estimators. For details on robust inference see Huber (1981), Basawa et al. (1985) and the references cited therein.

The robust estimating functions are now obtained using the procedure given in Basawa et d. (1985, p.563), i.e. obtain the estimating functions using the usual maximum likelihood method and then modify these estimating functions by using certain bounded functions such as Huber's $ function. Thus robust estimating functions for 81, fl2 and u2 axe given by '

and

where c; = u-l(Z; - OIZi'_, - 822;l), b = E(Z)* and gi formally satisfies (1) with ci in (1) replaced by Z i = $ ( ~ j ) , $ being Huber's $ function defined by

where K is a fixed positive constant. Let Sn = (Snl, Sn2, &)', where T denotes the transpose.

ESl’lMATlON A N 0 ’JXSl’S FOR THRESHOLD MODELS 101

Theorem. If 81, 82 and a2 satisfj. the conditions in (1) then, as n 4 00,

R-3Sn d. N(o,F) , where

F = a-2diag(Fl,F2,Fs)

(5 )

with F1 = [E(8!)]E(g2+), F2 = [E($)]E(g2-), Fa = f ~ - ~ v a r ( < ~ ) , z2+ = ( z + ) ~ and z2- = ( z - )~ . Proof. Petrucelli et d. (1984) show that, with 81 and 82 satisfying the conditions in (l), Zi is ergodic and

n

i s 1

The process {&} is a bounded modification of {Zi} and satisfies the relation

2i = 812i+_, + e2Bi_, + o$(Q).

Since the parameters and the errors of the new process still satisfy the conditions of ergodicity imposed in Petrucellj et d. (1984), the process (2;) is ergodic and thus we have

n

i=l

Then using the dominated convergence theorem

-1 - i First we consider S:l = n 2Snl = &Xi, for Xi = Zi2:1/(nzC). It is easy to see that .!?;l is a zero-mean, square-integrable martingale. Also note that Z i is independent of ij’ for all i > j . Now, to show the asymptotic normality of Szl, it suffices to show that

V: = xE(XiZlFi-1) - o - ~ F ~ in probability, i

and

~ E [ X ~ Z I ( I X ~ ~ > y)] .--) o for a~ 7 > 0, i

where Fi-1 is the a-field induced by XI,. . . , Xi-1 and I denotes the indicator function. If the above conditions hold then an application of Martingale central

102 P.M. KULKARNI

limit theorem guarantees the asymptotic normality of Sil. Now,

5 [E(Xt)] [ q] ' Y

= [E($)E(~~?l)E($)E(fii-l)] 2+ * * 3 [n u yl" ,

Since Zj and g:, are bounded random variables, al l their moments are bounded. Moreover, the model satisfies the ergodicity (stationarity) conditions. Therefore, there exist constants Ml(K,6) and Mz(K,b) such that, for 2 5 6 5 4, E(Z:)< M1(K,6)andE(Z,sfi)<M2(K,6)fori= 1, ..., n. Thenitisclearthat

Hence (see Hall & Heyde, 1980, Corollary 3.1, pp.58-59),

n-'Sn1 N ( o , ~ - ~ F I ) .

Similarly we get

7t-'Sn, 2 N ( O , U - ~ F , ) (t = 2,3).

The proof follows after noting that the covariance terms are zero because of the fact that Zi.2; is zero for each i.

Now, the partial derivatives of the estimating functions are given by

and so on; here, Z ' = a$(c)/&.

ESTIMATION AND TESTS FOR THRESHOLD MODELS 103

Using the ergodicity of & and noting that Zi is uncorrelated with 2, for i > j , it can be verified that

where

0 (202)"E( 3)]

It follows from the above that the conditions of Basawa et al. (1985) are satisfied, and thus the following Lemma can be verified easily.

Lemma. e'n is strongly consistent for 6 and

where 8n is the robust estimate of 8 obtained as a solution of the estimating equation Sn = 0 and

c = ( G ~ F - ~ G ) - ~ .

For the case that u = 1 in (l), matrices F, G, and C are obtained by taking the first 2 x 2 submatrices of F, G, and C above. Hence the 2 x 2 matrix C for this case can be written as

For the sake of simplicity let us'assume that u = 1, and consider testing the hypothesis of equality of the threshold parameters 01 and 62.

3. Testing of Hypotheses

Let us consider testing the hypothesis H:61 = 6 2 , against a sequence of alternatives K: 62 - 61 = n-3 h, where h is a known constant. This is equivalent to testing H:a2 = 0, against K : Q ~ = n - j h using a1 as a nuisance parameter, where ct = ( a 1 , c t ~ ) ~ = (01,92 - 61).

104 P.M. KULKARNI

Under a, 6r is the estimator of a1 obtained by solving n

i=l

Let 6 1 and 6 2 denote the unrestricted robust estimators of a1 and a2 obtained via ( 2 ) and (3). Then using the Lemma above it is easily verified, as n + 00, that

where

c12 - G 1 3 Cll

C1z - G I Cll + cz2 - 2ClZ B ( 4 = [

with Cij, i , j = 1,2 as above.

testing El are given by (a) Analogue of Wald statistic:

The robust versions of Wald and score statistics (Basawa et al. (1985)) for

W n = 7~6.1 BG' (c)G~ where 6 is the robust estimator of a and B22(*) = C11 + C22 - 2C12.

(b) Analogue of score statistic: Rn = n-l Sn2(hH)FF1 (GH).Sn2 (hH)

where 3,2(-) is the robust score function corresponding to a2, iiH is the estimator of a under R and F2 is as in the theorem given above.

These two statistics are asymptotically equivalent. Both are distributed asymp- totically as a central x2 with 1 d.f. under H and as a non-central x2 with 1 d.f. and the non-centrality parameter Xz = h2BG1 under the alternative hypothesis. Remark. For the non-robust case, i.e. when $(u) = u, the results given above reduce to the corresponding results obtained by using the method of maximum likelihood.

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GASTWIRTH, J.L. & RUBIN, H. (1975). The behavior of robust estimators on dependent data.

HUBER, P.J. (1981). Robust Statistics. New York: Wiley. KULKARNI, P.M. (1990). Robust Durbin and portmanteau goodness-of-fit test statistics and

- & HEYDE, C.C. (1987). Optima robust estimation for discrete time stochastic processes.

PETRUCELLI, J.D. k WOOLFORD, S.W. (1984). A threshold AR(1) model. J. Appl. Probab.

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