Statistical properties of the periodogram for two vector-valued stability series with missed observations

Introduction

Our method of proceeding is to derive the asymptotic moments of: the periodogram Our work is mainly based on the properties of the data windows and the matrix of second order smoothing modified periodogram. The properties of the smoothing periodogram using a weight function or data window were discussed in [1,3,7] . We see that the idea of smoothing the periodograms using data window is an important tool in the spectral analysis of time series. Our purpose is to compare the classic results in the spectral analysis of (r+s) vector valued strictly stability time series, where all observations are available and the case where some of the observations are randomly missed, using data window. The paper is organized as follows: Section1. Introduction, sstatistical properties of the periodogram for two vector-valued stability series will be in investigate in Section 2, in section 3 the Asymptotic properties of the Spectral Measure Matrix was discussed, the Asymptotic Moments of the Spectral Density Matrix were discussed in section 4, finally we will apply our theoretical study in the economy sector in section 5.

Statistical Properties of the Periodogram for Two Vector-Valued Stability Series

In this section we will investigate the sstatistical properties of the periodogram for two vector-In this section we will investigate the sstatistical properties of the periodogram for two vectorvalued stability series: Consider an

The second-order spectral densities can be derived as,

is a strictly stability series all of whose moments exist. For each

be a weight function that is bounded, is symmetric about 0 and has bounded first derivative such that

be bounded, is of bounded variation and vanishes for 0

be the discrete expanded finite Fourier transform is defined by , see [9][10][11] .

The theorem are proved [13] .

where the bar denotes the complex conjugate. Then,

Where, [2] If the data window function

is bounded and has bounded variation and vanishes for 0

Lemma 2.2. [2] Suppose that

be bounded by a constant L and satisfying the Lipschitz condition, [2] Suppose that

be bounded by a constant L and satisfying condition (2.17) then,

for some constants c L , and a point

Proof

The prove comes directly from (2.13) by taking the limits for both sides and then using the given conditions. In Corollary (2.2) below we make use of the Kroncker delta function

The statistical dependence of ) (

Under the conditions of theorem (2.2) then,

, and by using the Assumption (III) then we get from (2.14)

, then take the modulus for both sides of (2.15) and then using lemma (2.3) and the boundedness of

where, for some constant K , we have

. Hence, the corollary is obtained. In the case of     corollary (2.2) indicates the following corollary.

Corollary 2.3.

Under the conditions of theorem (2.2) and corollary (2.2) then,



Hence the proof is complete.

Asymptotic Properties of the Spectral Measure Matrix for Two Vector-Valued Stability Series

We will study the properties of the spectral measure matrix for two vector-valued stability Series in this section as follow, Let the series

), , ( min ..., , 1 ), ( ) ( have a spectral density matrix which is defined as

) ( the matrix of a spectral measure is given by :

in view of (3.1) we can consider estimating

by using (2.12) and (2.13) the prove comes directly.

proof Formula (3.5) comes directly by taking the limits for both sides of (3.3) and the proof is complete.

The statistical dependence of ) ( 1) (

Proof

Taking the modulus on both sides of (3.4), we get

Using Assumption III and the boundedness of

Then the corollary is obtained.

Asymptotic Moments of the Spectral Density Matrix for Two Vector-Valued Stability Series and Its Distributions

This section is concerned with constructing an estimate of ), (  zz f the matrix of second order spectral densities and study it's asymptotic moments.

Let

T B be a scale vector depending on .

which satisfies Assumption I with mean zero and let

Proof.

The proof comes directly by taking the limits for both sides of formula (4.2) as

Using Assumption III and the boundedness of

then the corollary is obtained.

Applications

We will apply our theoretical study on the production of "the national company for flour mills and feed".

Studying the quantity of wheat used and flour product

The data manipulated in this research are a monthly chronological series representing the average amount of wheat used and the amount of flour production in "the national company for flour mills and feed" for the period from January 2005 to September 2009.

Studying the quantity of wheat used

In this study we will comparison between our results, model of strictly stability time series (wheat) with some missing observations and the classical results, where all observations are available. Let ) (t X a be the data of the average amount of wheat used where all observations are available (classical case), suppose that there is some missing observations in a random way (our study), table 1 shows the comparison between our results, spectral analysis of strictly stability time series with some missing observations and the classical results, where all observations are available.

Table 1:

The comparison of the results with and without missed observations of the average amount of wheat used

Series with missed observations Series without missed observations

T i m e S e r i e s P l o t o f W h e a t X

The Average wheat used per month

Studying the quantity of flour production

In this study we will comparison between our results, model of strictly stability time series (flour) with some missing observations and the classical results, where all observations are available.

Let ) ( t Y a

be the data of the amount of flour production, where all observations are available (classical case), suppose that there is some missing observations in a random way (our study), table 2 shows the comparison between our results, spectral analysis of strictly stability time series with some missing observations and the classical results, where all observations are available. The average monthly production of flour

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