No abstract

In this paper, we study the estimation of the spectral density function and properties of the resulting estimator which called the periodogram. The statistical properties for this periodogram in case of actual observations and forecasted observations of the fuzzy time series are studied. Depending on MSEs which based on these statistical properties, the periodogram results can be compared in both cases. For this purpose a program that transfer the observed time series to fuzzy time series with large sizes is constructed.

Assume X1,X2,…,XN are realizations of N observations from a real-valued discrete parameter third-order stationary process Xt,t=0±1,±2,…, with bispectrum fXXX(λ1,λ2) where “−π≤λ1,λ2≤π”. Based on the previous assumption, L different multitapered biperiodograms IXXX(mt)j(λ1,λ2);j=1,2,…,L on overlapped segments (Xt(j);1≤t<N) can be constructed. Further, the mean and variance of the average of these different multitapered biperiodograms can be expressed as asymptotic expressions. According to different bispectral windows/kernels (Wβ(j)(α1,α2), where “−π⩽α1,α2⩽π” andβ is the bandwidth) and IXXX(mt)j(λ1,λ2), the bispectrum fXXX(λ1,λ2) can be estimated. The asymptotic expressions of the first- and second-ordered moments as well as the integrated relative mean squared error (IMSE) of this estimate are derived. Finally, some estimation results based on numerically generated data from the selected process “DCGINAR(1)” are presented and discussed in detail.