Asymptotic Properties of Spectral Estimates of Second-Order with Missed Observations

Abstract: Problem statement: As a complement of the periodogram study the asymp totic properties of the spectral density using data window for stati onary stochastic process are investigated. Some statistical properties of covariance estimation fun ction with missing observations are studied. Approach: The asymptotic normality was discussed. A numerica l example was discussed by using computer programming. Results: The study of time series with missed observations a d with the modified periodogram had the same results of the study of the classic time series. Conclusion: Modified periodogram with expanded finite Fourier t ransformation for time series with missed observation has improved the results of the classic time series.


INTRODUCTION
Our method of proceeding is to derive the asymptotic moments of: the matrix of sample measures necessary uniform error terms. Our work is mainly based on the properties of the data windows and the matrix of second order smoothing modified periodogram, Ghazal (2005). The properties of the smoothing periodogram using a weight function or data window were discussed in Brillinger (1969); Dahlhaus, 1985. 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-vector valued strictly stationary time series, where all observations are available and the case where some of the observations are randomly missed, using data window.
Related works: Dahlhaus (1985) considered the estimation of spectral measure of stationary process. He discussed the case where X(t), t is a strictly stationary Gaussian time series with real valued components and mean zero. He defined the periodograms, as in Brillinger (1969); Ghazal (2001;2005) and Ghazal and Farag (2000).
Let X(t); (t = 0, 1, …,T-1) are be some randomly missing observations. We construct the statistics ( ) xx C u denotes the matrix of absolute values, the bar denotes the complex conjugate. We defined f xx (λ) the r×r matrix of second order spectral densities by: and F xx (λ) the matrix of second order spectral measures by (Brillinger, 1969): λ , the matrix of second order smoothing modified periodograms. This last is derived from the finite Fourier transform of an observed stretch of data, X(t) (t = 0, 1, …,T-1) that have some randomly missing observations (Ghazal, 2005). We determine the asymptotic expressions for the cumulates of (T ) Assumption I: X(t) is a strictly stationary continuous time series and all its moments exist. For each j = 1,…,k-1 and k-tuple a 1 , a 2 ,…a k, we have: because cumulants are measures of the joint dependence of random variables, (2) is seen to be a form of mixing or asymptotic independence requirement for values of X(t) well separated in time. If X(t) satisfies assumption I we may define its cumulant spectral densities by:  ∫ Given B r >0 we then set that: Assumption III: Let U(T) is bounded variation and vanishes for t < 0, t >T-1. The function (T) a d (t) is called a data window and satisfies: Let B a (t) (t = 0, ±1,…) be a process independent of X(t) such that for every t: The success of recording an observation not depends on the fail of another and so it is independent. We may then define the modified series: Then Y(t) is a strictly stationary r-vector valued time series all of whose moments exist.
In this study Nr ( ) by deriving their mean, covariance and cumulate. We construct the expanded finite Fourier transform with data window with missed observations as: then the expanded smoothing modified periodogram is defined by: The bar denotes the complex conjugate and: λ∈R; a, b = 1,r and ( where the error terms are uniform in each case. and: The following lemma indicates that (  [ , ] = λ ∈ −π π . We shall prove that the asymptotic distribution of previous statistics is normal with mean zero the asymptotic normality has been demonstrated, under various conditions, by (Ghazal, 2005;Dahlhaus, 1985;Ghazal and Farag, 2000;Ghazal and Elhassanein, 2006).
for all j j a , b 1,r, = j [ , ], λ ∈ −π π j 1, k, = k 1, 2,... which satisfies assumption I with mean zero. Then: are asymptotically jointly multivariate normal with covariance structure given by: are asymptotically normal with mean zero and covariance structure indicated in Theorem 5. Example: In this example, we will comparison between our results, spectral analysis of strictly stationary time series with some missing observations and the classic results, where all observations are available. Data is available at www.egidegypt.com. Let Xa(t), (t = 0, ±1,…) be a strictly stationary rvector valued time series, we suppose know that the data Xa(t), (t = 1,2,…T) Which is the trade volume weakly of the Orascom Construction Industries since 1/1/2003 till 31/12/2008, where all observations are available of the series is available with some missing observations. H = 1, Y(t) = X(t), which is the classic case and then suppose that there is some missing observations in randomly way, i.e., H ≠ 1, to compare two cases shown in Fig. 6. The weakly trade volume data, X(t), 2003-2008 and The auto correlation function of the weakly trade volume data are shown in Fig. 1 and Fig. 2. The auto correlation and the partial auto correlation functions of the weakly trade volume data before smoothing are discussed and shown in Fig. 3.  Figure 4 shows The weakly trade volume data after adjustment X(t) .∇ 1 is used to smooth data before calculations. The weakly trade volume data after adjustment X(t) is mentioned. Figure 5 and 6 shows the imaginary part and real part of (T ) x f ( ) λ . where p = 0.99.   Table 1 shows the box-pierce (ljung-box) chi-square statistic and Table 2 shows Modified box-pierce (ljungbox) chi-square statistic.

Final estimates of parameters:
Differencing: 1 regular difference

MATERIALS AND METHODS
We used SPSS and matlab, the software programming to solve our numerical example.

RESULTS AND DISCUSSION
The study of time series with missed observations and with the modified periodogram had the same results of the study of the classical time series.

CONCLUSION
Modified periodogram with expanded finite Fourier transformation for time series with missed observation has improved the results of the classic time series.