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Estimation of Long Memory Linear Processes

Ichaou Mounirou*

Faculty of Management and Economics Sciences, Université de Parakou, Benin

*Corresponding Author:
Ichaou Mounirou
Faculty of Management and Economics Sciences Université de Parakou, Benin
E-mail: ichaou_bassir@yahoo.fr

Received date: 10/05/2016 Accepted date: 24/05/2016 Published date: 28/05/2016

Visit for more related articles at Research & Reviews: Journal of Statistics and Mathematical Sciences

Abstract

This paper studies asymptotic properties of the minimum distance Hellinger estimates for a stationary ultivariate linear gaussien long range dependent process of the form , where is a sequence of strictly stationary d-dimensional associated random vectors with E(Zt)= 0 and and {Au} is a sequence of coefficient matrices with and . By means of the properties of the kernel density estimate, the minimum istance Hellinger of this class are shown to be consistent, asymptotically normal and robust.

Keywords

Multivariate Processes, Kernel Density, Hellinger Distance, Linear Process, Parametric Estimation, Long Memory, Multivariate Processes

Introduction

Let equation bead-variate linear process independent of the form:


equation   (1)

Defined on a probability space (Ω equation (F), p), where {Zt} is a sequence of stationary d-variate associated random vectors with equation and positive definite covariance matrix ã: d×d . Throughout this paper we shall assume that


equation   (2)
equation   (3)
equation

where for any d d, d = 2, matrix A = (aij(θ)) whose components depend on the parameter θ, such as equation and 0d×d denotes the d×d zero matrix. Here θò Θ with Θ equation , with. Let


equation   (4)

where the prime denotes transpose, and the matrix г =(σkj) with


equation   (5)
equation

equation is assumed to be gaussian and have long rang dependent process. Fakhre-Zakeri and Lee proved a central theorem for multivariate linear processes generated by independent multivariate random vectors and Fakhre-Zakeri and Lee also derived a functional central limit theorem for multivariate linear processes generated by multivariate random vectors with martingale difference sequence. Tae-Sung Kim, Mi-HwaKo and Sung-Mo Chung [1] prove a central limit theorem for d-variate associated random vectors. The problem is how to estimate θ in order to investigate the fitting of the model to the data? An estimation of equation would have two essential properties: it would be efficient and its distribution would not be greatly pertubated.

{Xt} is a multivariate Gaussian process in dependent with density fθ (.) . We estimate the parameters in the general multivariate linear processes in (1).

In this paper is to prove a general estimation of the parameter vector θ by the minimum Hellinger distance Method (MHD). The only existing examples of MHD estimates are related to i.i.d. sequences of random variables’s [2-4]. For long memory univariate linear processes see Bitty and Hili [5]. The long memory concept appeared since 1950 from the works of Hurst in hydraulics. The process equation is said to be a long memory process if in (1), λ is a parameter of long memory, and 1 / 2 <λij < 1 for j = 1;…;d and i = 1;…;d.

The paper developers in section 2, some assumptions and lemmas, essentially based on the work of Tae-Sung Kim, Mi-Hwa Ko and Sung-Mo Chung [1] and the work of Theophilos Cacoullos [6]. Our main results arein section 3, based on work of Bitty and Hili [5] which show consistency and the asymptotic properties of the MHD estimators of the parameter θ. We conclude with some examples,

Parzen [7] gave the asymptotic properties of a class of estimates fn(x) an univariate density function f(x) on the basis of random sample X1,…,Xn from f(x). Motivated as in Parzen, we consider estimates fn(x) ofthe density function f(x) of the following form:

equation   (6)
equation   (7)

where Fn(x) denotes the empirical distribution function based on the sample of n independent observations X1,…, Xn on the random d-dimensional vector X with chosen to satisfy suitable conditions and {hn} is a sequence of positive constants which in the sequel will always satisfy hn→ 0, as n→8 .We suppose K(y) is a bore scalar function on Ed such that

equation   (8)
equation   (9)
equation   (10)

where y denotes the length of the vector.

And

equation   (11)
equation   (12)

also K(y) is absolutely integrable (hence f(x) is uniformly continuous).

equation   (13)

and equation   (14)

 

See Theopilos Cacoullos [6] and Bitty, Hili [5]

Notations and Assumptions: Let equation be a family of functions whereT is a compact parameter set equation of such that for all θ ε Θ , f(.,θ ) : equation is a positive integral function. Assume that f(.,θ) satisfies the following assumptions.

(A2): For all θ ,µ ε Θ,θ ≠ µ is a continuous differentiable function at θ εΘ .

(A2): (i) equation have a zero Lebesgue measure and f (.,θ)is bounded on equation

(ii) For θ ,µ ò Θ,θ ≠ µ implies that equation is a set of positive Lebesgue mesure, for all xò equation

(A3): K the kernel function such that

equation

(A4): The bandwidths {bn} satisfy natural conditions, equation for ι ≥ = 1 when n→∞

(A5): There exists a constant ß>0 such that equation

Let F denote the set of densities with respect to the Lebesgue measure on equation Define the functional equation in the following: Let equation .Denote by B(g) the set equation where H2 is the Hellinger distance.

If B(g) is reduced to a unique element, then define T(g) as the value of this element. Elsewhere, we choose an arbitrary but unique element of these minimums and call it T(g).

Lemma 1: Let equation be a strictly stationary associated sequence of d–dimensional random vectors with E(Zt) = 0, E(Zt) < +8 and positive definite covariance matrix г as (5). Let (Xt) be a d-variate linear process defined as in (1). Assume that

equation   (15)

then, the linear process(Xt) fulfills the limit central theorem, that is, equation(16)

Where →equation denotes the convergence in distribution and N (0, T) indicates an normal distribution with mean zero vector and covariance matrix T defined in (4).

