A thesis
Submitted to the council of the college of Administration
&Economics\ University of Karbala as partial fulfillment of the
requirements for the degree of Philosophy of Statistics Sciences
By
Maytham Abdul Wahab Saleh
Supervisor
Prof. Dr . Jassim Nassir Hussain

The time series is defined as the values of a particular phenomenon arranged according to time or the set of values that the phenomenon takes in consecutive and equal periods of time. There are two directions in the analysis of time series, the first in the field of time, in which the time series is a linear combination with sequential limits of random errors, and the second in the field of repetition, in which the time series is a weighted sum of sine and periodic cosine functions. The concept of long memory or long dependency (LRD) means that the process is made up of many historical correlations, and the sum of the autocorrelations is slowly decreasing. In order to give a description of the
long memory, it is recommended to describe the stationary series with the limits of the spectrum density function. Due to the length of the time series, hidden seasonal and periodic components arise in this type of series that
cannot be detected through the traditional method of analysis, but through the iterative method of analysis and by using non-parametric and semi-parametric methods to detect these components, the most important of which
is the method of the periodic scheme of the Fourier transform. Although there are many methods in this field, but there is room to add non-parametric methods to find non-parametric estimators to estimate the spectral density through which we can detect these components, so three non-parametric estimators have been proposed (Lomax Kernel estimator, Dirichlet Kernel) B ( estimator and Reciprocal inverse Gaussian Kernel estimator) And compare
it with the nonparametric beta estimator by RMAD statistic. To determine the best method, simulation study was used in the experimental side, and the results showed in the experimental side that the Lomax Kernel estimator was
the best for its ability to detect hidden components, and it also had the lowest value for the RMAD statistic. In order to find out the possibility of applying this method, it was applied the best method for real data for a time series
related to respiratory diseases and tested because they have a long memory. Kernel estimators are considered to have good advantages in the spectral analysis of time series in case the distribution is not known or the assumptions are difficult in the case of parameter estimation.