Asymptotic properties of the least squares estimators of a two ... - IITK

Metrika (1998) 48: 83±97

> Springer-Verlag 1998

Asymptotic properties of the least squares estimators of a two dimensional model Debasis Kundu1,*, Rameshwar D. Gupta2,** 1 Department of Mathematics, Indian Institute of Technology Kanpur, Pin 208016, India (e-mail: [email protected]) 2 Department of Mathematics, Statistics and Computer Science, The University of New Brunswick, Saint John, P.O. Box 5050, Canada, E2L 4L5 (e-mail: [email protected]) Received March 1997

Abstract. We consider a particular two dimensional model, which has a wide applications in statistical signal processing and texture classi®cations. We prove the consistency of the least squares estimators of the model parameters and also obtain the asymptotic distribution of the least squares estimators. We observe the strong consistency of the least squares estimators when the errors are independent and identically distributed double array random variables. We show that the asymptotic distribution of the least squares estimators are multivariate normal. It is observed that the asymptotic dispersion matrix coincides with the Cramer-Rao lower bound. This paper generalizes some of the existing one dimensional results to the two dimensional case. Some numerical experiments are performed to see how the asymptotic results work for ®nite samples. Key words: Strong consistency, texture classi®cation, statistical signal processing 1. Introduction We consider the following two dimensional model: y…m; n† ˆ

p X kˆ1

Ak0 cos…mlk0 ‡ nmk0 † ‡ X …m; n†;

for m ˆ 1; . . . ; M;

n ˆ 1; . . . ; N

…1†

* Part of the work has been supported by a grant of the Department of Science and Technology, Government of India, Grant No. SR/OY/M±06/93. ** Part of the work has been supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. OGP-0004850.

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D. Kundu, R. D. Gupta

where Ak0 's are unknown real numbers, lk0 's, mk0 's are unknown numbers, where lk0 A …ÿp; p† and mk0 A …0; p†. X …m; n† is a two dimensional (2-D) sequence of independent and identically distributed random variables. `p' is assumed to be a known integer. Given a sample y…m; n†; m ˆ 1; . . . ; M; n ˆ 1; . . . ; N, the problem is to estimate Ak0 's, lk0 's, mk0 's for k ˆ 1; . . . p and the error variance s 2. This is also a well discussed model in Multidimensional Signal Processing, when X …m; n†'s are independently and identically distributed (i.i.d.) random variables on a 2-D plane. See for example the works of Barbieri and Barone (1992), Cabrera and Bose (1993), Chun and Bose (1995), Hua (1995), Lang and McClellan (1982) and see the references there for the di¨erent estimation procedures. It is interesting to observe that the model (1) is the 2-D extension of the one dimensional frequency model, which was originally discussed by Hannan (1971) and Walker (1971) in the time series analysis. It is also observed that the model (1) can be used to model textures. To see how this model represents di¨erent textures the readers are referred to the work of Manderekar and Zhang (1995). They provided nice 2-D image plots of y…m; n†, whose grey level at …m; n† is proportional to the value of y…m; n† and when it is corrupted by independent Gaussian noise ®eld. Manderakar and Zhang (1995) considered this problem and obtained the consistency properties of the estimators of lk 's and mk 's when the estimators are obtained through the periodogram analysis and when X …m; n†'s are from a stationary random ®eld. Their results based on the work of Lai and Wei (1982) and they did not provide the asymptotic distribution of the estimators. But no where, at least not known to the authors, the properties of the least squares estimators have been discussed of the model (1). It is important to observe that the model (1) is a nonlinear regression model and unfortunately it does not satisfy the su½cient conditions stated by Jennrich (1969) or Wu (1981) for the least squares estimators (LSE) to be consistent. It may noted that when p ˆ 1, M ˆ 1 and lk0 ˆ 0, this model coincides with the one dimensional model discussed in Hannan (1971), Walker (1971), Kundu (1993) and Kundu and Mitra (1996). It was shown in Kundu (1993) that even the one dimensional model does not satisfy the su½cient conditions of Jennrich (1969) or Wu (1981). Therefore the consistency or the asymptotic normality of the LSE's is not immediate in this case. It is also worth mentioning at this stage that Bunke and Bunke (1989) also considered several nonlinear regression models in their book, but did not consider this particular model. One of the major di¨erence of this particular model with the other usual nonlinear models is in the rate of convergence. p Usually in the nonlinear model we observe the rate of convergence to be n but for this model, at least for the frequencies, the rate of convergence is n 3=2 . The main idea of this paper is to study the properties of the least squares estimators of the parameters of the model (1) and see how the asymptotic results behave for ®nite sample. We prove the strong consistency of the least squares estimators (LSE's) of Ak 's, lk 's and mk 's when the errors are i.i.d. double array random variables. We obtain that the asymptotic distribution of the least squares estimators are multivariate normal. The explicit expression of the asymptotic dispersion matrix of the LSE's are obtained, which may be useful to obtain the con®dence bounds. It is observed that the asymptotic dispersion matrix coincides with the Cramer Rao bound, which may not be

Two dimensional model

85

very surprising. We prove that the LSE of s 2 is strongly consistent when the error variance is ®nite and the asymptotic distribution of the LSE of s 2 can be obtained when the fourth order moment of error random variables are ®nite. Our approach is di¨erent from that of Mandrekar and Zhang (1995). Our results extend some of the existing one dimensional results of Walker (1971), Hannan (1971), Kundu (1993), and Kundu and Mitra (1996) to the two dimensional case. The rest of the papers is organized as follows. In Section 2, we prove the strong consistency of the LSE's of Ak 's, lk 's and mk 's when the errors are i.i.d. random variables. In Section 3, the asymptotic normality results of those estimators are established under the same set of assumptions. The consistency and the asymptotic normality results of the estimator of s 2 are obtained in section 4. A summary of numerical experiments is given in Section 5 and ®nally we draw conclusions from our work in Section 6.

2. Consistency of the LSE's of Ak , lk and mk We need the following lemma to prove the necessary results. We denote the set of positive integers by Z. Lemma 1. Let fX …m; n†; m; n A Zg be a i.i.d. sequence of double array random variables with mean zero and ®nite variance then; M X N 1 1 X X …m; n† cos…ma† cos…nb† sup a; b N M mˆ1 nˆ1 ! 0 a:s: when min fM; Ng ! y

…2†

where a.s. means almost surely. Proof of Lemma 1: Consider the following random variables; Z…m; n† ˆ X …m; n† ˆ0

if jX …m; n†j < …mn† 3=4 otherwise

First we will show that Z…m; n† and X …m; n† are equivalent sequences. Consider y X y X mˆ1 nˆ1

PfX …m; n† 6ˆ Z…m; n†g ˆ

y X y X

PfjX …m; n†j > …mn† 3=4 g

mˆ1 nˆ1

Now observe that there are at most 2 k k combinations of …m; n†'s such that mn < 2 k , therefore we have

86

D. Kundu, R. D. Gupta y X y X mˆ1 nˆ1

U U

PfjX …m; n†j V …mn† 3=4 g X

y X

PfjX …m; n†j V r 3=4 g

‰here r ˆ mnŠ

kˆ1 2 kÿ1Ur