For the proof of lemma 1, see theorem 1.1 of Tae-Sung Kim, Mi-Hwa Ko and Sung-Mo Chung [1]

Lemma 2: To remark 3.2 and theorem 3.5 of Tae-Sung Kim, Mi-Hwa Ko and Sung-Mo Chung [1], we have

equation   (17)

For the proof of lemma 2, see Tae-Sung Kim, Mi-Hwa Ko and Sung-Mo Chung [1]

Lemma 3: Assume that (A5) holds. If f1 is continuous on equation and if for almost all x, h is continuous on Φ , then

(i) for all equation

(ii) If B(g) is reduced to an unique element, then t is continuous on g Hellinger topology.

(iii) T (fθ ) =θ Uniquely on Θ

Proof: See Lemma 3.1 in Bitty and Hili [5].

Lemma 4: Assume that equation satisfies assumptions (A1),-(A3). Then, for all sequence of density equation converges to fθ in the Hellinger topology.

equation

where,

equation

With an a(q×q) - matrix whose components tends to zero n→∞

Proof: See Theorem 2 in Beran [2]

Lemma 5: Under assumptions (A3), if the bandwidth bn is an theorems 1 and 2, if f(.,θ) is continuous with a compact support. And if the density f(.,θ) of the observations satisfies assumptions(A1)-(A2). Then equation converges to f(.,θ) in the Hellinger topology.

Proof of lemma 8

Under assumption (A2), (A3) and (A5) and lemma 2, we have

equation

Then equation in Hellinger topology

Estimation of the Parameter

This method has been introduced by Beran [2] for independent samples, developed by Bitty and Hili [5] for linear univariate processes dependent in long memory. The present paper suppose the process independent multivariate with associated random vectors under same condition of Bittyand Hili [5] in long memory. The minimum Hellinger distance estimate of the parameter vector is obtained via a nonparametric estimate of the density of the process (Xt). We define equation as the value of θ ε Θ which minimizes the Hellinger distance equation

equation

where equation is the nonparametric estimate of f(.,θ) and

equation

There exist many methods of non parametric estimation in the literature. See for instance Rosenblatt [8] and therein. For computational reasons, we consider the kernel density estimate which is defined in section 2. Before analyzing the optimal properties of equation we need some assumptions.

equation

where equation is the nonparametric estimate of f(.,θ) and

equation

Asymptotic properties

Theorem 1 (Almost Sure Consistency): Assume that (A1)-(A5) hold. Then, equation almost surely converges toθ.

For the proof, see section 3.

Let denote by Jn as:equation

Let us denote by equation the following function.

equation

Where equation is a quantity which exists, and t denotes the transpose.

Condition 1:

We have the (q × q) matrix sequence vn in lemma 4 and the sequence Jn are such that Jnvn tend to zero as equation.

Theorem 2 (Asymptotic distribution): Assume that (A1)-(A6) and condition 1 hold. If

equation is a nonsingular (qq) -matrix,

equation admits a compact support then, we have equation

For the proof, see section 3.

Appendices

Proof of theorem 1

equation

From lemma 3,

equation

As equation uniquely, the remainder of proof follows from the continuity of the functional T(.) in lemma 1.

Proof of theorem 2

From lemma 2 and the proof of theorem 2 of Bitty and Hili [5], we have

equation

where an a (d×d)-matrix whose components tend to zero in probability when n→ 8 .

Under condition 1, we have

equation

So the limiting distribution of equation depends on the limiting distribution of equation, With

equation

For a = 0 , b = 0, we have the algebraic identity

equation

For equation we have

equation

With

equation

And

equation

From assumption (A6), then equation

equation

equation

equation

With

equation

And

equation

Under assumptions (A1)-(A2) we apply Taylor Lagrange in order 2 and assumption (A4) we have:

So

equation

So

equation

Furthermore, we have equation

where Fn(.) and F(.) are respectively the empirical distribution function and distribution function of the process. By integration by part, we have

equation

From Ho and Hsing [9,10] (theorem 2.1 and remark 2.2) and assumptions (A2) and (A4), we have

equation

where Φ is a standard Gaussian random variable and !D denotes convergence in distribution.

So

equation

where equation denotes the equivalence in distribution affinity. For all ξ >0,

equation

The convergence of equation depends on the convergence of equation

So under assumptions equation we have

equation

We have equation

equation
equation
equation
equation

With

equation

and

equation

Under assumptions (A1) - (A2) we apply Taylor-Lagrange formula in equation order 2 and assumption (A4), we have

Furthermore, from propositions 1, 2 and 3, we have

Part (a)

equation

or

equation

Where equation and U(x) take values according to the different points of the proof of lemma 3:

equation
equation

Here equation is the Multiple Wiener-Itô Integral defined in the relation (9) of section 1.1 and σ2 (x,c) is defined in the first point of proposition 3. Denote by equation

equation

or

equation

We deduce that

equation
equation

Were call that equation then equation So we conclude that D → 0 when n→∞

Part (b)

equation

with


equation

and

equation

From part (a), the proof of equation is the same as the proof of equation. Were place equation

Hence it suffices to prove that the limiting distribution of equation is the same as the limiting distribution of equation.Since

equation

then

equation

with

equation

and

equation

From the proof of lemma 3 (part (b)), we have:

equation

then


equation

From propositions 1, 2 and 3, we have:


equation

or

equation
equation
equation

or

equation

We deduce that

equation

or

equation

So,

equation

or


equation

Then,

equation

or

equation

Conclusion

We conclude that we have either an asymptotic normal distribution or an asymptotic process towards the Multiple Wiener- Itô Integral.

References